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254 lines
12 KiB
HTML
4 years ago
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual describes how to install and use the GNU multiple precision
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arithmetic library, version 6.1.0.
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Copyright 1991, 1993-2015 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this document under
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the terms of the GNU Free Documentation License, Version 1.3 or any later
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version published by the Free Software Foundation; with no Invariant Sections,
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with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover
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Texts being "You have freedom to copy and modify this GNU Manual, like GNU
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software". A copy of the license is included in
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GNU Free Documentation License. -->
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<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
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<head>
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<title>Efficiency (GNU MP 6.1.0)</title>
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<meta name="description" content="How to install and use the GNU multiple precision arithmetic library, version 6.1.0.">
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<meta name="keywords" content="Efficiency (GNU MP 6.1.0)">
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<meta name="resource-type" content="document">
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<meta name="distribution" content="global">
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<meta name="Generator" content="makeinfo">
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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
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<link href="index.html#Top" rel="start" title="Top">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="GMP-Basics.html#GMP-Basics" rel="up" title="GMP Basics">
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<link href="Debugging.html#Debugging" rel="next" title="Debugging">
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<link href="Demonstration-Programs.html#Demonstration-Programs" rel="prev" title="Demonstration Programs">
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<style type="text/css">
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ul.no-bullet {list-style: none}
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-->
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</style>
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</head>
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<body lang="en">
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<a name="Efficiency"></a>
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<div class="header">
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<p>
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Next: <a href="Debugging.html#Debugging" accesskey="n" rel="next">Debugging</a>, Previous: <a href="Demonstration-Programs.html#Demonstration-Programs" accesskey="p" rel="prev">Demonstration Programs</a>, Up: <a href="GMP-Basics.html#GMP-Basics" accesskey="u" rel="up">GMP Basics</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<a name="Efficiency-1"></a>
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<h3 class="section">3.11 Efficiency</h3>
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<a name="index-Efficiency"></a>
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<dl compact="compact">
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<dt>Small Operands</dt>
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<dd><a name="index-Small-operands"></a>
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<p>On small operands, the time for function call overheads and memory allocation
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can be significant in comparison to actual calculation. This is unavoidable
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in a general purpose variable precision library, although GMP attempts to be
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as efficient as it can on both large and small operands.
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</p>
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</dd>
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<dt>Static Linking</dt>
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<dd><a name="index-Static-linking"></a>
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<p>On some CPUs, in particular the x86s, the static <samp>libgmp.a</samp> should be
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used for maximum speed, since the PIC code in the shared <samp>libgmp.so</samp> will
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have a small overhead on each function call and global data address. For many
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programs this will be insignificant, but for long calculations there’s a gain
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to be had.
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</p>
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</dd>
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<dt>Initializing and Clearing</dt>
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<dd><a name="index-Initializing-and-clearing"></a>
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<p>Avoid excessive initializing and clearing of variables, since this can be
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quite time consuming, especially in comparison to otherwise fast operations
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like addition.
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</p>
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<p>A language interpreter might want to keep a free list or stack of
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initialized variables ready for use. It should be possible to integrate
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something like that with a garbage collector too.
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</p>
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</dd>
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<dt>Reallocations</dt>
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<dd><a name="index-Reallocations"></a>
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<p>An <code>mpz_t</code> or <code>mpq_t</code> variable used to hold successively increasing
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values will have its memory repeatedly <code>realloc</code>ed, which could be quite
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slow or could fragment memory, depending on the C library. If an application
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can estimate the final size then <code>mpz_init2</code> or <code>mpz_realloc2</code> can
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be called to allocate the necessary space from the beginning
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(see <a href="Initializing-Integers.html#Initializing-Integers">Initializing Integers</a>).
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</p>
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<p>It doesn’t matter if a size set with <code>mpz_init2</code> or <code>mpz_realloc2</code>
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is too small, since all functions will do a further reallocation if necessary.
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Badly overestimating memory required will waste space though.
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</p>
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</dd>
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<dt><code>2exp</code> Functions</dt>
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<dd><a name="index-2exp-functions"></a>
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<p>It’s up to an application to call functions like <code>mpz_mul_2exp</code> when
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appropriate. General purpose functions like <code>mpz_mul</code> make no attempt to
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identify powers of two or other special forms, because such inputs will
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usually be very rare and testing every time would be wasteful.
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</p>
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</dd>
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<dt><code>ui</code> and <code>si</code> Functions</dt>
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<dd><a name="index-ui-and-si-functions"></a>
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<p>The <code>ui</code> functions and the small number of <code>si</code> functions exist for
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convenience and should be used where applicable. But if for example an
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<code>mpz_t</code> contains a value that fits in an <code>unsigned long</code> there’s no
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need extract it and call a <code>ui</code> function, just use the regular <code>mpz</code>
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function.
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</p>
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</dd>
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<dt>In-Place Operations</dt>
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<dd><a name="index-In_002dplace-operations"></a>
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<p><code>mpz_abs</code>, <code>mpq_abs</code>, <code>mpf_abs</code>, <code>mpz_neg</code>, <code>mpq_neg</code>
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and <code>mpf_neg</code> are fast when used for in-place operations like
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<code>mpz_abs(x,x)</code>, since in the current implementation only a single field
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of <code>x</code> needs changing. On suitable compilers (GCC for instance) this is
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inlined too.
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</p>
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<p><code>mpz_add_ui</code>, <code>mpz_sub_ui</code>, <code>mpf_add_ui</code> and <code>mpf_sub_ui</code>
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benefit from an in-place operation like <code>mpz_add_ui(x,x,y)</code>, since
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usually only one or two limbs of <code>x</code> will need to be changed. The same
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applies to the full precision <code>mpz_add</code> etc if <code>y</code> is small. If
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<code>y</code> is big then cache locality may be helped, but that’s all.
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</p>
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<p><code>mpz_mul</code> is currently the opposite, a separate destination is slightly
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better. A call like <code>mpz_mul(x,x,y)</code> will, unless <code>y</code> is only one
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limb, make a temporary copy of <code>x</code> before forming the result. Normally
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that copying will only be a tiny fraction of the time for the multiply, so
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this is not a particularly important consideration.
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</p>
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<p><code>mpz_set</code>, <code>mpq_set</code>, <code>mpq_set_num</code>, <code>mpf_set</code>, etc, make
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no attempt to recognise a copy of something to itself, so a call like
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<code>mpz_set(x,x)</code> will be wasteful. Naturally that would never be written
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deliberately, but if it might arise from two pointers to the same object then
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a test to avoid it might be desirable.
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</p>
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<div class="example">
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<pre class="example">if (x != y)
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mpz_set (x, y);
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</pre></div>
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<p>Note that it’s never worth introducing extra <code>mpz_set</code> calls just to get
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in-place operations. If a result should go to a particular variable then just
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direct it there and let GMP take care of data movement.
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</p>
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</dd>
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<dt>Divisibility Testing (Small Integers)</dt>
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<dd><a name="index-Divisibility-testing"></a>
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<p><code>mpz_divisible_ui_p</code> and <code>mpz_congruent_ui_p</code> are the best functions
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for testing whether an <code>mpz_t</code> is divisible by an individual small
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integer. They use an algorithm which is faster than <code>mpz_tdiv_ui</code>, but
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which gives no useful information about the actual remainder, only whether
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it’s zero (or a particular value).
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</p>
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<p>However when testing divisibility by several small integers, it’s best to take
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a remainder modulo their product, to save multi-precision operations. For
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instance to test whether a number is divisible by any of 23, 29 or 31 take a
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remainder modulo <em>23*29*31 = 20677</em> and then test that.
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</p>
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<p>The division functions like <code>mpz_tdiv_q_ui</code> which give a quotient as well
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as a remainder are generally a little slower than the remainder-only functions
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like <code>mpz_tdiv_ui</code>. If the quotient is only rarely wanted then it’s
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probably best to just take a remainder and then go back and calculate the
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quotient if and when it’s wanted (<code>mpz_divexact_ui</code> can be used if the
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remainder is zero).
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</p>
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</dd>
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<dt>Rational Arithmetic</dt>
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<dd><a name="index-Rational-arithmetic"></a>
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<p>The <code>mpq</code> functions operate on <code>mpq_t</code> values with no common factors
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in the numerator and denominator. Common factors are checked-for and cast out
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as necessary. In general, cancelling factors every time is the best approach
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since it minimizes the sizes for subsequent operations.
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</p>
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<p>However, applications that know something about the factorization of the
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values they’re working with might be able to avoid some of the GCDs used for
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canonicalization, or swap them for divisions. For example when multiplying by
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a prime it’s enough to check for factors of it in the denominator instead of
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doing a full GCD. Or when forming a big product it might be known that very
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little cancellation will be possible, and so canonicalization can be left to
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the end.
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</p>
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<p>The <code>mpq_numref</code> and <code>mpq_denref</code> macros give access to the
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numerator and denominator to do things outside the scope of the supplied
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<code>mpq</code> functions. See <a href="Applying-Integer-Functions.html#Applying-Integer-Functions">Applying Integer Functions</a>.
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</p>
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<p>The canonical form for rationals allows mixed-type <code>mpq_t</code> and integer
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additions or subtractions to be done directly with multiples of the
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denominator. This will be somewhat faster than <code>mpq_add</code>. For example,
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</p>
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<div class="example">
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<pre class="example">/* mpq increment */
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mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
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/* mpq += unsigned long */
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mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
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/* mpq -= mpz */
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mpz_submul (mpq_numref(q), mpq_denref(q), z);
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</pre></div>
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</dd>
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<dt>Number Sequences</dt>
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<dd><a name="index-Number-sequences"></a>
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<p>Functions like <code>mpz_fac_ui</code>, <code>mpz_fib_ui</code> and <code>mpz_bin_uiui</code>
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are designed for calculating isolated values. If a range of values is wanted
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it’s probably best to call to get a starting point and iterate from there.
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</p>
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</dd>
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<dt>Text Input/Output</dt>
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<dd><a name="index-Text-input_002foutput"></a>
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<p>Hexadecimal or octal are suggested for input or output in text form.
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Power-of-2 bases like these can be converted much more efficiently than other
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bases, like decimal. For big numbers there’s usually nothing of particular
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interest to be seen in the digits, so the base doesn’t matter much.
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</p>
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<p>Maybe we can hope octal will one day become the normal base for everyday use,
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as proposed by King Charles XII of Sweden and later reformers.
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</p></dd>
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</dl>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Debugging.html#Debugging" accesskey="n" rel="next">Debugging</a>, Previous: <a href="Demonstration-Programs.html#Demonstration-Programs" accesskey="p" rel="prev">Demonstration Programs</a>, Up: <a href="GMP-Basics.html#GMP-Basics" accesskey="u" rel="up">GMP Basics</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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