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178 lines
7.5 KiB
HTML
178 lines
7.5 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>Assembly Floating Point (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="Assembly Floating Point (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 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="Assembly-Coding.html#Assembly-Coding" rel="up" title="Assembly Coding">
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<link href="Assembly-SIMD-Instructions.html#Assembly-SIMD-Instructions" rel="next" title="Assembly SIMD Instructions">
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<link href="Assembly-Functional-Units.html#Assembly-Functional-Units" rel="prev" title="Assembly Functional Units">
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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="Assembly-Floating-Point"></a>
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<div class="header">
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<p>
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Next: <a href="Assembly-SIMD-Instructions.html#Assembly-SIMD-Instructions" accesskey="n" rel="next">Assembly SIMD Instructions</a>, Previous: <a href="Assembly-Functional-Units.html#Assembly-Functional-Units" accesskey="p" rel="prev">Assembly Functional Units</a>, Up: <a href="Assembly-Coding.html#Assembly-Coding" accesskey="u" rel="up">Assembly Coding</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="Floating-Point"></a>
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<h4 class="subsection">15.8.6 Floating Point</h4>
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<a name="index-Assembly-floating-Point"></a>
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<p>Floating point arithmetic is used in GMP for multiplications on CPUs with poor
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integer multipliers. It’s mostly useful for <code>mpn_mul_1</code>,
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<code>mpn_addmul_1</code> and <code>mpn_submul_1</code> on 64-bit machines, and
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<code>mpn_mul_basecase</code> on both 32-bit and 64-bit machines.
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</p>
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<p>With IEEE 53-bit double precision floats, integer multiplications producing up
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to 53 bits will give exact results. Breaking a 64x64 multiplication
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into eight 16x<em>32->48</em> bit pieces is convenient. With
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some care though six 21x<em>32->53</em> bit products can be
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used, if one of the lower two 21-bit pieces also uses the sign bit.
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</p>
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<p>For the <code>mpn_mul_1</code> family of functions on a 64-bit machine, the
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invariant single limb is split at the start, into 3 or 4 pieces. Inside the
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loop, the bignum operand is split into 32-bit pieces. Fast conversion of
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these unsigned 32-bit pieces to floating point is highly machine-dependent.
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In some cases, reading the data into the integer unit, zero-extending to
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64-bits, then transferring to the floating point unit back via memory is the
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only option.
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</p>
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<p>Converting partial products back to 64-bit limbs is usually best done as a
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signed conversion. Since all values are smaller than <em>2^53</em>, signed
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and unsigned are the same, but most processors lack unsigned conversions.
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</p>
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<br>
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<br>
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<p>Here is a diagram showing 16x32 bit products for an <code>mpn_mul_1</code> or
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<code>mpn_addmul_1</code> with a 64-bit limb. The single limb operand V is split
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into four 16-bit parts. The multi-limb operand U is split in the loop into
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two 32-bit parts.
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</p>
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<div class="example">
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<pre class="example"> +---+---+---+---+
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|v48|v32|v16|v00| V operand
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+---+---+---+---+
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+-------+---+---+
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x | u32 | u00 | U operand (one limb)
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+---------------+
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---------------------------------
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+-----------+
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| u00 x v00 | p00 48-bit products
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+-----------+
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+-----------+
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| u00 x v16 | p16
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+-----------+
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+-----------+
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| u00 x v32 | p32
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+-----------+
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+-----------+
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| u00 x v48 | p48
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+-----------+
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+-----------+
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| u32 x v00 | r32
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+-----------+
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+-----------+
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| u32 x v16 | r48
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+-----------+
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+-----------+
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| u32 x v32 | r64
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+-----------+
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+-----------+
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| u32 x v48 | r80
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+-----------+
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</pre></div>
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<p><em>p32</em> and <em>r32</em> can be summed using floating-point addition, and
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likewise <em>p48</em> and <em>r48</em>. <em>p00</em> and <em>p16</em> can be summed
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with <em>r64</em> and <em>r80</em> from the previous iteration.
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</p>
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<p>For each loop then, four 49-bit quantities are transferred to the integer unit,
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aligned as follows,
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</p>
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<div class="example">
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<pre class="example">|-----64bits----|-----64bits----|
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+------------+
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| p00 + r64' | i00
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+------------+
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+------------+
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| p16 + r80' | i16
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+------------+
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+------------+
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| p32 + r32 | i32
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+------------+
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+------------+
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| p48 + r48 | i48
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+------------+
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</pre></div>
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<p>The challenge then is to sum these efficiently and add in a carry limb,
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generating a low 64-bit result limb and a high 33-bit carry limb (<em>i48</em>
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extends 33 bits into the high half).
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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="Assembly-SIMD-Instructions.html#Assembly-SIMD-Instructions" accesskey="n" rel="next">Assembly SIMD Instructions</a>, Previous: <a href="Assembly-Functional-Units.html#Assembly-Functional-Units" accesskey="p" rel="prev">Assembly Functional Units</a>, Up: <a href="Assembly-Coding.html#Assembly-Coding" accesskey="u" rel="up">Assembly Coding</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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</html>
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