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119 lines
5.7 KiB
HTML
119 lines
5.7 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>Radix to Binary (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="Radix to Binary (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="Radix-Conversion-Algorithms.html#Radix-Conversion-Algorithms" rel="up" title="Radix Conversion Algorithms">
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<link href="Other-Algorithms.html#Other-Algorithms" rel="next" title="Other Algorithms">
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<link href="Binary-to-Radix.html#Binary-to-Radix" rel="prev" title="Binary to Radix">
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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="Radix-to-Binary"></a>
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<div class="header">
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<p>
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Previous: <a href="Binary-to-Radix.html#Binary-to-Radix" accesskey="p" rel="prev">Binary to Radix</a>, Up: <a href="Radix-Conversion-Algorithms.html#Radix-Conversion-Algorithms" accesskey="u" rel="up">Radix Conversion Algorithms</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="Radix-to-Binary-1"></a>
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<h4 class="subsection">15.6.2 Radix to Binary</h4>
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<p><strong>This section needs to be rewritten, it currently describes the
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algorithms used before GMP 4.3.</strong>
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</p>
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<p>Conversions from a power-of-2 radix into binary use a simple and fast
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<em>O(N)</em> bitwise concatenation algorithm.
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</p>
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<p>Conversions from other radices use one of two algorithms. Sizes below
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<code>SET_STR_PRECOMPUTE_THRESHOLD</code> use a basic <em>O(N^2)</em> method. Groups
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of <em>n</em> digits are converted to limbs, where <em>n</em> is the biggest
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power of the base <em>b</em> which will fit in a limb, then those groups are
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accumulated into the result by multiplying by <em>b^n</em> and adding. This
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saves multi-precision operations, as per Knuth section 4.4 part E
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(see <a href="References.html#References">References</a>). Some special case code is provided for decimal, giving
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the compiler a chance to optimize multiplications by 10.
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</p>
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<p>Above <code>SET_STR_PRECOMPUTE_THRESHOLD</code> a sub-quadratic algorithm is used.
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First groups of <em>n</em> digits are converted into limbs. Then adjacent
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limbs are combined into limb pairs with <em>x*b^n+y</em>, where <em>x</em>
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and <em>y</em> are the limbs. Adjacent limb pairs are combined into quads
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similarly with <em>x*b^(2n)+y</em>. This continues until a single block
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remains, that being the result.
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</p>
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<p>The advantage of this method is that the multiplications for each <em>x</em> are
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big blocks, allowing Karatsuba and higher algorithms to be used. But the cost
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of calculating the powers <em>b^(n*2^i)</em> must be overcome.
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<code>SET_STR_PRECOMPUTE_THRESHOLD</code> usually ends up quite big, around 5000 digits, and on
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some processors much bigger still.
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</p>
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<p><code>SET_STR_PRECOMPUTE_THRESHOLD</code> is based on the input digits (and tuned
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for decimal), though it might be better based on a limb count, so as to be
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independent of the base. But that sort of count isn’t used by the base case
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and so would need some sort of initial calculation or estimate.
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</p>
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<p>The main reason <code>SET_STR_PRECOMPUTE_THRESHOLD</code> is so much bigger than the
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corresponding <code>GET_STR_PRECOMPUTE_THRESHOLD</code> is that <code>mpn_mul_1</code> is
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much faster than <code>mpn_divrem_1</code> (often by a factor of 5, or more).
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</p>
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<hr>
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<div class="header">
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<p>
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Previous: <a href="Binary-to-Radix.html#Binary-to-Radix" accesskey="p" rel="prev">Binary to Radix</a>, Up: <a href="Radix-Conversion-Algorithms.html#Radix-Conversion-Algorithms" accesskey="u" rel="up">Radix Conversion Algorithms</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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