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143 lines
7.6 KiB
HTML
143 lines
7.6 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>Binary to Radix (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="Binary to Radix (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="Radix-to-Binary.html#Radix-to-Binary" rel="next" title="Radix to Binary">
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<link href="Radix-Conversion-Algorithms.html#Radix-Conversion-Algorithms" rel="prev" title="Radix Conversion Algorithms">
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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="Binary-to-Radix"></a>
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<div class="header">
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<p>
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Next: <a href="Radix-to-Binary.html#Radix-to-Binary" accesskey="n" rel="next">Radix to Binary</a>, Previous: <a href="Radix-Conversion-Algorithms.html#Radix-Conversion-Algorithms" accesskey="p" rel="prev">Radix Conversion Algorithms</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="Binary-to-Radix-1"></a>
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<h4 class="subsection">15.6.1 Binary to Radix</h4>
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<p>Conversions from binary to a power-of-2 radix use a simple and fast
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<em>O(N)</em> bit extraction algorithm.
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</p>
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<p>Conversions from binary to other radices use one of two algorithms. Sizes
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below <code>GET_STR_PRECOMPUTE_THRESHOLD</code> use a basic <em>O(N^2)</em> method.
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Repeated divisions by <em>b^n</em> are made, where <em>b</em> is the radix and
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<em>n</em> is the biggest power that fits in a limb. But instead of simply
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using the remainder <em>r</em> from such divisions, an extra divide step is done
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to give a fractional limb representing <em>r/b^n</em>. The digits of <em>r</em>
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can then be extracted using multiplications by <em>b</em> rather than divisions.
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Special case code is provided for decimal, allowing multiplications by 10 to
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optimize to shifts and adds.
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</p>
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<p>Above <code>GET_STR_PRECOMPUTE_THRESHOLD</code> a sub-quadratic algorithm is used.
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For an input <em>t</em>, powers <em>b^(n*2^i)</em> of the radix are
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calculated, until a power between <em>t</em> and <em>sqrt(t)</em> is
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reached. <em>t</em> is then divided by that largest power, giving a quotient
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which is the digits above that power, and a remainder which is those below.
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These two parts are in turn divided by the second highest power, and so on
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recursively. When a piece has been divided down to less than
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<code>GET_STR_DC_THRESHOLD</code> limbs, the basecase algorithm described above is
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used.
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</p>
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<p>The advantage of this algorithm is that big divisions can make use of the
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sub-quadratic divide and conquer division (see <a href="Divide-and-Conquer-Division.html#Divide-and-Conquer-Division">Divide and Conquer Division</a>), and big divisions tend to have less overheads than lots of
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separate single limb divisions anyway. But in any case the cost of
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calculating the powers <em>b^(n*2^i)</em> must first be overcome.
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</p>
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<p><code>GET_STR_PRECOMPUTE_THRESHOLD</code> and <code>GET_STR_DC_THRESHOLD</code> represent
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the same basic thing, the point where it becomes worth doing a big division to
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cut the input in half. <code>GET_STR_PRECOMPUTE_THRESHOLD</code> includes the cost
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of calculating the radix power required, whereas <code>GET_STR_DC_THRESHOLD</code>
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assumes that’s already available, which is the case when recursing.
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</p>
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<p>Since the base case produces digits from least to most significant but they
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want to be stored from most to least, it’s necessary to calculate in advance
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how many digits there will be, or at least be sure not to underestimate that.
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For GMP the number of input bits is multiplied by <code>chars_per_bit_exactly</code>
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from <code>mp_bases</code>, rounding up. The result is either correct or one too
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big.
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</p>
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<p>Examining some of the high bits of the input could increase the chance of
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getting the exact number of digits, but an exact result every time would not
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be practical, since in general the difference between numbers 100… and
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99… is only in the last few bits and the work to identify 99…
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might well be almost as much as a full conversion.
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</p>
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<p>The <em>r/b^n</em> scheme described above for using multiplications to bring out
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digits might be useful for more than a single limb. Some brief experiments
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with it on the base case when recursing didn’t give a noticeable improvement,
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but perhaps that was only due to the implementation. Something similar would
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work for the sub-quadratic divisions too, though there would be the cost of
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calculating a bigger radix power.
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</p>
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<p>Another possible improvement for the sub-quadratic part would be to arrange
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for radix powers that balanced the sizes of quotient and remainder produced,
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i.e. the highest power would be an <em>b^(n*k)</em> approximately equal to
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<em>sqrt(t)</em>, not restricted to a <em>2^i</em> factor. That ought to
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smooth out a graph of times against sizes, but may or may not be a net
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speedup.
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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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Next: <a href="Radix-to-Binary.html#Radix-to-Binary" accesskey="n" rel="next">Radix to Binary</a>, Previous: <a href="Radix-Conversion-Algorithms.html#Radix-Conversion-Algorithms" accesskey="p" rel="prev">Radix Conversion Algorithms</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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