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<h1>Module <a href="type_Q.html">Q</a></h1>
<pre><span class="keyword">module</span> Q: <code class="code">sig</code> <a href="Q.html">..</a> <code class="code">end</code></pre><div class="info module top">
Rationals.
<p>
This modules builds arbitrary precision rationals on top of arbitrary
integers from module Z.
<p>
This file is part of the Zarith library
http://forge.ocamlcore.org/projects/zarith .
It is distributed under LGPL 2 licensing, with static linking exception.
See the LICENSE file included in the distribution.
<p>
Copyright (c) 2010-2011 Antoine Miné, Abstraction project.
Abstraction is part of the LIENS (Laboratoire d'Informatique de l'ENS),
a joint laboratory by:
CNRS (Centre national de la recherche scientifique, France),
ENS (École normale supérieure, Paris, France),
INRIA Rocquencourt (Institut national de recherche en informatique, France).<br>
</div>
<hr width="100%">
<br>
<h1 id="1_Types">Types</h1><br>
<pre><code><span id="TYPEt"><span class="keyword">type</span> <code class="type"></code>t</span> = {</code></pre><table class="typetable">
<tr>
<td align="left" valign="top" >
<code> </code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTt.num">num</span> : <code class="type"><a href="Z.html#TYPEt">Z.t</a></code>;</code></td>
<td class="typefieldcomment" align="left" valign="top" ><code>(*</code></td><td class="typefieldcomment" align="left" valign="top" >Numerator.</td><td class="typefieldcomment" align="left" valign="bottom" ><code>*)</code></td>
</tr>
<tr>
<td align="left" valign="top" >
<code> </code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTt.den">den</span> : <code class="type"><a href="Z.html#TYPEt">Z.t</a></code>;</code></td>
<td class="typefieldcomment" align="left" valign="top" ><code>(*</code></td><td class="typefieldcomment" align="left" valign="top" >Denominator, >= 0</td><td class="typefieldcomment" align="left" valign="bottom" ><code>*)</code></td>
</tr></table>
}
<div class="info ">
A rational is represented as a pair numerator/denominator, reduced to
have a non-negative denominator and no common factor.
This form is canonical (enabling polymorphic equality and hashing).
The representation allows three special numbers: <code class="code">inf</code> (1/0), <code class="code">-inf</code> (-1/0)
and <code class="code">undef</code> (0/0).<br>
</div>
<br>
<h1 id="1_Construction">Construction</h1><br>
<pre><span id="VALmake"><span class="keyword">val</span> make</span> : <code class="type"><a href="Z.html#TYPEt">Z.t</a> -> <a href="Z.html#TYPEt">Z.t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
<code class="code">make num den</code> constructs a new rational equal to <code class="code">num</code>/<code class="code">den</code>.
It takes care of putting the rational in canonical form.<br>
</div>
<pre><span id="VALzero"><span class="keyword">val</span> zero</span> : <code class="type"><a href="Q.html#TYPEt">t</a></code></pre>
<pre><span id="VALone"><span class="keyword">val</span> one</span> : <code class="type"><a href="Q.html#TYPEt">t</a></code></pre>
<pre><span id="VALminus_one"><span class="keyword">val</span> minus_one</span> : <code class="type"><a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
0, 1, -1.<br>
</div>
<pre><span id="VALinf"><span class="keyword">val</span> inf</span> : <code class="type"><a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
1/0.<br>
</div>
<pre><span id="VALminus_inf"><span class="keyword">val</span> minus_inf</span> : <code class="type"><a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
-1/0.<br>
</div>
<pre><span id="VALundef"><span class="keyword">val</span> undef</span> : <code class="type"><a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
0/0.<br>
</div>
<pre><span id="VALof_bigint"><span class="keyword">val</span> of_bigint</span> : <code class="type"><a href="Z.html#TYPEt">Z.t</a> -> <a href="Q.html#TYPEt">t</a></code></pre>
<pre><span id="VALof_int"><span class="keyword">val</span> of_int</span> : <code class="type">int -> <a href="Q.html#TYPEt">t</a></code></pre>
<pre><span id="VALof_int32"><span class="keyword">val</span> of_int32</span> : <code class="type">int32 -> <a href="Q.html#TYPEt">t</a></code></pre>
<pre><span id="VALof_int64"><span class="keyword">val</span> of_int64</span> : <code class="type">int64 -> <a href="Q.html#TYPEt">t</a></code></pre>
<pre><span id="VALof_nativeint"><span class="keyword">val</span> of_nativeint</span> : <code class="type">nativeint -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Conversions from various integer types.<br>
</div>
<pre><span id="VALof_ints"><span class="keyword">val</span> of_ints</span> : <code class="type">int -> int -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Conversion from an <code class="code">int</code> numerator and an <code class="code">int</code> denominator.<br>
</div>
<pre><span id="VALof_float"><span class="keyword">val</span> of_float</span> : <code class="type">float -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Conversion from a <code class="code">float</code>.
The conversion is exact, and maps NaN to <code class="code">undef</code>.<br>
</div>
<pre><span id="VALof_string"><span class="keyword">val</span> of_string</span> : <code class="type">string -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Converts a string to a rational.
Plain decimals, and <code class="code">/</code> separated decimal ratios (with optional sign) are
understood.
Additionally, the special <code class="code">inf</code>, <code class="code">-inf</code>, and <code class="code">undef</code> are recognized
(they can also be typeset respectively as <code class="code">1/0</code>, <code class="code">-1/0</code>, <code class="code">0/0</code>).<br>
</div>
<br>
<h1 id="1_Inspection">Inspection</h1><br>
<pre><span id="VALnum"><span class="keyword">val</span> num</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Z.html#TYPEt">Z.t</a></code></pre><div class="info ">
Get the numerator.<br>
</div>
<pre><span id="VALden"><span class="keyword">val</span> den</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Z.html#TYPEt">Z.t</a></code></pre><div class="info ">
Get the denominator.<br>
</div>
<br>
<h1 id="1_Testing">Testing</h1><br>
<pre><code><span id="TYPEkind"><span class="keyword">type</span> <code class="type"></code>kind</span> = </code></pre><table class="typetable">
<tr>
<td align="left" valign="top" >
<code><span class="keyword">|</span></code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTkind.ZERO"><span class="constructor">ZERO</span></span></code></td>
<td class="typefieldcomment" align="left" valign="top" ><code>(*</code></td><td class="typefieldcomment" align="left" valign="top" >0</td><td class="typefieldcomment" align="left" valign="bottom" ><code>*)</code></td>
</tr>
<tr>
<td align="left" valign="top" >
<code><span class="keyword">|</span></code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTkind.INF"><span class="constructor">INF</span></span></code></td>
<td class="typefieldcomment" align="left" valign="top" ><code>(*</code></td><td class="typefieldcomment" align="left" valign="top" >infinity, i.e. 1/0</td><td class="typefieldcomment" align="left" valign="bottom" ><code>*)</code></td>
</tr>
<tr>
<td align="left" valign="top" >
<code><span class="keyword">|</span></code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTkind.MINF"><span class="constructor">MINF</span></span></code></td>
<td class="typefieldcomment" align="left" valign="top" ><code>(*</code></td><td class="typefieldcomment" align="left" valign="top" >minus infinity, i.e. -1/0</td><td class="typefieldcomment" align="left" valign="bottom" ><code>*)</code></td>
</tr>
<tr>
<td align="left" valign="top" >
<code><span class="keyword">|</span></code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTkind.UNDEF"><span class="constructor">UNDEF</span></span></code></td>
<td class="typefieldcomment" align="left" valign="top" ><code>(*</code></td><td class="typefieldcomment" align="left" valign="top" >undefined, i.e., 0/0</td><td class="typefieldcomment" align="left" valign="bottom" ><code>*)</code></td>
</tr>
<tr>
<td align="left" valign="top" >
<code><span class="keyword">|</span></code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTkind.NZERO"><span class="constructor">NZERO</span></span></code></td>
<td class="typefieldcomment" align="left" valign="top" ><code>(*</code></td><td class="typefieldcomment" align="left" valign="top" >well-defined, non-infinity, non-zero number</td><td class="typefieldcomment" align="left" valign="bottom" ><code>*)</code></td>
</tr></table>
<div class="info ">
Rationals can be categorized into different kinds, depending mainly on
whether the numerator and/or denominator is null.<br>
</div>
<pre><span id="VALclassify"><span class="keyword">val</span> classify</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEkind">kind</a></code></pre><div class="info ">
Determines the kind of a rational.<br>
</div>
<pre><span id="VALis_real"><span class="keyword">val</span> is_real</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> bool</code></pre><div class="info ">
Whether the argument is non-infinity and non-undefined.<br>
</div>
<pre><span id="VALsign"><span class="keyword">val</span> sign</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int</code></pre><div class="info ">
Returns 1 if the argument is positive (including inf), -1 if it is
negative (including -inf), and 0 if it is null or undefined.<br>
</div>
<pre><span id="VALcompare"><span class="keyword">val</span> compare</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> int</code></pre><div class="info ">
<code class="code">compare x y</code> compares <code class="code">x</code> to <code class="code">y</code> and returns 1 if <code class="code">x</code> is strictly
greater that <code class="code">y</code>, -1 if it is strictly smaller, and 0 if they are
equal.
This is a total ordering.
Infinities are ordered in the natural way, while undefined is considered
the smallest of all: undef = undef < -inf <= -inf < x < inf <= inf.
This is consistent with OCaml's handling of floating-point infinities
and NaN.
<p>
OCaml's polymorphic comparison will NOT return a result consistent with
the ordering of rationals.<br>
</div>
<pre><span id="VALequal"><span class="keyword">val</span> equal</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> bool</code></pre><div class="info ">
Equality testing.
This is consistent with <code class="code">compare</code>; in particular, <code class="code">undef</code>=<code class="code">undef</code>.<br>
</div>
<pre><span id="VALmin"><span class="keyword">val</span> min</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Returns the smallest of its arguments.<br>
</div>
<pre><span id="VALmax"><span class="keyword">val</span> max</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Returns the largest of its arguments.<br>
</div>
<pre><span id="VALleq"><span class="keyword">val</span> leq</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> bool</code></pre><div class="info ">
Less than or equal.<br>
</div>
<pre><span id="VALgeq"><span class="keyword">val</span> geq</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> bool</code></pre><div class="info ">
Greater than or equal.<br>
</div>
<pre><span id="VALlt"><span class="keyword">val</span> lt</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> bool</code></pre><div class="info ">
Less than (not equal).<br>
</div>
<pre><span id="VALgt"><span class="keyword">val</span> gt</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> bool</code></pre><div class="info ">
Greater than (not equal).<br>
</div>
<br>
<h1 id="1_Conversions">Conversions</h1><br>
<pre><span id="VALto_bigint"><span class="keyword">val</span> to_bigint</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Z.html#TYPEt">Z.t</a></code></pre>
<pre><span id="VALto_int"><span class="keyword">val</span> to_int</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int</code></pre>
<pre><span id="VALto_int32"><span class="keyword">val</span> to_int32</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int32</code></pre>
<pre><span id="VALto_int64"><span class="keyword">val</span> to_int64</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int64</code></pre>
<pre><span id="VALto_nativeint"><span class="keyword">val</span> to_nativeint</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> nativeint</code></pre><div class="info ">
Convert to integer by truncation.
Raises a <code class="code">Divide_by_zero</code> if the argument is an infinity or undefined.
Raises a <code class="code">Z.Overflow</code> if the result does not fit in the destination
type.<br>
</div>
<pre><span id="VALto_string"><span class="keyword">val</span> to_string</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> string</code></pre><div class="info ">
Converts to human-readable, decimal, <code class="code">/</code>-separated rational.<br>
</div>
<br>
<h1 id="1_Arithmeticoperations">Arithmetic operations</h1><br>
<br>
In all operations, the result is <code class="code">undef</code> if one argument is <code class="code">undef</code>.
Other operations can return <code class="code">undef</code>: such as <code class="code">inf</code>-<code class="code">inf</code>, <code class="code">inf</code>*0, 0/0.<br>
<pre><span id="VALneg"><span class="keyword">val</span> neg</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Negation.<br>
</div>
<pre><span id="VALabs"><span class="keyword">val</span> abs</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Absolute value.<br>
</div>
<pre><span id="VALadd"><span class="keyword">val</span> add</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Addition.<br>
</div>
<pre><span id="VALsub"><span class="keyword">val</span> sub</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Subtraction. We have <code class="code">sub x y</code> = <code class="code">add x (neg y)</code>.<br>
</div>
<pre><span id="VALmul"><span class="keyword">val</span> mul</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Multiplication.<br>
</div>
<pre><span id="VALinv"><span class="keyword">val</span> inv</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Inverse.
Note that <code class="code">inv 0</code> is defined, and equals <code class="code">inf</code>.<br>
</div>
<pre><span id="VALdiv"><span class="keyword">val</span> div</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Division.
We have <code class="code">div x y</code> = <code class="code">mul x (inv y)</code>, and <code class="code">inv x</code> = <code class="code">div one x</code>.<br>
</div>
<pre><span id="VALmul_2exp"><span class="keyword">val</span> mul_2exp</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
<code class="code">mul_2exp x n</code> multiplies <code class="code">x</code> by 2 to the power of <code class="code">n</code>.<br>
</div>
<pre><span id="VALdiv_2exp"><span class="keyword">val</span> div_2exp</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
<code class="code">div_2exp x n</code> divides <code class="code">x</code> by 2 to the power of <code class="code">n</code>.<br>
</div>
<br>
<h1 id="1_Printing">Printing</h1><br>
<pre><span id="VALprint"><span class="keyword">val</span> print</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> unit</code></pre><div class="info ">
Prints the argument on the standard output.<br>
</div>
<pre><span id="VALoutput"><span class="keyword">val</span> output</span> : <code class="type">Pervasives.out_channel -> <a href="Q.html#TYPEt">t</a> -> unit</code></pre><div class="info ">
Prints the argument on the specified channel.
Also intended to be used as <code class="code">%a</code> format printer in <code class="code">Printf.printf</code>.<br>
</div>
<pre><span id="VALsprint"><span class="keyword">val</span> sprint</span> : <code class="type">unit -> <a href="Q.html#TYPEt">t</a> -> string</code></pre><div class="info ">
To be used as <code class="code">%a</code> format printer in <code class="code">Printf.sprintf</code>.<br>
</div>
<pre><span id="VALbprint"><span class="keyword">val</span> bprint</span> : <code class="type">Buffer.t -> <a href="Q.html#TYPEt">t</a> -> unit</code></pre><div class="info ">
To be used as <code class="code">%a</code> format printer in <code class="code">Printf.bprintf</code>.<br>
</div>
<pre><span id="VALpp_print"><span class="keyword">val</span> pp_print</span> : <code class="type">Format.formatter -> <a href="Q.html#TYPEt">t</a> -> unit</code></pre><div class="info ">
Prints the argument on the specified formatter.
Also intended to be used as <code class="code">%a</code> format printer in <code class="code">Format.printf</code>.<br>
</div>
<br>
<h1 id="1_Prefixandinfixoperators">Prefix and infix operators</h1><br>
<br>
Classic prefix and infix <code class="code">int</code> operators are redefined on <code class="code">t</code>.<br>
<pre><span id="VAL(~-)"><span class="keyword">val</span> (~-)</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Negation <code class="code">neg</code>.<br>
</div>
<pre><span id="VAL(~+)"><span class="keyword">val</span> (~+)</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Identity.<br>
</div>
<pre><span id="VAL(+)"><span class="keyword">val</span> (+)</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Addition <code class="code">add</code>.<br>
</div>
<pre><span id="VAL(-)"><span class="keyword">val</span> (-)</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Subtraction <code class="code">sub</code>.<br>
</div>
<pre><span id="VAL( * )"><span class="keyword">val</span> ( * )</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Multiplication <code class="code">mul</code>.<br>
</div>
<pre><span id="VAL(/)"><span class="keyword">val</span> (/)</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Division <code class="code">div</code>.<br>
</div>
<pre><span id="VAL(lsl)"><span class="keyword">val</span> (lsl)</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Multiplication by a power of two <code class="code">mul_2exp</code>.<br>
</div>
<pre><span id="VAL(asr)"><span class="keyword">val</span> (asr)</span> : <code class="type"><a href="Q.html#TYPEt">t</a> -> int -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Division by a power of two <code class="code">shift_right</code>.<br>
</div>
<pre><span id="VAL(~$)"><span class="keyword">val</span> (~$)</span> : <code class="type">int -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Conversion from <code class="code">int</code>.<br>
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<pre><span id="VAL(//)"><span class="keyword">val</span> (//)</span> : <code class="type">int -> int -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Creates a rational from two <code class="code">int</code>s.<br>
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<pre><span id="VAL(~$$)"><span class="keyword">val</span> (~$$)</span> : <code class="type"><a href="Z.html#TYPEt">Z.t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Conversion from <code class="code">Z.t</code>.<br>
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<pre><span id="VAL(///)"><span class="keyword">val</span> (///)</span> : <code class="type"><a href="Z.html#TYPEt">Z.t</a> -> <a href="Z.html#TYPEt">Z.t</a> -> <a href="Q.html#TYPEt">t</a></code></pre><div class="info ">
Creates a rational from two <code class="code">Z.t</code>.<br>
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