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<!--Weave of '2/dmn' generated by 7-->
<ul class="crumbs"><li><a href="../webs.html">Source</a></li><li><a href="../compiler.html">Compiler Modules</a></li><li><a href="index.html">kinds</a></li><li><a href="index.html#2">Chapter 2: Kinds</a></li><li><b>Dimensions</b></li></ul><p class="purpose">To keep a small database indicating the physical dimensions of numerical values, and how they combine: for instance, allowing us to specify that a length times a length is an area.</p>
<ul class="toc"><li><a href="#SP1">&#167;1. Definitions</a></li><li><a href="#SP14">&#167;14. Prior kinds</a></li><li><a href="#SP15">&#167;15. Multiplication lists</a></li><li><a href="#SP19">&#167;19. Unary operations</a></li><li><a href="#SP20">&#167;20. Euclid's algorithm</a></li><li><a href="#SP22">&#167;22. Unit sequences</a></li><li><a href="#SP30">&#167;30. Performing derivations</a></li><li><a href="#SP31">&#167;31. Classifying the units</a></li><li><a href="#SP34">&#167;34. Scaling</a></li><li><a href="#SP40">&#167;40. Arithmetic on kinds</a></li></ul><hr class="tocbar">
<p class="inwebparagraph"><a id="SP1"></a><b>&#167;1. Definitions. </b></p>
<p class="inwebparagraph"><a id="SP2"></a><b>&#167;2. </b>Dimension in this sense is a term drawn from physics. The idea is that when
quantities are multiplied together, their natures are combined as well as
the actual numbers involved. For instance, in
v = fλ
if the frequency f of a wave is measured in Hz (counts per second), and
the wavelength λ in m, then the velocity v must be measured
in m/s: and that is indeed a measure of velocity, so this looks right.
We can tell that the formula
v = f^2λ
must be wrong because it would result in an acceleration. Physicists use the
term "dimensions" much as Inform uses the term "kinds of value".
</p>
<p class="inwebparagraph">Inform applies dimension-checking to all "quasinumerical" kinds &mdash; those
which can be expressed numerically. The choice of which kinds are quasinumerical
is all done in the I6 template, not built into Inform at the compiler level,
but the standard setup makes number, time, intermediate results of calculations
(see below), and what the Inform documentation calls "units" &mdash; kinds of
value specified by literal patterns.
</p>
<p class="inwebparagraph"><a id="SP3"></a><b>&#167;3. </b>Inform divides quasinumerical kinds into three: fundamental units, derived units
with dimensions, and dimensionless units. In the default setup provided by
the template, a typical run has one fundamental unit ("time"), one
dimensionless unit ("number") and &mdash; unless the source text does
something very strange &mdash; no derived units.
</p>
<p class="inwebparagraph">It would no doubt be cool to distinguish these by applying Buckingham's
π-theorem to all the equations we need to use, but this is a tricky
technique and does not always produce the "natural" results which people
expect. Instead, Inform requires the writer to specify explicitly how units
combine.
</p>
<p class="inwebparagraph">Number and time are built-in special cases. Further fundamental units are created
every time source text like this is read:
</p>
<blockquote>
<p>Mass is a kind of value. 1kg specifies a mass.</p>
</blockquote>
<p class="inwebparagraph">Derived units only come about when the source text specifies a multiplication
rule. For instance, when Inform reads
</p>
<blockquote>
<p>A mass times an acceleration specifies a force.</p>
</blockquote>
<p class="inwebparagraph">it chooses one of the three units &mdash; say, force &mdash; and derives that from the
others.
</p>
<p class="inwebparagraph">Multiplication rules are stored in a linked list associated with left operand;
so that the rule A times B specifies C causes (B, C) to be stored
in the list of <code class="display"><span class="extract">multiplications</span></code> belonging to A.
</p>
<pre class="display">
<span class="reserved">typedef</span><span class="plain"> </span><span class="reserved">struct</span><span class="plain"> </span><span class="reserved">dimensional_rules</span><span class="plain"> {</span>
<span class="reserved">struct</span><span class="plain"> </span><span class="reserved">dimensional_rule</span><span class="plain"> *</span><span class="identifier">multiplications</span><span class="plain">;</span>
<span class="plain">} </span><span class="reserved">dimensional_rules</span><span class="plain">;</span>
<span class="reserved">typedef</span><span class="plain"> </span><span class="reserved">struct</span><span class="plain"> </span><span class="reserved">dimensional_rule</span><span class="plain"> {</span>
<span class="reserved">struct</span><span class="plain"> </span><span class="identifier">wording</span><span class="plain"> </span><span class="identifier">name</span><span class="plain">;</span>
<span class="reserved">struct</span><span class="plain"> </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">right</span><span class="plain">;</span>
<span class="reserved">struct</span><span class="plain"> </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">outcome</span><span class="plain">;</span>
<span class="reserved">struct</span><span class="plain"> </span><span class="reserved">dimensional_rule</span><span class="plain"> *</span><span class="identifier">next</span><span class="plain">;</span>
<span class="plain">} </span><span class="reserved">dimensional_rule</span><span class="plain">;</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The structure dimensional_rules is private to this section.</p>
<p class="endnote">The structure dimensional_rule is accessed in 2/kc and here.</p>
<p class="inwebparagraph"><a id="SP4"></a><b>&#167;4. </b>The derivation process can be seen in action by feeding Inform
definitions of the SI units (see the test case <code class="display"><span class="extract">SIUnits-G</span></code>) and looking at
the output of:
</p>
<blockquote>
<p>Test dimensions (internal) with --.</p>
</blockquote>
<p class="inwebparagraph">(The dash is meaningless &mdash; this is a test with no input.) In the output, we
see that
</p>
<p class="inwebparagraph"></p>
<pre class="display">
<span class="plain">Base units: time, length, mass, elapsed time, electric current, temperature, luminosity</span>
<span class="plain">Derived units:</span>
<span class="plain">frequency = (elapsed time)-1</span>
<span class="plain">force = (length).(mass).(elapsed time)-2</span>
<span class="plain">energy = (length)2.(mass).(elapsed time)-2</span>
<span class="plain">pressure = (length)-1.(mass).(elapsed time)-2</span>
<span class="plain">power = (length)2.(mass).(elapsed time)-3</span>
<span class="plain">electric charge = (elapsed time).(electric current)</span>
<span class="plain">voltage = (length)2.(mass).(elapsed time)-3.(electric current)-1</span>
</pre>
<p class="inwebparagraph">...and so on. Those expressions on the right hand sides are "derived units",
where the numbers are powers, so that negative numbers mean division.
It's easy to see why we want to give names and notations for some of
these derived units &mdash; imagine going into a cycle shop and asking for a
5 m^2.kg.s^{-3}.A^{-1} battery.
</p>
<p class="inwebparagraph"><a id="SP5"></a><b>&#167;5. </b>A "dimensionless" quantity is one which is just a number, and is not a
physical measurement as such. In an equation like
K = ((mv^2) / (2))
the 2 is clearly dimensionless, but other possibilities also exist. The
arc length of part of a circle at radius r drawn out to angle θ
(if measured in radians) is given by:
A = θ r
Here A and r are both lengths, so the angle θ must be dimensionless.
But clearly it's not quite conceptually the same thing as an ordinary number.
Inform creates new dimensionless quantities this way, too:
</p>
<blockquote>
<p>Angle is a kind of value. 1 rad specifies an angle. Length times angle specifies a length.</p>
</blockquote>
<p class="inwebparagraph">Inform is not quite so careful about distinguishing dimensionless quantities
as some physicists might be. The official SI units distinguish angle, measured
in radians, and solid angle, in steradians, writing them as having units
m.m^{-1} and m^2.m^{-2} respectively &mdash;
one is a ratio of lengths, the other of areas. Inform cancels the units
and sees them as dimensionally equal. So if we write:
</p>
<blockquote>
<p>Solid angle is a kind of value. 1 srad specifies an solid angle. Area times solid angle specifies an area.</p>
</blockquote>
<p class="inwebparagraph">then Inform treats angle and solid angle as having the same multiplicative
properties &mdash; but it still allows variables to have either one as a kind of
value, and prints them differently.
</p>
<p class="inwebparagraph">Note that a dimensionless unit (other than number) can only get that way
by derivation, so it is always a derived unit, never a fundamental unit.
</p>
<p class="inwebparagraph"><a id="SP6"></a><b>&#167;6. </b>In the process of calculations, we often need to create other and nameless
units as partial answers of calculations. Consider the kinetic energy equation
K = ((mv^2) / (2))
being evaluated the way a computer does it, one step at a time. One way takes
the mass, multiplies by the velocity to get a momentum, multiplies by the
velocity again to get energy, then divides by a dimensionless constant. But
another way would be to square the velocity first, then multiply by mass
to get energy, then halve. If we do it that way, what units are the squared
velocity in? The answer has to be
</p>
<p class="inwebparagraph"></p>
<pre class="display">
<span class="plain">(length)2.(elapsed time)-2</span>
</pre>
<p class="inwebparagraph">but that's a unit which isn't useful for much, and doesn't have any everyday
name. Inform creates what are called "intermediate kinds" like this in
order to be able to represent the kinds of intermediate values which turn
up in calculation. They use the special <code class="display"><span class="extract">CON_INTERMEDIATE</span></code> construction, they
are nameless, and the user isn't allowed to store the results permanently.
(They can't be the kind of a global variable, a table column, and so on.)
If the user wants to deal with such values on a long-term basis, he must give
them a name, like this:
</p>
<blockquote>
<p>Funkiness is a kind of value. 1 Claude is a funkiness. A velocity times a velocity specifies a funkiness.</p>
</blockquote>
<p class="inwebparagraph"><a id="SP7"></a><b>&#167;7. </b>Expressions like m^2.kg are stored inside Inform as
sequences of ordered pairs in the form
((B_1, p_1), (B_2, p_2), ..., (B_k, p_k))
where each B_i is the type ID of a fundamental unit, each p_i is a non-zero
integer, and B_1 &lt; B_2 &lt; ... &lt; B_k. For instance, energy would be
((length, 2), (mass, 1), (elapsed~time, -2)).
</p>
<p class="inwebparagraph">Every physically different derived unit has a unique and distinct sequence.
This is only true because a unit sequence is forbidden to contain derived
units. For instance, specific heat capacity looks as if it is written with
two different units in physics:
J. K^{-1}. kg^{-1} =
m^2.s^{-2}.K^{-1}
But this is because the Joule is a derived unit. Substituting
J = m^2.kg.s^{-2}
to get back to fundamental units shows that both sides would be computed as the
same unit sequence.
</p>
<p class="inwebparagraph">The case k=0, the empty sequence, is not only legal but important: it is
the derivation for a dimensionless unit. (As discussed above, Inform doesn't
see different dimensionless units as being physically different.)
</p>
<pre class="display">
<span class="reserved">typedef</span><span class="plain"> </span><span class="reserved">struct</span><span class="plain"> </span><span class="reserved">unit_pair</span><span class="plain"> {</span>
<span class="reserved">struct</span><span class="plain"> </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">fund_unit</span><span class="plain">; </span> <span class="comment">and this really must be a fundamental kind</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">power</span><span class="plain">; </span> <span class="comment">a non-zero integer</span>
<span class="plain">} </span><span class="reserved">unit_pair</span><span class="plain">;</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The structure unit_pair is private to this section.</p>
<p class="inwebparagraph"><a id="SP8"></a><b>&#167;8. </b>The following is a hard limit, but really not a problematic one. The
entire SI system has only 7 fundamental units, and the only named scientific
unit I've seen which has even 5 terms in its derivation is molar entropy, a
less than everyday chemical measure
(kg.m^2.s^{-2}.K^{-1}.mol^{-1},
if you're taking notes).
</p>
<pre class="definitions">
<span class="definitionkeyword">define</span> <span class="constant">MAX_BASE_UNITS_IN_SEQUENCE</span><span class="plain"> 16</span>
</pre>
<pre class="display">
<span class="reserved">typedef</span><span class="plain"> </span><span class="reserved">struct</span><span class="plain"> </span><span class="reserved">unit_sequence</span><span class="plain"> {</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">no_unit_pairs</span><span class="plain">; </span> <span class="comment">in range 0 to <code class="display"><span class="extract">MAX_BASE_UNITS_IN_SEQUENCE</span></code></span>
<span class="reserved">struct</span><span class="plain"> </span><span class="reserved">unit_pair</span><span class="plain"> </span><span class="identifier">unit_pairs</span><span class="plain">[</span><span class="constant">MAX_BASE_UNITS_IN_SEQUENCE</span><span class="plain">];</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">scaling_factor</span><span class="plain">; </span> <span class="comment">see discussion of scaling below</span>
<span class="plain">} </span><span class="reserved">unit_sequence</span><span class="plain">;</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The structure unit_sequence is private to this section.</p>
<p class="inwebparagraph"><a id="SP9"></a><b>&#167;9. </b>Manipulating units like m^2.kg.s^{-2} looks
a little like manipulating formal polynomials in several variables, and of
course that isn't an accident. Another way of thinking of the above is that
we have a ring R of underlying numbers but that all arithmetic is done
in a larger ring. For each unit extend by R by a pair of formal
variables U_i and U_i^{-1}, and then quotient by the ideal generated
by U_jU_j^{-1} (so that they are indeed reciprocals of each other, as
the notation suggests) and also by all of the derivations we know of. Thus
Inform calculates in the ring:
I = R[U_1, U_2, ..., U_n, U_1^{-1}, ..., U_n^{-1}] /
(U_1U_1^{-1}, U_2U_2^{-1}, ..., U_nU_n^{-1}, D_1, D_2, ..., D_i).
It does that in practice by eliminating all of the U_i and U_i^{-1}
which are derived, so that it's left with just
I = R[U_1, U_2, ..., U_k, U_1^{-1}, ..., U_k^{-1}] /
(U_1U_1^{-1}, U_2U_2^{-1}, ..., U_kU_k^{-1}).
</p>
<p class="inwebparagraph">For instance, given seconds, Watts and Joules,
I = R[s, s^{-1}, W, W^{-1}, J, J^{-1}]/
(ss^{-1} = 1, WW^{-1}=1, JJ^{-1} = 1,
sW = J)
which by substituting all occurrences of J can be reduced to:
I = R[s, s^{-1}, W, W^{-1}]/
(ss^{-1} = 1, WW^{-1}=1).
Of course there are other ways to calculate I &mdash; we could have
eliminated any of the three units and kept the other two.
</p>
<p class="inwebparagraph">If the derivations were ever more complex than AB=C, we might have to
use some elegant algorithms for calculating Gröbner bases in order to
determine I. But Inform's syntax is such that the writer of the source
text gives us the simplest possible description of the ideal, so no such
fun occurs.
</p>
<p class="inwebparagraph">What does this ring look like? Because we are not allowed to add terms with
different powers of the variables, we only ever deal with monomials. Thus
we can form 2Ws^{-1} + 7Ws^{-1}, but Inform forbids
us to form (say) 6J + 7W. We can therefore picture the ring I
as a great mass of parallel copies of R. Dimensionless values all live
in R itself, while energies all live in R.s.W, powers in R.W
and so on. Addition and subtraction slide values around within their own
parallel copies, but multiplication and division move them from one to
another. The computation v_1v_2 is done in general by calculating the
numerical part (in R) at run-time, and the units (the choice of which
copy of R) at compile-time.
</p>
<p class="inwebparagraph"><a id="SP10"></a><b>&#167;10. </b>But enough abstraction: time for some arithmetic. Inform performs
checking whenever values from two different kinds are combined by any of
eight arithmetic operations, numbered as follows. The numbers must not
be changed without amending the definitions of "plus" and so on
in the Standard Rules.
</p>
<pre class="definitions">
<span class="definitionkeyword">define</span> <span class="constant">NO_OPERATIONS</span><span class="plain"> 9</span>
<span class="definitionkeyword">define</span> <span class="constant">PLUS_OPERATION</span><span class="plain"> 0 </span> <span class="comment">addition</span>
<span class="definitionkeyword">define</span> <span class="constant">MINUS_OPERATION</span><span class="plain"> 1 </span> <span class="comment">subtraction</span>
<span class="definitionkeyword">define</span> <span class="constant">TIMES_OPERATION</span><span class="plain"> 2 </span> <span class="comment">multiplication</span>
<span class="definitionkeyword">define</span> <span class="constant">DIVIDE_OPERATION</span><span class="plain"> 3 </span> <span class="comment">division</span>
<span class="definitionkeyword">define</span> <span class="constant">REMAINDER_OPERATION</span><span class="plain"> 4 </span> <span class="comment">remainder after division</span>
<span class="definitionkeyword">define</span> <span class="constant">APPROXIMATION_OPERATION</span><span class="plain"> 5 </span> <span class="comment">"X to the nearest Y"</span>
<span class="definitionkeyword">define</span> <span class="constant">ROOT_OPERATION</span><span class="plain"> 6 </span> <span class="comment">square root &mdash; a unary operation</span>
<span class="definitionkeyword">define</span> <span class="constant">REALROOT_OPERATION</span><span class="plain"> 7 </span> <span class="comment">real-valued square root &mdash; a unary operation</span>
<span class="definitionkeyword">define</span> <span class="constant">CUBEROOT_OPERATION</span><span class="plain"> 8 </span> <span class="comment">cube root &mdash; similarly unary</span>
<span class="definitionkeyword">define</span> <span class="constant">EQUALS_OPERATION</span><span class="plain"> 9 </span> <span class="comment">set equal &mdash; used only in equations</span>
<span class="definitionkeyword">define</span> <span class="constant">POWER_OPERATION</span><span class="plain"> 10 </span> <span class="comment">raise to integer power &mdash; used only in equations</span>
<span class="definitionkeyword">define</span> <span class="constant">UNARY_MINUS_OPERATION</span><span class="plain"> 11 </span> <span class="comment">unary minus &mdash; used only in equations</span>
</pre>
<p class="inwebparagraph"><a id="SP11"></a><b>&#167;11. </b>The following is associated with "total...", as in "the total weight
of things on the table", but that's a dodge used in the Standard Rules,
and for dimensional purposes we ignore it.
</p>
<pre class="definitions">
<span class="definitionkeyword">define</span> <span class="constant">TOTAL_OPERATION</span><span class="plain"> 12 </span> <span class="comment">not really one of the above</span>
</pre>
<p class="inwebparagraph"><a id="SP12"></a><b>&#167;12. </b>There are two reasons why Inform monitors arithmetic: to keep track of
how it changes kinds, and to preserve scaling factors.
</p>
<p class="inwebparagraph">We start from the principle that not every arithmetic operation can be done,
and that even when it can, the result may have a different kind than the
operand(s) had. For one thing, every arithmetic operation requires that its
operands are quasinumerical &mdash; Inform won't allow a text to be multiplied by
a sound effect. (Occasionally we have thought about allowing text to be
duplicated by multiplication &mdash; 2 times "zig" would be "zigzig", and
maybe similarly for lists &mdash; but it always seemed more likely to be used by
mistake than intentionally.)
</p>
<p class="inwebparagraph">Other restrictions are also applied. For instance, a time cannot be added
to a number, or vice versa; addition, subtraction and approximation require
both operands to have the same units.
</p>
<p class="inwebparagraph"><a id="SP13"></a><b>&#167;13. </b>Finally, scaling. Number is straightforwardly an integer kind:
it holds whole numbers. But other quasinumerical kinds can be stored
using scaled, fixed-point arithmetic. In general for each named unit
U (fundamental or derived) there is a positive integer k_U such that the
true value v is stored at run-time as the I6 integer k_U v. We call
this the scaled value.
</p>
<p class="inwebparagraph">For example, if the text reads:
</p>
<blockquote>
<p>Force is a kind of value. 1N specifies a force scaled up by 1000.</p>
</blockquote>
<p class="inwebparagraph">then k = 1000 and the value 1N will be stored at run-time as <code class="display"><span class="extract">1000</span></code>;
forces can thus be calculated to a true value accuracy of at best 0.001N,
stored at run-time as <code class="display"><span class="extract">1</span></code>.
</p>
<p class="inwebparagraph">It must be emphasised that this is scaled, fixed-point arithmetic: there
are no mantissas or exponents. In such schemes the scale factor is usually
2^{16} or some similar power of 2, but here we want to use exactly the
scale factors laid out by the source text &mdash; partly because the user
knows best, partly so that it is unambiguous how to print values, partly
so that source text like "0.001N" determines an exact value rather than
being approximated by a binary equivalent.
</p>
<p class="inwebparagraph"><a id="SP14"></a><b>&#167;14. Prior kinds. </b>It turns out to be convenient to have a definition ordering of fundamental kinds,
which is completely unlike the &lt;= relation; it just places them in
order of creation.
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_prior</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">A</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">B</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">A</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">B</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">B</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">A</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::get_construct</span><span class="plain">(</span><span class="identifier">A</span><span class="plain">)-&gt;</span><span class="identifier">allocation_id</span><span class="plain"> &lt; </span><span class="functiontext">Kinds::get_construct</span><span class="plain">(</span><span class="identifier">B</span><span class="plain">)-&gt;</span><span class="identifier">allocation_id</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::kind_prior is used in <a href="#SP24_4">&#167;24.4</a>, <a href="#SP30_1">&#167;30.1</a>.</p>
<p class="inwebparagraph"><a id="SP15"></a><b>&#167;15. Multiplication lists. </b>The linked lists of multiplication rules begin empty for every kind:
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::dim_initialise</span><span class="plain">(</span><span class="reserved">dimensional_rules</span><span class="plain"> *</span><span class="identifier">dimrs</span><span class="plain">) {</span>
<span class="identifier">dimrs</span><span class="plain">-</span><span class="element">&gt;multiplications</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::dim_initialise is used in 2/kc2 (<a href="2-kc2.html#SP7_1">&#167;7.1</a>).</p>
<p class="inwebparagraph"><a id="SP16"></a><b>&#167;16. </b>And this adds a new one to the relevant list:
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::record_multiplication_rule</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">left</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">right</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">outcome</span><span class="plain">) {</span>
<span class="reserved">dimensional_rules</span><span class="plain"> *</span><span class="identifier">dimrs</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dim_rules</span><span class="plain">(</span><span class="identifier">left</span><span class="plain">);</span>
<span class="reserved">dimensional_rule</span><span class="plain"> *</span><span class="identifier">dimr</span><span class="plain">;</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">dimr</span><span class="plain"> = </span><span class="identifier">dimrs</span><span class="plain">-</span><span class="element">&gt;multiplications</span><span class="plain">; </span><span class="identifier">dimr</span><span class="plain">; </span><span class="identifier">dimr</span><span class="plain"> = </span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;next</span><span class="plain">)</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;right</span><span class="plain"> == </span><span class="identifier">right</span><span class="plain">) {</span>
<span class="identifier">KINDS_PROBLEM_HANDLER</span><span class="plain">(</span><span class="constant">DimensionRedundant_KINDERROR</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">);</span>
<span class="reserved">return</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">dimensional_rule</span><span class="plain"> *</span><span class="identifier">dimr_new</span><span class="plain"> = </span><span class="identifier">CREATE</span><span class="plain">(</span><span class="reserved">dimensional_rule</span><span class="plain">);</span>
<span class="identifier">dimr_new</span><span class="plain">-</span><span class="element">&gt;right</span><span class="plain"> = </span><span class="identifier">right</span><span class="plain">;</span>
<span class="identifier">dimr_new</span><span class="plain">-</span><span class="element">&gt;outcome</span><span class="plain"> = </span><span class="identifier">outcome</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">current_sentence</span><span class="plain">)</span>
<span class="identifier">dimr_new</span><span class="plain">-</span><span class="element">&gt;name</span><span class="plain"> = </span><span class="identifier">ParseTree::get_text</span><span class="plain">(</span><span class="identifier">current_sentence</span><span class="plain">);</span>
<span class="reserved">else</span>
<span class="identifier">dimr_new</span><span class="plain">-</span><span class="element">&gt;name</span><span class="plain"> = </span><span class="identifier">EMPTY_WORDING</span><span class="plain">;</span>
<span class="identifier">dimr_new</span><span class="plain">-</span><span class="element">&gt;next</span><span class="plain"> = </span><span class="identifier">dimrs</span><span class="plain">-</span><span class="element">&gt;multiplications</span><span class="plain">;</span>
<span class="identifier">dimrs</span><span class="plain">-</span><span class="element">&gt;multiplications</span><span class="plain"> = </span><span class="identifier">dimr_new</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::record_multiplication_rule is used in <a href="#SP18">&#167;18</a>.</p>
<p class="inwebparagraph"><a id="SP17"></a><b>&#167;17. </b>The following loop-header macro iterates through the possible triples
(L, R, O) of multiplication rules L* R = O.
</p>
<pre class="definitions">
<span class="definitionkeyword">define</span> <span class="identifier">LOOP_OVER_MULTIPLICATIONS</span><span class="plain">(</span><span class="identifier">left_operand</span><span class="plain">, </span><span class="identifier">right_operand</span><span class="plain">, </span><span class="identifier">outcome_type</span><span class="plain">, </span><span class="identifier">wn</span><span class="plain">)</span>
<span class="reserved">dimensional_rules</span><span class="plain"> *</span><span class="identifier">dimrs</span><span class="plain">;</span>
<span class="reserved">dimensional_rule</span><span class="plain"> *</span><span class="identifier">dimr</span><span class="plain">;</span>
<span class="identifier">LOOP_OVER_BASE_KINDS</span><span class="plain">(</span><span class="identifier">left_operand</span><span class="plain">)</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">dimrs</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dim_rules</span><span class="plain">(</span><span class="identifier">left_operand</span><span class="plain">),</span>
<span class="identifier">dimr</span><span class="plain"> = (</span><span class="identifier">dimrs</span><span class="plain">)?(</span><span class="identifier">dimrs</span><span class="plain">-</span><span class="element">&gt;multiplications</span><span class="plain">):</span><span class="identifier">NULL</span><span class="plain">,</span>
<span class="identifier">wn</span><span class="plain"> = (</span><span class="identifier">dimr</span><span class="plain">)?(</span><span class="identifier">Wordings::first_wn</span><span class="plain">(</span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;name</span><span class="plain">)):-1,</span>
<span class="identifier">right_operand</span><span class="plain"> = (</span><span class="identifier">dimr</span><span class="plain">)?(</span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;right</span><span class="plain">):0,</span>
<span class="identifier">outcome_type</span><span class="plain"> = (</span><span class="identifier">dimr</span><span class="plain">)?(</span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;outcome</span><span class="plain">):0;</span>
<span class="identifier">dimr</span><span class="plain">;</span>
<span class="identifier">dimr</span><span class="plain"> = </span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;next</span><span class="plain">,</span>
<span class="identifier">wn</span><span class="plain"> = (</span><span class="identifier">dimr</span><span class="plain">)?(</span><span class="identifier">Wordings::first_wn</span><span class="plain">(</span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;name</span><span class="plain">)):-1,</span>
<span class="identifier">right_operand</span><span class="plain"> = (</span><span class="identifier">dimr</span><span class="plain">)?(</span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;right</span><span class="plain">):0,</span>
<span class="identifier">outcome_type</span><span class="plain"> = (</span><span class="identifier">dimr</span><span class="plain">)?(</span><span class="identifier">dimr</span><span class="plain">-</span><span class="element">&gt;outcome</span><span class="plain">):0)</span>
</pre>
<p class="inwebparagraph"><a id="SP18"></a><b>&#167;18. </b>And this is where the user asks for a multiplication to come out in a
particular way:
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::dim_set_multiplication</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">left</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">right</span><span class="plain">,</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">outcome</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::is_proper_constructor</span><span class="plain">(</span><span class="identifier">left</span><span class="plain">)) ||</span>
<span class="plain">(</span><span class="functiontext">Kinds::is_proper_constructor</span><span class="plain">(</span><span class="identifier">right</span><span class="plain">)) ||</span>
<span class="plain">(</span><span class="functiontext">Kinds::is_proper_constructor</span><span class="plain">(</span><span class="identifier">outcome</span><span class="plain">))) {</span>
<span class="identifier">KINDS_PROBLEM_HANDLER</span><span class="plain">(</span><span class="constant">DimensionNotBaseKOV_KINDERROR</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">);</span>
<span class="reserved">return</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">left</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) ||</span>
<span class="plain">(</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">right</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) ||</span>
<span class="plain">(</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">outcome</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">)) {</span>
<span class="identifier">KINDS_PROBLEM_HANDLER</span><span class="plain">(</span><span class="constant">NonDimensional_KINDERROR</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">);</span>
<span class="reserved">return</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="functiontext">Kinds::Dimensions::record_multiplication_rule</span><span class="plain">(</span><span class="identifier">left</span><span class="plain">, </span><span class="identifier">right</span><span class="plain">, </span><span class="identifier">outcome</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">left</span><span class="plain">, </span><span class="identifier">outcome</span><span class="plain">)) &amp;&amp; (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">right</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">))) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">right</span><span class="plain">, </span><span class="identifier">outcome</span><span class="plain">)) &amp;&amp; (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">left</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">))) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="functiontext">Kinds::Dimensions::make_unit_derivation</span><span class="plain">(</span><span class="identifier">left</span><span class="plain">, </span><span class="identifier">right</span><span class="plain">, </span><span class="identifier">outcome</span><span class="plain">);</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::dim_set_multiplication appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP19"></a><b>&#167;19. Unary operations. </b>All we need to know is which ones are unary, in fact, and:
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::arithmetic_op_is_unary</span><span class="plain">(</span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">op</span><span class="plain">) {</span>
<span class="reserved">switch</span><span class="plain"> (</span><span class="identifier">op</span><span class="plain">) {</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">CUBEROOT_OPERATION</span><span class="plain">:</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">ROOT_OPERATION</span><span class="plain">:</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">REALROOT_OPERATION</span><span class="plain">:</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">UNARY_MINUS_OPERATION</span><span class="plain">:</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::arithmetic_op_is_unary is used in <a href="#SP40">&#167;40</a>, <a href="#SP40_1">&#167;40.1</a>, <a href="#SP40_2">&#167;40.2</a>.</p>
<p class="inwebparagraph"><a id="SP20"></a><b>&#167;20. Euclid's algorithm. </b>In my entire life, I believe this is the only time I have ever actually
used Euclid's algorithm for the GCD of two natural numbers. I've never
quite understood why textbooks take this as somehow the typical algorithm.
My maths students always find it a little oblique, despite the almost
trivial proof that it works. I find it hard to visualise myself, for that
matter. And then, consider that the average number of iterations τ_n,
in effect its running time, is known to be
τ_n = ((12log 2) / (π^2))log n + (4P + 5/2) + O(n^{-1/6+ε)
for any ε&gt;0, where P is defined in terms of an integral, Euler's
constant, and an evaluation of the derivative of the Riemann zeta function
&mdash; see D. E. Knuth, `Evaluation of Porter's Constant', reprinted in
"Selected Papers on Analysis of Algorithms" (Stanford: CSLI Lecture Notes
102, 2000). In practice, a shade under log n steps, then, which is nicely
quick. But I don't look at the code and immediately see this, myself.
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::gcd</span><span class="plain">(</span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">m</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">n</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">m</span><span class="plain">&lt;1) || (</span><span class="identifier">n</span><span class="plain">&lt;1)) </span><span class="identifier">internal_error</span><span class="plain">(</span><span class="string">"applied Kinds::Dimensions::gcd outside natural numbers"</span><span class="plain">);</span>
<span class="reserved">while</span><span class="plain"> (</span><span class="identifier">TRUE</span><span class="plain">) {</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">rem</span><span class="plain"> = </span><span class="identifier">m</span><span class="plain">%</span><span class="identifier">n</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">rem</span><span class="plain"> == 0) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">n</span><span class="plain">;</span>
<span class="identifier">m</span><span class="plain"> = </span><span class="identifier">n</span><span class="plain">; </span><span class="identifier">n</span><span class="plain"> = </span><span class="identifier">rem</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::gcd is used in <a href="#SP21">&#167;21</a>, 2/sav (<a href="2-sav.html#SP9_2_2">&#167;9.2.2</a>, <a href="2-sav.html#SP21_1_2">&#167;21.1.2</a>).</p>
<p class="inwebparagraph"><a id="SP21"></a><b>&#167;21. </b>The sequence of operation here is to reduce the risk of integer overflows
when multiplying <code class="display"><span class="extract">m</span></code> by <code class="display"><span class="extract">n</span></code>.
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::lcm</span><span class="plain">(</span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">m</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">n</span><span class="plain">) {</span>
<span class="reserved">return</span><span class="plain"> (</span><span class="identifier">m</span><span class="plain">/</span><span class="functiontext">Kinds::Dimensions::gcd</span><span class="plain">(</span><span class="identifier">m</span><span class="plain">, </span><span class="identifier">n</span><span class="plain">))*</span><span class="identifier">n</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::lcm is used in <a href="#SP40_4">&#167;40.4</a>.</p>
<p class="inwebparagraph"><a id="SP22"></a><b>&#167;22. Unit sequences. </b>Given a fundamental type B, convert it to a unit sequence: B = B^1, so we
get a sequence with a single pair: ((B, 1)). Uniquely, "number" is born
derived and dimensionless, though, so that comes out as the empty sequence.
</p>
<pre class="display">
<span class="reserved">unit_sequence</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::fundamental_unit_sequence</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">B</span><span class="plain">) {</span>
<span class="reserved">unit_sequence</span><span class="plain"> </span><span class="identifier">us</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">B</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) {</span>
<span class="identifier">us</span><span class="element">.no_unit_pairs</span><span class="plain"> = 0;</span>
<span class="identifier">us</span><span class="element">.unit_pairs</span><span class="plain">[0]</span><span class="element">.fund_unit</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span><span class="identifier">us</span><span class="element">.unit_pairs</span><span class="plain">[0]</span><span class="element">.power</span><span class="plain"> = 0; </span> <span class="comment">redundant, but appeases <code class="display"><span class="extract">gcc -O2</span></code></span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> {</span>
<span class="identifier">us</span><span class="element">.no_unit_pairs</span><span class="plain"> = 1;</span>
<span class="identifier">us</span><span class="element">.unit_pairs</span><span class="plain">[0]</span><span class="element">.fund_unit</span><span class="plain"> = </span><span class="identifier">B</span><span class="plain">; </span><span class="identifier">us</span><span class="element">.unit_pairs</span><span class="plain">[0]</span><span class="element">.power</span><span class="plain"> = 1;</span>
<span class="plain">}</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">us</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::fundamental_unit_sequence is used in 2/kc2 (<a href="2-kc2.html#SP7_1">&#167;7.1</a>).</p>
<p class="inwebparagraph"><a id="SP23"></a><b>&#167;23. </b>As noted above, two units represent dimensionally equivalent physical
quantities if and only if they are identical, which makes comparison easy:
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::compare_unit_sequences</span><span class="plain">(</span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">ik1</span><span class="plain">, </span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">ik2</span><span class="plain">) {</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">i</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">ik1</span><span class="plain"> == </span><span class="identifier">ik2</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">ik1</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || (</span><span class="identifier">ik2</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">ik1</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> != </span><span class="identifier">ik2</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">i</span><span class="plain">=0; </span><span class="identifier">i</span><span class="plain">&lt;</span><span class="identifier">ik1</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">; </span><span class="identifier">i</span><span class="plain">++)</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">ik1</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain">, </span><span class="identifier">ik2</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) ||</span>
<span class="plain">(</span><span class="identifier">ik1</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.power</span><span class="plain"> != </span><span class="identifier">ik2</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.power</span><span class="plain">))</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::compare_unit_sequences is used in <a href="#SP30_4">&#167;30.4</a>, <a href="#SP40">&#167;40</a>, <a href="#SP40_3">&#167;40.3</a>, 2/kc (<a href="2-kc.html#SP10">&#167;10</a>).</p>
<p class="inwebparagraph"><a id="SP24"></a><b>&#167;24. </b>We now have three fundamental operations we can perform on unit sequences.
First, we can multiply them: that is, we store in <code class="display"><span class="extract">result</span></code> the unit
sequence representing X_1^{s_1}X_2^{s_2}, where X_1 and X_2 are
represented by unit sequences <code class="display"><span class="extract">us1</span></code> and <code class="display"><span class="extract">us2</span></code>.
</p>
<p class="inwebparagraph">So the case s_1 = s_2 = 1 represents multiplying X_1 by X_2, while
s_1 = 1, s_2 = -1 represents dividing X_1 by X_2. But we can also
raise to higher powers.
</p>
<p class="inwebparagraph">Our method relies on noting that
X_1 = T_{11}^{p_{11}}. T_{12}^{p_{12}}... T_{1n}^{p_{1n}},
X_2 = T_{21}^{p_{21}}. T_{22}^{p_{22}}... T_{2m}^{p_{2m}}
where T_{11} &lt; T_{12} &lt; ... &lt; T_{1n} and T_{21}&lt;T_{22}&lt;...&lt;T_{2m}. We
can therefore merge the two in a single pass.
</p>
<p class="inwebparagraph">On each iteration of the loop the variables <code class="display"><span class="extract">i1</span></code> and <code class="display"><span class="extract">i2</span></code> are our current
read positions in each sequence, while we are currently looking at the
unit pairs (<code class="display"><span class="extract">t1</span></code>, <code class="display"><span class="extract">m1</span></code>) and (<code class="display"><span class="extract">t2</span></code>, <code class="display"><span class="extract">m2</span></code>). The following symmetrical
algorithm holds on to each pair until the one from the other sequence has had
a chance to catch up with it, because we always deal with the pair with the
numerically lower <code class="display"><span class="extract">t</span></code> first. This also proves that the <code class="display"><span class="extract">results</span></code> sequence comes
out in numerical order.
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::multiply_unit_sequences</span><span class="plain">(</span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">us1</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">s1</span><span class="plain">, </span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">us2</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">s2</span><span class="plain">,</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">result</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">result</span><span class="plain"> == </span><span class="identifier">us1</span><span class="plain">) || (</span><span class="identifier">result</span><span class="plain"> == </span><span class="identifier">us2</span><span class="plain">)) </span><span class="identifier">internal_error</span><span class="plain">(</span><span class="string">"result must be different structure"</span><span class="plain">);</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> = 0;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">i1</span><span class="plain"> = 0, </span><span class="identifier">i2</span><span class="plain"> = 0; </span> <span class="comment">read position in sequences 1, 2</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">t1</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">p1</span><span class="plain"> = 0; </span> <span class="comment">start with no current term from sequence 1</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">t2</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">p2</span><span class="plain"> = 0; </span> <span class="comment">start with no current term from sequence 2</span>
<span class="reserved">while</span><span class="plain"> (</span><span class="identifier">TRUE</span><span class="plain">) {</span>
&lt;<span class="cwebmacro">If we have no current term from sequence 1, and it hasn't run out, fetch a new one</span> <span class="cwebmacronumber">24.1</span>&gt;<span class="character">;</span>
&lt;<span class="cwebmacro">If we have no current term from sequence 2, and it hasn't run out, fetch a new one</span> <span class="cwebmacronumber">24.2</span>&gt;<span class="character">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">t1</span><span class="plain">, </span><span class="identifier">t2</span><span class="plain">)) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">t1</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">break</span><span class="plain">; </span> <span class="comment">both sequences have now run out</span>
&lt;<span class="cwebmacro">Both terms refer to the same fundamental unit, so combine these into the result</span> <span class="cwebmacronumber">24.3</span>&gt;<span class="plain">;</span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> {</span>
&lt;<span class="cwebmacro">The terms refer to different fundamental units, so copy the numerically lower one into the result</span> <span class="cwebmacronumber">24.4</span>&gt;<span class="plain">;</span>
<span class="plain">}</span>
<span class="plain">}</span>
<span class="identifier">LOGIF</span><span class="plain">(</span><span class="identifier">KIND_CREATIONS</span><span class="plain">, </span><span class="string">"Multiplication: $Q * $Q = $Q\</span><span class="plain">n</span><span class="string">"</span><span class="plain">, </span><span class="identifier">us1</span><span class="plain">, </span><span class="identifier">us2</span><span class="plain">, </span><span class="identifier">result</span><span class="plain">);</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::multiply_unit_sequences is used in <a href="#SP26_2">&#167;26.2</a>, <a href="#SP30_2">&#167;30.2</a>, <a href="#SP30_4">&#167;30.4</a>, <a href="#SP40">&#167;40</a>.</p>
<p class="inwebparagraph"><a id="SP24_1"></a><b>&#167;24.1. </b><code class="display">
&lt;<span class="cwebmacrodefn">If we have no current term from sequence 1, and it hasn't run out, fetch a new one</span> <span class="cwebmacronumber">24.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">t1</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) &amp;&amp; (</span><span class="identifier">us1</span><span class="plain">) &amp;&amp; (</span><span class="identifier">i1</span><span class="plain"> &lt; </span><span class="identifier">us1</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">)) {</span>
<span class="identifier">t1</span><span class="plain"> = </span><span class="identifier">us1</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i1</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain">; </span><span class="identifier">p1</span><span class="plain"> = </span><span class="identifier">us1</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i1</span><span class="plain">]</span><span class="element">.power</span><span class="plain">; </span><span class="identifier">i1</span><span class="plain">++;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP24">&#167;24</a>.</p>
<p class="inwebparagraph"><a id="SP24_2"></a><b>&#167;24.2. </b><code class="display">
&lt;<span class="cwebmacrodefn">If we have no current term from sequence 2, and it hasn't run out, fetch a new one</span> <span class="cwebmacronumber">24.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">t2</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) &amp;&amp; (</span><span class="identifier">us2</span><span class="plain">) &amp;&amp; (</span><span class="identifier">i2</span><span class="plain"> &lt; </span><span class="identifier">us2</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">)) {</span>
<span class="identifier">t2</span><span class="plain"> = </span><span class="identifier">us2</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i2</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain">; </span><span class="identifier">p2</span><span class="plain"> = </span><span class="identifier">us2</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i2</span><span class="plain">]</span><span class="element">.power</span><span class="plain">; </span><span class="identifier">i2</span><span class="plain">++;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP24">&#167;24</a>.</p>
<p class="inwebparagraph"><a id="SP24_3"></a><b>&#167;24.3. </b>So here the head of one sequence is T^{p_1} and the head of the other
is T^{p_2}, so in the product we ought to see (T^{p_1})^{s_1}.
(T^{p_2})^{s_2} = T^{p_1s_1+p_2s_2}. But we don't enter terms that have
cancelled out, that is, where p_1s_1+p_2s_2 = 0.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Both terms refer to the same fundamental unit, so combine these into the result</span> <span class="cwebmacronumber">24.3</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">p</span><span class="plain"> = </span><span class="identifier">p1</span><span class="plain">*</span><span class="identifier">s1</span><span class="plain"> + </span><span class="identifier">p2</span><span class="plain">*</span><span class="identifier">s2</span><span class="plain">; </span> <span class="comment">combined power of <code class="display"><span class="extract">t1</span></code> = <code class="display"><span class="extract">t2</span></code></span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">p</span><span class="plain"> != 0) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> == </span><span class="constant">MAX_BASE_UNITS_IN_SEQUENCE</span><span class="plain">)</span>
&lt;<span class="cwebmacro">Trip a unit sequence overflow</span> <span class="cwebmacronumber">24.3.1</span>&gt;<span class="plain">;</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain"> = </span><span class="identifier">t1</span><span class="plain">;</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">++]</span><span class="element">.power</span><span class="plain"> = </span><span class="identifier">p</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="identifier">t1</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span><span class="identifier">t2</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span> <span class="comment">dispose of both terms as dealt with</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP24">&#167;24</a>.</p>
<p class="inwebparagraph"><a id="SP24_4"></a><b>&#167;24.4. </b>Otherwise we copy. By copying the numerically lower term, we can be sure
that it will never occur again in either sequence. So we can copy it straight
into the results.
</p>
<p class="inwebparagraph">The code is slightly warped by the fact that <code class="display"><span class="extract">UNKNOWN_NT</span></code>, representing the
end of the sequence, happens to be numerically lower than all the valid
kinds. We don't want to make use of facts like that, so we write code
to deal with <code class="display"><span class="extract">UNKNOWN_NT</span></code> explicitly.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">The terms refer to different fundamental units, so copy the numerically lower one into the result</span> <span class="cwebmacronumber">24.4</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">t2</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || ((</span><span class="identifier">t1</span><span class="plain"> != </span><span class="identifier">NULL</span><span class="plain">) &amp;&amp; (</span><span class="functiontext">Kinds::Dimensions::kind_prior</span><span class="plain">(</span><span class="identifier">t1</span><span class="plain">, </span><span class="identifier">t2</span><span class="plain">)))) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> == </span><span class="constant">MAX_BASE_UNITS_IN_SEQUENCE</span><span class="plain">)</span>
&lt;<span class="cwebmacro">Trip a unit sequence overflow</span> <span class="cwebmacronumber">24.3.1</span>&gt;<span class="plain">;</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain"> = </span><span class="identifier">t1</span><span class="plain">;</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">++]</span><span class="element">.power</span><span class="plain"> = </span><span class="identifier">p1</span><span class="plain">*</span><span class="identifier">s1</span><span class="plain">;</span>
<span class="identifier">t1</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span> <span class="comment">dispose of the head of sequence 1 as dealt with</span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> </span><span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">t1</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || ((</span><span class="identifier">t2</span><span class="plain"> != </span><span class="identifier">NULL</span><span class="plain">) &amp;&amp; (</span><span class="functiontext">Kinds::Dimensions::kind_prior</span><span class="plain">(</span><span class="identifier">t2</span><span class="plain">, </span><span class="identifier">t1</span><span class="plain">)))) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> == </span><span class="constant">MAX_BASE_UNITS_IN_SEQUENCE</span><span class="plain">)</span>
&lt;<span class="cwebmacro">Trip a unit sequence overflow</span> <span class="cwebmacronumber">24.3.1</span>&gt;<span class="plain">;</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain"> = </span><span class="identifier">t2</span><span class="plain">;</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">++]</span><span class="element">.power</span><span class="plain"> = </span><span class="identifier">p2</span><span class="plain">*</span><span class="identifier">s2</span><span class="plain">;</span>
<span class="identifier">t2</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span> <span class="comment">dispose of the head of sequence 1 as dealt with</span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> </span><span class="identifier">internal_error</span><span class="plain">(</span><span class="string">"unit pairs disarrayed"</span><span class="plain">);</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP24">&#167;24</a>.</p>
<p class="inwebparagraph"><a id="SP24_3_1"></a><b>&#167;24.3.1. </b>For reasons explained above, this is really never going to happen by
accident, but we'll be careful:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Trip a unit sequence overflow</span> <span class="cwebmacronumber">24.3.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="identifier">KINDS_PROBLEM_HANDLER</span><span class="plain">(</span><span class="constant">UnitSequenceOverflow_KINDERROR</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">);</span>
<span class="reserved">return</span><span class="plain">;</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP24_3">&#167;24.3</a>, <a href="#SP24_4">&#167;24.4</a> (twice).</p>
<p class="inwebparagraph"><a id="SP25"></a><b>&#167;25. </b>The second operation is taking roots.
</p>
<p class="inwebparagraph">Surprisingly, perhaps, it's much easier to compute sqrt(X) or
^{3} sqrt(X) for any unit X &mdash; it's just that it can't always be done.
Inform does not permit non-integer powers of units, so for instance
sqrt(time) does not exist, whereas sqrt(length^2.mass^{-2})
does. Square roots exist if each power in the sequence is even, cube roots
exist if each is divisible by 3. We return <code class="display"><span class="extract">TRUE</span></code> or <code class="display"><span class="extract">FALSE</span></code> according to
whether the root could be taken, and if <code class="display"><span class="extract">FALSE</span></code> then the contents of
<code class="display"><span class="extract">result</span></code> are undefined.
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::root_unit_sequence</span><span class="plain">(</span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">us</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">pow</span><span class="plain">, </span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">result</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">us</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="plain">*</span><span class="identifier">result</span><span class="plain"> = *</span><span class="identifier">us</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">i</span><span class="plain">;</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">i</span><span class="plain">=0; </span><span class="identifier">i</span><span class="plain">&lt;</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">; </span><span class="identifier">i</span><span class="plain">++) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.power</span><span class="plain">) % </span><span class="identifier">pow</span><span class="plain"> != 0) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.power</span><span class="plain"> = (</span><span class="identifier">result</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.power</span><span class="plain">)/</span><span class="identifier">pow</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::root_unit_sequence is used in <a href="#SP40">&#167;40</a>, <a href="#SP41">&#167;41</a>.</p>
<p class="inwebparagraph"><a id="SP26"></a><b>&#167;26. </b>The final operation on unit sequences is substitution. Given a fundamental type
B, we substitute B = K_D into an existing unit sequence K_E. (This is
used when B is becoming a derived type &mdash; once we discover that B=K_D,
we are no longer allowed to keep B in any unit sequence.)
</p>
<p class="inwebparagraph">We simply search for B^p, and if we find it, we remove it and then
multiply by K_D^p.
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::dim_substitute</span><span class="plain">(</span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">existing</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">fundamental</span><span class="plain">, </span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">derived</span><span class="plain">) {</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">i</span><span class="plain">, </span><span class="identifier">j</span><span class="plain">, </span><span class="identifier">p</span><span class="plain"> = 0, </span><span class="identifier">found</span><span class="plain"> = </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">existing</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">i</span><span class="plain">=0; </span><span class="identifier">i</span><span class="plain">&lt;</span><span class="identifier">existing</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">; </span><span class="identifier">i</span><span class="plain">++)</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">existing</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain">, </span><span class="identifier">fundamental</span><span class="plain">)) {</span>
<span class="identifier">p</span><span class="plain"> = </span><span class="identifier">existing</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]</span><span class="element">.power</span><span class="plain">;</span>
<span class="identifier">found</span><span class="plain"> = </span><span class="identifier">TRUE</span><span class="plain">;</span>
&lt;<span class="cwebmacro">Remove the B term from the existing sequence</span> <span class="cwebmacronumber">26.1</span>&gt;<span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">found</span><span class="plain">)</span>
&lt;<span class="cwebmacro">Multiply the existing sequence by a suitable power of B's derivation</span> <span class="cwebmacronumber">26.2</span>&gt;<span class="character">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::dim_substitute is used in <a href="#SP30_3">&#167;30.3</a>.</p>
<p class="inwebparagraph"><a id="SP26_1"></a><b>&#167;26.1. </b>We shuffle the remaining terms in the sequence down by one, overwriting B:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Remove the B term from the existing sequence</span> <span class="cwebmacronumber">26.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">j</span><span class="plain">=</span><span class="identifier">i</span><span class="plain">; </span><span class="identifier">j</span><span class="plain">&lt;</span><span class="identifier">existing</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">-1; </span><span class="identifier">j</span><span class="plain">++)</span>
<span class="identifier">existing</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">j</span><span class="plain">] = </span><span class="identifier">existing</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">j</span><span class="plain">+1];</span>
<span class="identifier">existing</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">--;</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP26">&#167;26</a>.</p>
<p class="inwebparagraph"><a id="SP26_2"></a><b>&#167;26.2. </b>We now multiply by K_D^p.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Multiply the existing sequence by a suitable power of B's derivation</span> <span class="cwebmacronumber">26.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">unit_sequence</span><span class="plain"> </span><span class="identifier">result</span><span class="plain">;</span>
<span class="functiontext">Kinds::Dimensions::multiply_unit_sequences</span><span class="plain">(</span><span class="identifier">existing</span><span class="plain">, 1, </span><span class="identifier">derived</span><span class="plain">, </span><span class="identifier">p</span><span class="plain">, &amp;</span><span class="identifier">result</span><span class="plain">);</span>
<span class="plain">*</span><span class="identifier">existing</span><span class="plain"> = </span><span class="identifier">result</span><span class="plain">;</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP26">&#167;26</a>.</p>
<p class="inwebparagraph"><a id="SP27"></a><b>&#167;27. </b>For reasons which will be explained below, a unit sequence also has
a scale factor associated with it:
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::us_get_scaling_factor</span><span class="plain">(</span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">us</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">us</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> 1;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">us</span><span class="plain">-</span><span class="element">&gt;scaling_factor</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::us_get_scaling_factor is used in 2/uk (<a href="2-uk.html#SP19">&#167;19</a>).</p>
<p class="inwebparagraph"><a id="SP28"></a><b>&#167;28. </b>That just leaves, as usual, indexing...
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::index_unit_sequence</span><span class="plain">(</span><span class="identifier">OUTPUT_STREAM</span><span class="plain">, </span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">deriv</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">briefly</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">deriv</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> == 0) { </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"dimensionless"</span><span class="plain">); </span><span class="reserved">return</span><span class="plain">; }</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">j</span><span class="plain">;</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">j</span><span class="plain">=0; </span><span class="identifier">j</span><span class="plain">&lt;</span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">; </span><span class="identifier">j</span><span class="plain">++) {</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">fundamental</span><span class="plain"> = </span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">j</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">power</span><span class="plain"> = </span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">j</span><span class="plain">]</span><span class="element">.power</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">briefly</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">j</span><span class="plain">&gt;0) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"."</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"("</span><span class="plain">);</span>
<span class="plain">#</span><span class="identifier">ifdef</span><span class="plain"> </span><span class="identifier">CORE_MODULE</span>
<span class="identifier">Kinds::Index::index_kind</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">fundamental</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">);</span>
<span class="plain">#</span><span class="reserved">else</span>
<span class="functiontext">Kinds::Textual::write</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">fundamental</span><span class="plain">);</span>
<span class="plain">#</span><span class="identifier">endif</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">")"</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> != 1) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&lt;sup&gt;%d&lt;/sup&gt;"</span><span class="plain">, </span><span class="identifier">power</span><span class="plain">);</span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">j</span><span class="plain">&gt;0) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" times "</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> &lt; 0) { </span><span class="identifier">power</span><span class="plain"> = -</span><span class="identifier">power</span><span class="plain">; </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"reciprocal of "</span><span class="plain">); }</span>
<span class="identifier">wording</span><span class="plain"> </span><span class="identifier">W</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_name</span><span class="plain">(</span><span class="identifier">fundamental</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"%W"</span><span class="plain">, </span><span class="identifier">W</span><span class="plain">);</span>
<span class="reserved">switch</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain">) {</span>
<span class="reserved">case</span><span class="plain"> 1: </span><span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> 2: </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" squared"</span><span class="plain">); </span><span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> 3: </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" cubed"</span><span class="plain">); </span><span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">default</span><span class="plain">: </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" to the power %d"</span><span class="plain">, </span><span class="identifier">power</span><span class="plain">); </span><span class="reserved">break</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="plain">}</span>
<span class="plain">}</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::index_unit_sequence is used in <a href="#SP32_2">&#167;32.2</a>, 2/dk (<a href="2-dk.html#SP24_1">&#167;24.1</a>).</p>
<p class="inwebparagraph"><a id="SP29"></a><b>&#167;29. </b>...and logging.
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::log_unit_sequence</span><span class="plain">(</span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">deriv</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">deriv</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) { </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">"&lt;null-us&gt;"</span><span class="plain">); </span><span class="reserved">return</span><span class="plain">; }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> == 0) { </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">"dimensionless"</span><span class="plain">); </span><span class="reserved">return</span><span class="plain">; }</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">j</span><span class="plain">;</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">j</span><span class="plain">=0; </span><span class="identifier">j</span><span class="plain">&lt;</span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain">; </span><span class="identifier">j</span><span class="plain">++) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">j</span><span class="plain">&gt;0) </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">"."</span><span class="plain">);</span>
<span class="identifier">LOG</span><span class="plain">(</span><span class="string">"($u)"</span><span class="plain">, </span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">j</span><span class="plain">]</span><span class="element">.fund_unit</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">j</span><span class="plain">]</span><span class="element">.power</span><span class="plain"> != 1) </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">"%d"</span><span class="plain">, </span><span class="identifier">deriv</span><span class="plain">-</span><span class="element">&gt;unit_pairs</span><span class="plain">[</span><span class="identifier">j</span><span class="plain">]</span><span class="element">.power</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::log_unit_sequence appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP30"></a><b>&#167;30. Performing derivations. </b>The following is called when the user specifies that L times R specifies
an O. These are required all to be quasinumerical: any of the three might
be either a fundamental unit (so far) or a derived unit (already).
</p>
<p class="inwebparagraph">If two or more are fundamental units, we have a choice. That is, suppose we have
created three kinds already: mass, acceleration, force. Then we read:
</p>
<blockquote>
<p>Mass times acceleration specifies a force.</p>
</blockquote>
<p class="inwebparagraph">We could make this true in any of three ways: keep M and A as fundamental units
and derive F from them, keep A and F as fundamental units and derive M from those,
or keep M and F while deriving A. Inform always chooses the most recently
created unit as the one to derive, on the grounds that the source text has
probably set things out with what the user thinks are the most fundamental
units first.
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::make_unit_derivation</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">left</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">right</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">outcome</span><span class="plain">) {</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">terms</span><span class="plain">[3];</span>
<span class="identifier">terms</span><span class="plain">[0] = </span><span class="identifier">left</span><span class="plain">; </span><span class="identifier">terms</span><span class="plain">[1] = </span><span class="identifier">right</span><span class="plain">; </span><span class="identifier">terms</span><span class="plain">[2] = </span><span class="identifier">outcome</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">newest_term</span><span class="plain"> = -1;</span>
&lt;<span class="cwebmacro">Find which (if any) of the three units is the newest-made fundamental unit</span> <span class="cwebmacronumber">30.1</span>&gt;<span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">newest_term</span><span class="plain"> &gt;= 0) {</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">derivation</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">;</span>
&lt;<span class="cwebmacro">Derive the newest one by rearranging the equation in terms of the other two</span> <span class="cwebmacronumber">30.2</span>&gt;<span class="plain">;</span>
&lt;<span class="cwebmacro">Substitute this new derivation to eliminate this fundamental unit from other sequences</span> <span class="cwebmacronumber">30.3</span>&gt;<span class="plain">;</span>
<span class="plain">} </span><span class="reserved">else</span>
&lt;<span class="cwebmacro">Check this derivation to make sure it is redundant, not contradictory</span> <span class="cwebmacronumber">30.4</span>&gt;<span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::make_unit_derivation is used in <a href="#SP18">&#167;18</a>.</p>
<p class="inwebparagraph"><a id="SP30_1"></a><b>&#167;30.1. </b>Data type IDs are allocated in creation order, so "newest" means largest ID.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Find which (if any) of the three units is the newest-made fundamental unit</span> <span class="cwebmacronumber">30.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">i</span><span class="plain">; </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">max</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">for</span><span class="plain"> (</span><span class="identifier">i</span><span class="plain">=0; </span><span class="identifier">i</span><span class="plain">&lt;3; </span><span class="identifier">i</span><span class="plain">++)</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Dimensions::kind_prior</span><span class="plain">(</span><span class="identifier">max</span><span class="plain">, </span><span class="identifier">terms</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">])) &amp;&amp; (</span><span class="functiontext">Kinds::Behaviour::test_if_derived</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">]) == </span><span class="identifier">FALSE</span><span class="plain">)) {</span>
<span class="identifier">newest_term</span><span class="plain"> = </span><span class="identifier">i</span><span class="plain">; </span><span class="identifier">max</span><span class="plain"> = </span><span class="identifier">terms</span><span class="plain">[</span><span class="identifier">i</span><span class="plain">];</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP30">&#167;30</a>.</p>
<p class="inwebparagraph"><a id="SP30_2"></a><b>&#167;30.2. </b>We need to ensure that the user's multiplication rule is henceforth true,
and we do that by fixing the newest unit to make it so.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Derive the newest one by rearranging the equation in terms of the other two</span> <span class="cwebmacronumber">30.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">kx</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">, *</span><span class="identifier">ky</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">; </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">sx</span><span class="plain"> = 0, </span><span class="identifier">sy</span><span class="plain"> = 0;</span>
<span class="reserved">switch</span><span class="plain"> (</span><span class="identifier">newest_term</span><span class="plain">) {</span>
<span class="reserved">case</span><span class="plain"> 0: </span> <span class="comment">here L is newest and we derive L = R^{-1}O</span>
<span class="identifier">kx</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[1]); </span><span class="identifier">sx</span><span class="plain"> = -1;</span>
<span class="identifier">ky</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[2]); </span><span class="identifier">sy</span><span class="plain"> = 1;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> 1: </span> <span class="comment">here R is newest and we derive R = L^{-1}O</span>
<span class="identifier">kx</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[0]); </span><span class="identifier">sx</span><span class="plain"> = -1;</span>
<span class="identifier">ky</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[2]); </span><span class="identifier">sy</span><span class="plain"> = 1;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> 2: </span> <span class="comment">here O is newest and we derive O = LR</span>
<span class="identifier">kx</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[0]); </span><span class="identifier">sx</span><span class="plain"> = 1;</span>
<span class="identifier">ky</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[1]); </span><span class="identifier">sy</span><span class="plain"> = 1;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="identifier">derivation</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[</span><span class="identifier">newest_term</span><span class="plain">]);</span>
<span class="reserved">unit_sequence</span><span class="plain"> </span><span class="identifier">result</span><span class="plain">;</span>
<span class="functiontext">Kinds::Dimensions::multiply_unit_sequences</span><span class="plain">(</span><span class="identifier">kx</span><span class="plain">, </span><span class="identifier">sx</span><span class="plain">, </span><span class="identifier">ky</span><span class="plain">, </span><span class="identifier">sy</span><span class="plain">, &amp;</span><span class="identifier">result</span><span class="plain">);</span>
<span class="plain">*</span><span class="identifier">derivation</span><span class="plain"> = </span><span class="identifier">result</span><span class="plain">;</span>
<span class="functiontext">Kinds::Behaviour::now_derived</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[</span><span class="identifier">newest_term</span><span class="plain">]);</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP30">&#167;30</a>.</p>
<p class="inwebparagraph"><a id="SP30_3"></a><b>&#167;30.3. </b>Later in Inform's run, when we start compiling code, many more unit sequences
will exist on a temporary basis &mdash; as part of the kinds for intermediate results
in calculations &mdash; but early on, when we're here, the only unit sequences made
are the derivations of the units. So it is easy to cover all of them.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Substitute this new derivation to eliminate this fundamental unit from other sequences</span> <span class="cwebmacronumber">30.3</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">R</span><span class="plain">;</span>
<span class="identifier">LOOP_OVER_BASE_KINDS</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)) {</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">existing</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">);</span>
<span class="functiontext">Kinds::Dimensions::dim_substitute</span><span class="plain">(</span><span class="identifier">existing</span><span class="plain">, </span><span class="identifier">terms</span><span class="plain">[</span><span class="identifier">newest_term</span><span class="plain">], </span><span class="identifier">derivation</span><span class="plain">);</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP30">&#167;30</a>.</p>
<p class="inwebparagraph"><a id="SP30_4"></a><b>&#167;30.4. </b>If we have AB = C but all three of A, B, C are already derived,
that puts us in a bind. Their definitions are fixed already, so we can't
simply force the equation to come true by fixing one of them. That means
either the derivation is redundant &mdash; because it's already true that
AB = C &mdash; or contradictory &mdash; because we know AB!= C. We silently
allow a redundancy, as it may have been put in for clarity, or so that
the user can check the consistency of his own definitions, or to make
the Kinds index page more helpful. But we must reject a contradiction.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Check this derivation to make sure it is redundant, not contradictory</span> <span class="cwebmacronumber">30.4</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">unit_sequence</span><span class="plain"> </span><span class="identifier">product</span><span class="plain">;</span>
<span class="functiontext">Kinds::Dimensions::multiply_unit_sequences</span><span class="plain">(</span>
<span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[0]), 1,</span>
<span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[1]), 1, &amp;</span><span class="identifier">product</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::compare_unit_sequences</span><span class="plain">(&amp;</span><span class="identifier">product</span><span class="plain">,</span>
<span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">terms</span><span class="plain">[2])) == </span><span class="identifier">FALSE</span><span class="plain">)</span>
<span class="identifier">KINDS_PROBLEM_HANDLER</span><span class="plain">(</span><span class="constant">DimensionsInconsistent_KINDERROR</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">);</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP30">&#167;30</a>.</p>
<p class="inwebparagraph"><a id="SP31"></a><b>&#167;31. Classifying the units. </b>Some of the derived units are dimensionless, others not. Number
is always dimensionless; and any unit whose derivation is the empty
unit sequence must be dimensionless.
</p>
<pre class="display">
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">K</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">K</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">, </span><span class="identifier">K_real_number</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::us_dimensionless</span><span class="plain">(</span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">));</span>
<span class="plain">}</span>
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::us_dimensionless</span><span class="plain">(</span><span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">us</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">us</span><span class="plain">) &amp;&amp; (</span><span class="identifier">us</span><span class="plain">-</span><span class="element">&gt;no_unit_pairs</span><span class="plain"> == 0)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::dimensionless is used in <a href="#SP32_2">&#167;32.2</a>, <a href="#SP33">&#167;33</a>, <a href="#SP40_1">&#167;40.1</a>, <a href="#SP41">&#167;41</a>.</p>
<p class="endnote">The function Kinds::Dimensions::us_dimensionless is used in <a href="#SP40_2">&#167;40.2</a>.</p>
<p class="inwebparagraph"><a id="SP32"></a><b>&#167;32. </b>Using these definitions, we can now print analyses of the units into the
index and the debugging log.
</p>
<pre class="display">
<span class="plain">#</span><span class="identifier">ifdef</span><span class="plain"> </span><span class="identifier">CORE_MODULE</span>
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::index_dimensional_rules</span><span class="plain">(</span><span class="identifier">OUTPUT_STREAM</span><span class="plain">) {</span>
<span class="identifier">HTML_TAG</span><span class="plain">(</span><span class="string">"hr"</span><span class="plain">);</span>
&lt;<span class="cwebmacro">Index the rubric about quasinumerical kinds</span> <span class="cwebmacronumber">32.1</span>&gt;<span class="plain">;</span>
&lt;<span class="cwebmacro">Index the table of quasinumerical kinds</span> <span class="cwebmacronumber">32.2</span>&gt;<span class="plain">;</span>
&lt;<span class="cwebmacro">Index the table of multiplication rules</span> <span class="cwebmacronumber">32.3</span>&gt;<span class="plain">;</span>
<span class="plain">}</span>
<span class="plain">#</span><span class="identifier">endif</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::index_dimensional_rules appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP32_1"></a><b>&#167;32.1. </b><code class="display">
&lt;<span class="cwebmacrodefn">Index the rubric about quasinumerical kinds</span> <span class="cwebmacronumber">32.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="identifier">HTML_OPEN</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">);</span>
<span class="identifier">HTML_TAG_WITH</span><span class="plain">(</span><span class="string">"a"</span><span class="plain">, </span><span class="string">"calculator"</span><span class="plain">);</span>
<span class="identifier">HTML::begin_plain_html_table</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">HTML::first_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">HTML_TAG_WITH</span><span class="plain">(</span><span class="string">"img"</span><span class="plain">, </span><span class="string">"border=0 src=inform:/doc_images/calc2.png"</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&amp;nbsp;"</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"Kinds of value marked with the &lt;b&gt;calculator symbol&lt;/b&gt; are numerical - "</span>
<span class="string">"these are values we can add, multiply and so on. The range of these "</span>
<span class="string">"numbers depends on the Format setting for the project (Glulx format "</span>
<span class="string">"supports much higher numbers than Z-code)."</span><span class="plain">);</span>
<span class="identifier">HTML::end_html_row</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">HTML::end_html_table</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">HTML_CLOSE</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">);</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP32">&#167;32</a>.</p>
<p class="inwebparagraph"><a id="SP32_2"></a><b>&#167;32.2. </b><code class="display">
&lt;<span class="cwebmacrodefn">Index the table of quasinumerical kinds</span> <span class="cwebmacronumber">32.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="identifier">HTML_OPEN</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">);</span>
<span class="identifier">HTML::begin_plain_html_table</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">HTML::first_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&lt;b&gt;kind of value&lt;/b&gt;"</span><span class="plain">);</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&lt;b&gt;minimum&lt;/b&gt;"</span><span class="plain">);</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&lt;b&gt;maximum&lt;/b&gt;"</span><span class="plain">);</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&lt;b&gt;dimensions&lt;/b&gt;"</span><span class="plain">);</span>
<span class="identifier">HTML::end_html_row</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">R</span><span class="plain">;</span>
<span class="identifier">LOOP_OVER_BASE_KINDS</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::is_intermediate</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)) </span><span class="reserved">continue</span><span class="plain">;</span>
<span class="identifier">HTML::first_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">Kinds::Index::index_kind</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">R</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">);</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
&lt;<span class="cwebmacro">Index the minimum positive value for a quasinumerical kind</span> <span class="cwebmacronumber">32.2.1</span>&gt;<span class="plain">;</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
&lt;<span class="cwebmacro">Index the maximum positive value for a quasinumerical kind</span> <span class="cwebmacronumber">32.2.2</span>&gt;<span class="plain">;</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&lt;i&gt;dimensionless&lt;/i&gt;"</span><span class="plain">);</span>
<span class="reserved">else</span><span class="plain"> {</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">deriv</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">);</span>
<span class="functiontext">Kinds::Dimensions::index_unit_sequence</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">deriv</span><span class="plain">, </span><span class="identifier">TRUE</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="identifier">HTML::end_html_row</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="identifier">HTML::end_html_table</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">HTML_CLOSE</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">);</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP32">&#167;32</a>.</p>
<p class="inwebparagraph"><a id="SP32_2_1"></a><b>&#167;32.2.1. </b>At run-time, the minimum positive value is of course <code class="display"><span class="extract">1</span></code>, but because of
scaling this can appear to be much lower.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Index the minimum positive value for a quasinumerical kind</span> <span class="cwebmacronumber">32.2.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">)) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"1"</span><span class="plain">);</span>
<span class="reserved">else</span><span class="plain"> {</span>
<span class="identifier">text_stream</span><span class="plain"> *</span><span class="identifier">p</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_index_minimum_value</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">Str::len</span><span class="plain">(</span><span class="identifier">p</span><span class="plain">) &gt; 0) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"%S"</span><span class="plain">, </span><span class="identifier">p</span><span class="plain">);</span>
<span class="reserved">else</span><span class="plain"> </span><span class="identifier">LiteralPatterns::index_value</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">,</span>
<span class="identifier">LiteralPatterns::list_of_literal_forms</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">), 1);</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP32_2">&#167;32.2</a>.</p>
<p class="inwebparagraph"><a id="SP32_2_2"></a><b>&#167;32.2.2. </b><code class="display">
&lt;<span class="cwebmacrodefn">Index the maximum positive value for a quasinumerical kind</span> <span class="cwebmacronumber">32.2.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">)) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">TargetVMs::is_16_bit</span><span class="plain">(</span><span class="identifier">Task::vm</span><span class="plain">())) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"32767"</span><span class="plain">);</span>
<span class="reserved">else</span><span class="plain"> </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"2147483647"</span><span class="plain">);</span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> {</span>
<span class="identifier">text_stream</span><span class="plain"> *</span><span class="identifier">p</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_index_maximum_value</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">Str::len</span><span class="plain">(</span><span class="identifier">p</span><span class="plain">) &gt; 0) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"%S"</span><span class="plain">, </span><span class="identifier">p</span><span class="plain">);</span>
<span class="reserved">else</span><span class="plain"> {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">TargetVMs::is_16_bit</span><span class="plain">(</span><span class="identifier">Task::vm</span><span class="plain">()))</span>
<span class="identifier">LiteralPatterns::index_value</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">,</span>
<span class="identifier">LiteralPatterns::list_of_literal_forms</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">), 32767);</span>
<span class="reserved">else</span>
<span class="identifier">LiteralPatterns::index_value</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">,</span>
<span class="identifier">LiteralPatterns::list_of_literal_forms</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">), 2147483647);</span>
<span class="plain">}</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP32_2">&#167;32.2</a>.</p>
<p class="inwebparagraph"><a id="SP32_3"></a><b>&#167;32.3. </b>This is simply a table of all the multiplications declared in the source
text, sorted into kind order of left and then right operand.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Index the table of multiplication rules</span> <span class="cwebmacronumber">32.3</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">L</span><span class="plain">, *</span><span class="identifier">R</span><span class="plain">, *</span><span class="identifier">O</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">NP</span><span class="plain"> = 0, </span><span class="identifier">wn</span><span class="plain">;</span>
<span class="identifier">LOOP_OVER_MULTIPLICATIONS</span><span class="plain">(</span><span class="identifier">L</span><span class="plain">, </span><span class="identifier">R</span><span class="plain">, </span><span class="identifier">O</span><span class="plain">, </span><span class="identifier">wn</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">NP</span><span class="plain">++ == 0) {</span>
<span class="identifier">HTML_OPEN</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"This is how multiplication changes kinds:"</span><span class="plain">);</span>
<span class="identifier">HTML_CLOSE</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">);</span>
<span class="identifier">HTML_OPEN</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">);</span>
<span class="identifier">HTML::begin_plain_html_table</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="identifier">HTML::first_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">wn</span><span class="plain"> &gt;= 0) </span><span class="identifier">Index::link</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">wn</span><span class="plain">);</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">Kinds::Index::index_kind</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">L</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">);</span>
<span class="identifier">HTML::begin_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">I</span><span class="string">"808080"</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" x "</span><span class="plain">);</span>
<span class="identifier">HTML::end_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">Kinds::Index::index_kind</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">R</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">);</span>
<span class="identifier">HTML::begin_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">I</span><span class="string">"808080"</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" = "</span><span class="plain">);</span>
<span class="identifier">HTML::end_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">Kinds::Index::index_kind</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">O</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">, </span><span class="identifier">FALSE</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;"</span><span class="plain">);</span>
<span class="identifier">HTML::next_html_column</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, 0);</span>
<span class="identifier">LiteralPatterns::index_benchmark_value</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">L</span><span class="plain">);</span>
<span class="identifier">HTML::begin_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">I</span><span class="string">"808080"</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" x "</span><span class="plain">);</span>
<span class="identifier">HTML::end_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">LiteralPatterns::index_benchmark_value</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">R</span><span class="plain">);</span>
<span class="identifier">HTML::begin_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">I</span><span class="string">"808080"</span><span class="plain">);</span>
<span class="identifier">WRITE</span><span class="plain">(</span><span class="string">" = "</span><span class="plain">);</span>
<span class="identifier">HTML::end_colour</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="identifier">LiteralPatterns::index_benchmark_value</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">, </span><span class="identifier">O</span><span class="plain">);</span>
<span class="identifier">HTML::end_html_row</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">NP</span><span class="plain"> &gt; 0) { </span><span class="identifier">HTML::end_html_table</span><span class="plain">(</span><span class="identifier">OUT</span><span class="plain">); </span><span class="identifier">HTML_CLOSE</span><span class="plain">(</span><span class="string">"p"</span><span class="plain">); }</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP32">&#167;32</a>.</p>
<p class="inwebparagraph"><a id="SP33"></a><b>&#167;33. </b>A simpler format for the debugging log, which is printed when we ask for
the internal "dimensions" test:
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::log_unit_analysis</span><span class="plain">(</span><span class="reserved">void</span><span class="plain">) {</span>
<span class="identifier">LOG</span><span class="plain">(</span><span class="string">"Dimensionless: "</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">c</span><span class="plain"> = 0; </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">R</span><span class="plain">;</span>
<span class="identifier">LOOP_OVER_BASE_KINDS</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)) { </span><span class="reserved">if</span><span class="plain"> (</span><span class="identifier">c</span><span class="plain">++ &gt; 0) </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">", "</span><span class="plain">); </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">"$u"</span><span class="plain">, </span><span class="identifier">R</span><span class="plain">); }</span>
<span class="identifier">LOG</span><span class="plain">(</span><span class="string">"\</span><span class="plain">n</span><span class="string">Base units: "</span><span class="plain">);</span>
<span class="identifier">c</span><span class="plain"> = 0;</span>
<span class="identifier">LOOP_OVER_BASE_KINDS</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) &amp;&amp;</span>
<span class="plain">(</span><span class="functiontext">Kinds::Dimensions::kind_is_derived</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) &amp;&amp;</span>
<span class="plain">(</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)))</span>
<span class="plain">{ </span><span class="reserved">if</span><span class="plain"> (</span><span class="identifier">c</span><span class="plain">++ &gt; 0) </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">", "</span><span class="plain">); </span><span class="identifier">LOG</span><span class="plain">(</span><span class="string">"$u"</span><span class="plain">, </span><span class="identifier">R</span><span class="plain">); }</span>
<span class="identifier">LOG</span><span class="plain">(</span><span class="string">"\</span><span class="plain">n</span><span class="string">Derived units:\</span><span class="plain">n</span><span class="string">"</span><span class="plain">);</span>
<span class="identifier">LOOP_OVER_BASE_KINDS</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Dimensions::kind_is_derived</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)) &amp;&amp; (</span><span class="functiontext">Kinds::is_intermediate</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">)) {</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">deriv</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">);</span>
<span class="identifier">LOG</span><span class="plain">(</span><span class="string">"$u = $Q\</span><span class="plain">n</span><span class="string">"</span><span class="plain">, </span><span class="identifier">R</span><span class="plain">, </span><span class="identifier">deriv</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="plain">}</span>
<span class="reserved">int</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_is_derived</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">K</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::is_intermediate</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Behaviour::is_quasinumerical</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">)) &amp;&amp;</span>
<span class="plain">(</span><span class="functiontext">Kinds::Behaviour::test_if_derived</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">) == </span><span class="identifier">TRUE</span><span class="plain">) &amp;&amp;</span>
<span class="plain">(</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">TRUE</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">FALSE</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::log_unit_analysis appears nowhere else.</p>
<p class="endnote">The function Kinds::Dimensions::kind_is_derived appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP34"></a><b>&#167;34. Scaling. </b>Recall that every quasinumerical kind U has a scale factor k_U, by
default 1 and always &gt;= 1, such that the true value v is represented
at runtime as the I6 integer k_U v. For instance, if length is measured
in metres but has a scale factor of 1000 then the I6 integer 1 means the
true value 1mm, 10 means 1cm, 1000 means 1m. This I6 integer is called
the scaled value s, and to reiterate, s = k_U v.
</p>
<p class="inwebparagraph">Scaled values are convenient, and they have no effect on how we add, subtract,
approximate (that is, round off) or take remainder after division. If we
have true values v_1 and v_2 with scaled values s_1 and s_2,
and s_o is the scaled value for true value v_1+v_2, then
</p>
<p class="inwebparagraph"> s_1 + s_2 = k_Uv_1 + k_Uv_2 = k_U(v_1+v_2) = s_o.
</p>
<p class="inwebparagraph">So ordinary I6 <code class="display"><span class="extract">+</span></code> at run-time correctly adds scaled values. But that's
not true for all operations, and this is where we deal with that.
</p>
<p class="inwebparagraph"><a id="SP35"></a><b>&#167;35. </b>First, multiplication. This time the values v_1 and v_2 may have
different kinds, which we'll call X and Y, and the result in general
will be a third kind, which we'll call O (for outcome). Then:
</p>
<p class="inwebparagraph"> s_1s_2 = k_Xv_1. k_Yv_2 = k_Ov_1v_2.(((k_Xk_Y) / (k_O))) = s_o.(((k_Xk_Y) / (k_O)))
</p>
<p class="inwebparagraph">so that simply multiplying the scaled values produces an answer which is
too large by a factor of k_Xk_Y/k_O. We need to correct for that, which
we do either by dividing by this factor or multiplying by its reciprocal.
</p>
<p class="inwebparagraph">This is all a little delicate since rounding errors may be an issue and
since k_Xk_Y/k_O is itself evaluated in integer arithmetic. In an ideal
world we might use the same k for many units (e.g., k=1000 throughout)
and then of course this cancels to just 1000. But in practice people
won't always do this &mdash; they may use some Babylonian, base 60, units, such
as minutes and degrees, for instance, where k=3600 would be more natural.
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_multiplication</span><span class="plain">(</span><span class="identifier">OUTPUT_STREAM</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindy</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || (</span><span class="identifier">kindy</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">kindy</span><span class="plain">, </span><span class="constant">TIMES_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_Y</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindy</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &gt; </span><span class="identifier">k_O</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"/%d"</span><span class="plain">, (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">));</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &lt; </span><span class="identifier">k_O</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"*%d"</span><span class="plain">, (</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_Y</span><span class="plain">));</span>
<span class="plain">}</span>
<span class="plain">#</span><span class="identifier">ifdef</span><span class="plain"> </span><span class="identifier">CORE_MODULE</span>
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_multiplication_emit_op</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindy</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || (</span><span class="identifier">kindy</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">kindy</span><span class="plain">, </span><span class="constant">TIMES_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_Y</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindy</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &gt; </span><span class="identifier">k_O</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">DIVIDE_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &lt; </span><span class="identifier">k_O</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">TIMES_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="plain">}</span>
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_multiplication_emit_factor</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindy</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || (</span><span class="identifier">kindy</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">kindy</span><span class="plain">, </span><span class="constant">TIMES_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_Y</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindy</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &gt; </span><span class="identifier">k_O</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_X</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &lt; </span><span class="identifier">k_O</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_Y</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="plain">}</span>
<span class="plain">#</span><span class="identifier">endif</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_multiplication appears nowhere else.</p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_multiplication_emit_op appears nowhere else.</p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_multiplication_emit_factor appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP36"></a><b>&#167;36. </b>Second, division, which is similar.
((s_1) / (s_2)) = ((k_Xv_1) / (k_Yv_2)) = k_O((v_1) / (v_2)).(((k_X) / (k_Ok_Y)))
= s_o.(((k_X) / (k_Ok_Y))right)
so this time the excess to correct is a factor of k_X/k_Ok_Y.
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_division</span><span class="plain">(</span><span class="identifier">OUTPUT_STREAM</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindy</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || (</span><span class="identifier">kindy</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">kindy</span><span class="plain">, </span><span class="constant">DIVIDE_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_Y</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindy</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"*%d"</span><span class="plain">, (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">));</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"/%d"</span><span class="plain">, (</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_Y</span><span class="plain">));</span>
<span class="plain">}</span>
<span class="plain">#</span><span class="identifier">ifdef</span><span class="plain"> </span><span class="identifier">CORE_MODULE</span>
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_division_emit_op</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindy</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || (</span><span class="identifier">kindy</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">kindy</span><span class="plain">, </span><span class="constant">DIVIDE_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_Y</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindy</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">TIMES_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">DIVIDE_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="plain">}</span>
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_division_emit_factor</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindy</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) || (</span><span class="identifier">kindy</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">kindy</span><span class="plain">, </span><span class="constant">DIVIDE_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_Y</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindy</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_Y</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_Y</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="plain">}</span>
<span class="plain">#</span><span class="identifier">endif</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_division appears nowhere else.</p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_division_emit_op appears nowhere else.</p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_division_emit_factor appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP37"></a><b>&#167;37. </b>Third, the taking of pth roots, at any rate for p=2 or p=3.
</p>
<pre class="display">
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_root</span><span class="plain">(</span><span class="identifier">OUTPUT_STREAM</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">power</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 2) </span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="constant">ROOT_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 3) </span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="constant">CUBEROOT_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 2) </span>&lt;<span class="cwebmacro">Apply a scaling correction for square roots</span> <span class="cwebmacronumber">37.1</span>&gt;
<span class="reserved">else</span><span class="plain"> </span><span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 3) </span>&lt;<span class="cwebmacro">Apply a scaling correction for cube roots</span> <span class="cwebmacronumber">37.2</span>&gt;
<span class="reserved">else</span><span class="plain"> </span><span class="identifier">internal_error</span><span class="plain">(</span><span class="string">"can only scale square and cube roots"</span><span class="plain">);</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_root appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP37_1"></a><b>&#167;37.1. </b>For square roots,
sqrt(s) = sqrt(k_Xv) = sqrt(k_X) sqrt(v) = k_O sqrt(v).
((( sqrt(k_X)) / (k_O))) = s_o .
((( sqrt(k_X)) / (k_O)))
and now the overestimate is a factor of k = sqrt(k_X)/k_O. However,
rather than calculating k sqrt(x) we calculate sqrt(k^2 x), since
this way accuracy losses in taking the square root are much reduced.
Therefore this scaling operating is to be performed inside the root
function, not outside, and it scales by k^2 not k:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Apply a scaling correction for square roots</span> <span class="cwebmacronumber">37.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"*%d"</span><span class="plain">, (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">));</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"/%d"</span><span class="plain">, (</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">));</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP37">&#167;37</a>.</p>
<p class="inwebparagraph"><a id="SP37_2"></a><b>&#167;37.2. </b>For cube roots,
curt(s) = curt(k_Xv) = curt(k_X) curt(v) = k_O curt(v).
((( curt(k_X)) / (k_O))) = s_o.
((( curt(k_X)) / (k_O)))
and the overestimate is k = curt(k_X)/k_O. Scaling once again within
the rooting function, we scale by k^3:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Apply a scaling correction for cube roots</span> <span class="cwebmacronumber">37.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"*%d"</span><span class="plain">, (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">));</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) </span><span class="identifier">WRITE</span><span class="plain">(</span><span class="string">"/%d"</span><span class="plain">, (</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">));</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP37">&#167;37</a>.</p>
<p class="inwebparagraph"><a id="SP38"></a><b>&#167;38. </b></p>
<pre class="display">
<span class="plain">#</span><span class="identifier">ifdef</span><span class="plain"> </span><span class="identifier">CORE_MODULE</span>
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_root_emit_op</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">power</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 2) </span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="constant">ROOT_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 3) </span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="constant">CUBEROOT_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 2) </span>&lt;<span class="cwebmacro">Emit a scaling correction for square roots</span> <span class="cwebmacronumber">38.1</span>&gt;
<span class="reserved">else</span><span class="plain"> </span><span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 3) </span>&lt;<span class="cwebmacro">Emit a scaling correction for cube roots</span> <span class="cwebmacronumber">38.2</span>&gt;
<span class="reserved">else</span><span class="plain"> </span><span class="identifier">internal_error</span><span class="plain">(</span><span class="string">"can only scale square and cube roots"</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="plain">#</span><span class="identifier">endif</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_root_emit_op appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP38_1"></a><b>&#167;38.1. </b>For square roots,
sqrt(s) = sqrt(k_Xv) = sqrt(k_X) sqrt(v) = k_O sqrt(v).
((( sqrt(k_X)) / (k_O))) = s_o .
((( sqrt(k_X)) / (k_O)))
and now the overestimate is a factor of k = sqrt(k_X)/k_O. However,
rather than calculating k sqrt(x) we calculate sqrt(k^2 x), since
this way accuracy losses in taking the square root are much reduced.
Therefore this scaling operating is to be performed inside the root
function, not outside, and it scales by k^2 not k:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Emit a scaling correction for square roots</span> <span class="cwebmacronumber">38.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">TIMES_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">DIVIDE_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP38">&#167;38</a>.</p>
<p class="inwebparagraph"><a id="SP38_2"></a><b>&#167;38.2. </b>For cube roots,
curt(s) = curt(k_Xv) = curt(k_X) curt(v) = k_O curt(v).
((( curt(k_X)) / (k_O))) = s_o.
((( curt(k_X)) / (k_O)))
and the overestimate is k = curt(k_X)/k_O. Scaling once again within
the rooting function, we scale by k^3:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Emit a scaling correction for cube roots</span> <span class="cwebmacronumber">38.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">TIMES_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::inv_primitive</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">DIVIDE_BIP</span><span class="plain">); </span><span class="identifier">Produce::down</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP38">&#167;38</a>.</p>
<p class="inwebparagraph"><a id="SP39"></a><b>&#167;39. </b></p>
<pre class="display">
<span class="plain">#</span><span class="identifier">ifdef</span><span class="plain"> </span><span class="identifier">CORE_MODULE</span>
<span class="reserved">void</span><span class="plain"> </span><span class="functiontext">Kinds::Dimensions::kind_rescale_root_emit_factor</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindx</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">power</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindx</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">kindo</span><span class="plain"> = </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 2) </span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="constant">ROOT_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 3) </span><span class="identifier">kindo</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">, </span><span class="identifier">NULL</span><span class="plain">, </span><span class="constant">CUBEROOT_OPERATION</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">kindo</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_X</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindx</span><span class="plain">);</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">k_O</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">kindo</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 2) </span>&lt;<span class="cwebmacro">Emit factor for a scaling correction for square roots</span> <span class="cwebmacronumber">39.1</span>&gt;
<span class="reserved">else</span><span class="plain"> </span><span class="reserved">if</span><span class="plain"> (</span><span class="identifier">power</span><span class="plain"> == 3) </span>&lt;<span class="cwebmacro">Emit factor for a scaling correction for cube roots</span> <span class="cwebmacronumber">39.2</span>&gt;
<span class="reserved">else</span><span class="plain"> </span><span class="identifier">internal_error</span><span class="plain">(</span><span class="string">"can only scale square and cube roots"</span><span class="plain">);</span>
<span class="plain">}</span>
<span class="plain">#</span><span class="identifier">endif</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::kind_rescale_root_emit_factor appears nowhere else.</p>
<p class="inwebparagraph"><a id="SP39_1"></a><b>&#167;39.1. </b>For square roots,
sqrt(s) = sqrt(k_Xv) = sqrt(k_X) sqrt(v) = k_O sqrt(v).
((( sqrt(k_X)) / (k_O))) = s_o .
((( sqrt(k_X)) / (k_O)))
and now the overestimate is a factor of k = sqrt(k_X)/k_O. However,
rather than calculating k sqrt(x) we calculate sqrt(k^2 x), since
this way accuracy losses in taking the square root are much reduced.
Therefore this scaling operating is to be performed inside the root
function, not outside, and it scales by k^2 not k:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Emit factor for a scaling correction for square roots</span> <span class="cwebmacronumber">39.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP39">&#167;39</a>.</p>
<p class="inwebparagraph"><a id="SP39_2"></a><b>&#167;39.2. </b>For cube roots,
curt(s) = curt(k_Xv) = curt(k_X) curt(v) = k_O curt(v).
((( curt(k_X)) / (k_O))) = s_o.
((( curt(k_X)) / (k_O)))
and the overestimate is k = curt(k_X)/k_O. Scaling once again within
the rooting function, we scale by k^3:
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Emit factor for a scaling correction for cube roots</span> <span class="cwebmacronumber">39.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &gt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_X</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain">*</span><span class="identifier">k_O</span><span class="plain"> &lt; </span><span class="identifier">k_X</span><span class="plain">) { </span><span class="identifier">Produce::val</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">(), </span><span class="identifier">K_number</span><span class="plain">, </span><span class="identifier">LITERAL_IVAL</span><span class="plain">, (</span><span class="identifier">inter_t</span><span class="plain">) (</span><span class="identifier">k_X</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">/</span><span class="identifier">k_O</span><span class="plain">)); </span><span class="identifier">Produce::up</span><span class="plain">(</span><span class="identifier">Emit::tree</span><span class="plain">()); }</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP39">&#167;39</a>.</p>
<p class="inwebparagraph"><a id="SP40"></a><b>&#167;40. Arithmetic on kinds. </b>We are finally able to provide our central routine, the one providing a service
for the rest of Inform. Given <code class="display"><span class="extract">K1</span></code> and <code class="display"><span class="extract">K2</span></code>, we return the kind resulting
from applying arithmetic operation <code class="display"><span class="extract">op</span></code>, or <code class="display"><span class="extract">NULL</span></code> if the operation cannot
meaningfully be applied. In the case where <code class="display"><span class="extract">op</span></code> is a unary operation, <code class="display"><span class="extract">K2</span></code>
has no significance and should be <code class="display"><span class="extract">NULL</span></code>.
</p>
<p class="inwebparagraph">The kinds here are called operands, because they are what will be operated on.
</p>
<pre class="display">
<span class="reserved">kind</span><span class="plain"> *</span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">K1</span><span class="plain">, </span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">K2</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">op</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">K1</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Dimensions::arithmetic_op_is_unary</span><span class="plain">(</span><span class="identifier">op</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) &amp;&amp; (</span><span class="identifier">K2</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">operand1</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">operand1</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">operand2</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">K2</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Dimensions::arithmetic_op_is_unary</span><span class="plain">(</span><span class="identifier">op</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) &amp;&amp; (</span><span class="identifier">operand2</span><span class="plain"> == </span><span class="identifier">NULL</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">unit_sequence</span><span class="plain"> </span><span class="identifier">result</span><span class="plain">;</span>
<span class="reserved">switch</span><span class="plain"> (</span><span class="identifier">op</span><span class="plain">) {</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">PLUS_OPERATION</span><span class="plain">:</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">MINUS_OPERATION</span><span class="plain">:</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">EQUALS_OPERATION</span><span class="plain">:</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">APPROXIMATION_OPERATION</span><span class="plain">:</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::compare_unit_sequences</span><span class="plain">(</span><span class="identifier">operand1</span><span class="plain">, </span><span class="identifier">operand2</span><span class="plain">)) {</span>
<span class="identifier">result</span><span class="plain"> = *</span><span class="identifier">operand1</span><span class="plain">;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">UNARY_MINUS_OPERATION</span><span class="plain">:</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">REMAINDER_OPERATION</span><span class="plain">:</span>
<span class="identifier">result</span><span class="plain"> = *</span><span class="identifier">operand1</span><span class="plain">;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">ROOT_OPERATION</span><span class="plain">:</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::root_unit_sequence</span><span class="plain">(</span><span class="identifier">operand1</span><span class="plain">, 2, &amp;</span><span class="identifier">result</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">)</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">REALROOT_OPERATION</span><span class="plain">:</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::root_unit_sequence</span><span class="plain">(</span><span class="identifier">operand1</span><span class="plain">, 2, &amp;</span><span class="identifier">result</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">)</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">CUBEROOT_OPERATION</span><span class="plain">:</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::root_unit_sequence</span><span class="plain">(</span><span class="identifier">operand1</span><span class="plain">, 3, &amp;</span><span class="identifier">result</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">)</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">TIMES_OPERATION</span><span class="plain">:</span>
<span class="functiontext">Kinds::Dimensions::multiply_unit_sequences</span><span class="plain">(</span><span class="identifier">operand1</span><span class="plain">, 1, </span><span class="identifier">operand2</span><span class="plain">, 1, &amp;</span><span class="identifier">result</span><span class="plain">);</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">case</span><span class="plain"> </span><span class="constant">DIVIDE_OPERATION</span><span class="plain">:</span>
<span class="functiontext">Kinds::Dimensions::multiply_unit_sequences</span><span class="plain">(</span><span class="identifier">operand1</span><span class="plain">, 1, </span><span class="identifier">operand2</span><span class="plain">, -1, &amp;</span><span class="identifier">result</span><span class="plain">);</span>
<span class="reserved">break</span><span class="plain">;</span>
<span class="reserved">default</span><span class="plain">: </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="plain">}</span>
&lt;<span class="cwebmacro">Handle calculations entirely between dimensionless units more delicately</span> <span class="cwebmacronumber">40.1</span>&gt;<span class="plain">;</span>
&lt;<span class="cwebmacro">Handle the case of a dimensionless result</span> <span class="cwebmacronumber">40.2</span>&gt;<span class="plain">;</span>
&lt;<span class="cwebmacro">Identify the result as a known kind, if possible</span> <span class="cwebmacronumber">40.3</span>&gt;<span class="plain">;</span>
&lt;<span class="cwebmacro">And otherwise create a kind as the intermediate result of a calculation</span> <span class="cwebmacronumber">40.4</span>&gt;<span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::arithmetic_on_kinds is used in <a href="#SP35">&#167;35</a>, <a href="#SP36">&#167;36</a>, <a href="#SP37">&#167;37</a>, <a href="#SP38">&#167;38</a>, <a href="#SP39">&#167;39</a>, <a href="#SP41">&#167;41</a>.</p>
<p class="inwebparagraph"><a id="SP40_1"></a><b>&#167;40.1. </b>If <code class="display"><span class="extract">result</span></code> is the empty unit sequence, we'll identify it as a number,
because number is the lowest type ID representing a dimensionless unit.
Usually that's good: for instance, it says that a frequency times a time
is a number, and not some more exotic dimensionless quantity like an angle.
</p>
<p class="inwebparagraph">But it's not so good when the calculation is not really physical at all, but
purely mathematical, and all we are doing is working on dimensionless units.
For instance, if take an angle θ and double it to 2θ, we don't
want Inform to say the result is number &mdash; we want 2θ to be
another angle. So we make an exception.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Handle calculations entirely between dimensionless units more delicately</span> <span class="cwebmacronumber">40.1</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::arithmetic_op_is_unary</span><span class="plain">(</span><span class="identifier">op</span><span class="plain">)) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">op</span><span class="plain"> == </span><span class="constant">REALROOT_OPERATION</span><span class="plain">) &amp;&amp; (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">)))</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">K_real_number</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">K1</span><span class="plain">;</span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">)) &amp;&amp;</span>
<span class="plain">(</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">K2</span><span class="plain">))) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">K2</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">K1</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">, </span><span class="identifier">K_number</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">K2</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Compare::eq</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">, </span><span class="identifier">K2</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">K1</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP40">&#167;40</a>.</p>
<p class="inwebparagraph"><a id="SP40_2"></a><b>&#167;40.2. </b>It's also possible to get a dimensionless result by, for example, dividing
a mass by another mass, and we need to be careful to keep track of whether
we're using real or integer arithmetic: 1500.0m divided by 10.0m must be
150.0, not 150.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Handle the case of a dimensionless result</span> <span class="cwebmacronumber">40.2</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::us_dimensionless</span><span class="plain">(&amp;</span><span class="identifier">result</span><span class="plain">)) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::arithmetic_op_is_unary</span><span class="plain">(</span><span class="identifier">op</span><span class="plain">)) {</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::FloatingPoint::uses_floating_point</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">K_real_number</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">K_number</span><span class="plain">;</span>
<span class="plain">} </span><span class="reserved">else</span><span class="plain"> {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="functiontext">Kinds::FloatingPoint::uses_floating_point</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">)) ||</span>
<span class="plain">(</span><span class="functiontext">Kinds::FloatingPoint::uses_floating_point</span><span class="plain">(</span><span class="identifier">K2</span><span class="plain">))) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">K_real_number</span><span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">K_number</span><span class="plain">;</span>
<span class="plain">}</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP40">&#167;40</a>.</p>
<p class="inwebparagraph"><a id="SP40_3"></a><b>&#167;40.3. </b>If we've produced the right combination of fundamental units to make one of the
named units, then we return that as an atomic kind. For instance, maybe we
divided a velocity by a time, and now we find that we have m.
s^{-2}, which turns out to have a name: acceleration.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">Identify the result as a known kind, if possible</span> <span class="cwebmacronumber">40.3</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">R</span><span class="plain">;</span>
<span class="identifier">LOOP_OVER_BASE_KINDS</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::compare_unit_sequences</span><span class="plain">(&amp;</span><span class="identifier">result</span><span class="plain">,</span>
<span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">R</span><span class="plain">)))</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">R</span><span class="plain">;</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP40">&#167;40</a>, <a href="#SP41">&#167;41</a>.</p>
<p class="inwebparagraph"><a id="SP40_4"></a><b>&#167;40.4. </b>Otherwise the <code class="display"><span class="extract">result</span></code> is a unit sequence which doesn't have a name, so
we store it as an intermediate kind, representing a temporary value living
only for the duration of a calculation.
</p>
<p class="inwebparagraph">A last little wrinkle is: how we should scale this? For results like an
acceleration, something defined in the source text, we know how accurate the
author wants us to be. But these intermediate kinds are not defined, and we
don't know for sure what the author would want. It seems wise to set
k &gt;= k_X and k&gt;= k_Y, so that we have at least as much detail as
the calculation would have had within each operand kind. So perhaps we should
put k = max(k_X, k_Y). But in fact we will choose k = <code class="display"><span class="extract">Kinds::Dimensions::lcm(k_X, k_Y)</span></code>,
the least common multiple, so that any subsequent divisions will cancel
correctly and we won't lose too much information through integer rounding.
(In practice this will probably either be the same as max(k_X, k_Y)
or will multiply by 6, since <code class="display"><span class="extract">Kinds::Dimensions::lcm(60, 1000) == 6000</span></code> and so on.)
</p>
<p class="inwebparagraph">The same unit sequence can have different scalings each time it appears as
an intermediate calculation. We could get to m^2. kg
either as m.kg times m, or as m^2 times
kg, or many other ways, and we'll get different scalings depending
on the route. This is why the <code class="display"><span class="extract">unit_sequence</span></code> structure has a
<code class="display"><span class="extract">scaling_factor</span></code> field; the choice of scale factor does not depend on
the physics but on the arithmetic method being used.
</p>
<p class="macrodefinition"><code class="display">
&lt;<span class="cwebmacrodefn">And otherwise create a kind as the intermediate result of a calculation</span> <span class="cwebmacronumber">40.4</span>&gt; =
</code></p>
<pre class="displaydefn">
<span class="identifier">result</span><span class="element">.scaling_factor</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::lcm</span><span class="plain">(</span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">K1</span><span class="plain">), </span><span class="functiontext">Kinds::Behaviour::scale_factor</span><span class="plain">(</span><span class="identifier">K2</span><span class="plain">));</span>
<span class="reserved">return</span><span class="plain"> </span><span class="functiontext">Kinds::intermediate_construction</span><span class="plain">(&amp;</span><span class="identifier">result</span><span class="plain">);</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">This code is used in <a href="#SP40">&#167;40</a>.</p>
<p class="inwebparagraph"><a id="SP41"></a><b>&#167;41. </b></p>
<pre class="display">
<span class="reserved">kind</span><span class="plain"> *</span><span class="functiontext">Kinds::Dimensions::to_rational_power</span><span class="plain">(</span><span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">F</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">n</span><span class="plain">, </span><span class="reserved">int</span><span class="plain"> </span><span class="identifier">m</span><span class="plain">) {</span>
<span class="reserved">if</span><span class="plain"> ((</span><span class="identifier">n</span><span class="plain"> &lt; 1) || (</span><span class="identifier">m</span><span class="plain"> &lt; 1)) </span><span class="identifier">internal_error</span><span class="plain">(</span><span class="string">"bad rational power"</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::dimensionless</span><span class="plain">(</span><span class="identifier">F</span><span class="plain">)) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">F</span><span class="plain">;</span>
<span class="reserved">kind</span><span class="plain"> *</span><span class="identifier">K</span><span class="plain"> = </span><span class="identifier">K_number</span><span class="plain">;</span>
<span class="reserved">int</span><span class="plain"> </span><span class="identifier">op</span><span class="plain"> = </span><span class="constant">TIMES_OPERATION</span><span class="plain">;</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">n</span><span class="plain"> &lt; 0) { </span><span class="identifier">n</span><span class="plain"> = -</span><span class="identifier">n</span><span class="plain">; </span><span class="identifier">op</span><span class="plain"> = </span><span class="constant">DIVIDE_OPERATION</span><span class="plain">; }</span>
<span class="reserved">while</span><span class="plain"> (</span><span class="identifier">n</span><span class="plain"> &gt; 0) {</span>
<span class="identifier">K</span><span class="plain"> = </span><span class="functiontext">Kinds::Dimensions::arithmetic_on_kinds</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">, </span><span class="identifier">F</span><span class="plain">, </span><span class="identifier">op</span><span class="plain">);</span>
<span class="identifier">n</span><span class="plain">--;</span>
<span class="plain">}</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="identifier">m</span><span class="plain"> == 1) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">K</span><span class="plain">;</span>
<span class="reserved">unit_sequence</span><span class="plain"> </span><span class="identifier">result</span><span class="plain">;</span>
<span class="reserved">unit_sequence</span><span class="plain"> *</span><span class="identifier">operand</span><span class="plain"> = </span><span class="functiontext">Kinds::Behaviour::get_dimensional_form</span><span class="plain">(</span><span class="identifier">K</span><span class="plain">);</span>
<span class="reserved">if</span><span class="plain"> (</span><span class="functiontext">Kinds::Dimensions::root_unit_sequence</span><span class="plain">(</span><span class="identifier">operand</span><span class="plain">, </span><span class="identifier">m</span><span class="plain">, &amp;</span><span class="identifier">result</span><span class="plain">) == </span><span class="identifier">FALSE</span><span class="plain">) </span><span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
&lt;<span class="cwebmacro">Identify the result as a known kind, if possible</span> <span class="cwebmacronumber">40.3</span>&gt;<span class="plain">;</span>
<span class="reserved">return</span><span class="plain"> </span><span class="identifier">NULL</span><span class="plain">;</span>
<span class="plain">}</span>
</pre>
<p class="inwebparagraph"></p>
<p class="endnote">The function Kinds::Dimensions::to_rational_power appears nowhere else.</p>
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