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7.0 KiB
HTML
<!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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<title>Exact Division (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="Exact Division (GNU MP 6.1.0)">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="Division-Algorithms.html#Division-Algorithms" rel="up" title="Division Algorithms">
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<link href="Exact-Remainder.html#Exact-Remainder" rel="next" title="Exact Remainder">
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<link href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" rel="prev" title="Block-Wise Barrett Division">
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</head>
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<body lang="en">
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<a name="Exact-Division"></a>
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<div class="header">
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<p>
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Next: <a href="Exact-Remainder.html#Exact-Remainder" accesskey="n" rel="next">Exact Remainder</a>, Previous: <a href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" accesskey="p" rel="prev">Block-Wise Barrett Division</a>, Up: <a href="Division-Algorithms.html#Division-Algorithms" accesskey="u" rel="up">Division 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="Exact-Division-1"></a>
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<h4 class="subsection">15.2.5 Exact Division</h4>
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<p>A so-called exact division is when the dividend is known to be an exact
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multiple of the divisor. Jebelean’s exact division algorithm uses this
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knowledge to make some significant optimizations (see <a href="References.html#References">References</a>).
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</p>
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<p>The idea can be illustrated in decimal for example with 368154 divided by
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543. Because the low digit of the dividend is 4, the low digit of the
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quotient must be 8. This is arrived at from <em>4*7 mod 10</em>, using the fact 7 is the modular inverse of 3 (the low digit of
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the divisor), since <em>3*7
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≡ 1 mod 10</em>. So <em>8*543=4344</em> can be
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subtracted from the dividend leaving 363810. Notice the low digit has become
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zero.
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</p>
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<p>The procedure is repeated at the second digit, with the next quotient digit 7
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(<em>7 ≡ 1*7 mod 10</em>), subtracting
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<em>7*543=3801</em>, leaving 325800. And finally at
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the third digit with quotient digit 6 (<em>8*7
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mod 10</em>), subtracting <em>6*543=3258</em> leaving 0.
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So the quotient is 678.
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</p>
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<p>Notice however that the multiplies and subtractions don’t need to extend past
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the low three digits of the dividend, since that’s enough to determine the
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three quotient digits. For the last quotient digit no subtraction is needed
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at all. On a 2NxN division like this one, only about half the work of
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a normal basecase division is necessary.
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</p>
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<p>For an NxM exact division producing Q=N-M quotient limbs, the
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saving over a normal basecase division is in two parts. Firstly, each of the
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Q quotient limbs needs only one multiply, not a 2x1 divide and
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multiply. Secondly, the crossproducts are reduced when <em>Q>M</em> to
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<em>Q*M-M*(M+1)/2</em>, or when <em>Q<=M</em> to <em>Q*(Q-1)/2</em>. Notice the savings are complementary. If Q is big then many
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divisions are saved, or if Q is small then the crossproducts reduce to a small
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number.
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</p>
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<p>The modular inverse used is calculated efficiently by <code>binvert_limb</code> in
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<samp>gmp-impl.h</samp>. This does four multiplies for a 32-bit limb, or six for a
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64-bit limb. <samp>tune/modlinv.c</samp> has some alternate implementations that
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might suit processors better at bit twiddling than multiplying.
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</p>
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<p>The sub-quadratic exact division described by Jebelean in “Exact Division
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with Karatsuba Complexity” is not currently implemented. It uses a
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rearrangement similar to the divide and conquer for normal division
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(see <a href="Divide-and-Conquer-Division.html#Divide-and-Conquer-Division">Divide and Conquer Division</a>), but operating from low to high. A
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further possibility not currently implemented is “Bidirectional Exact Integer
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Division” by Krandick and Jebelean which forms quotient limbs from both the
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high and low ends of the dividend, and can halve once more the number of
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crossproducts needed in a 2NxN division.
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</p>
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<p>A special case exact division by 3 exists in <code>mpn_divexact_by3</code>,
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supporting Toom-3 multiplication and <code>mpq</code> canonicalizations. It forms
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quotient digits with a multiply by the modular inverse of 3 (which is
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<code>0xAA..AAB</code>) and uses two comparisons to determine a borrow for the next
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limb. The multiplications don’t need to be on the dependent chain, as long as
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the effect of the borrows is applied, which can help chips with pipelined
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multipliers.
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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="Exact-Remainder.html#Exact-Remainder" accesskey="n" rel="next">Exact Remainder</a>, Previous: <a href="Block_002dWise-Barrett-Division.html#Block_002dWise-Barrett-Division" accesskey="p" rel="prev">Block-Wise Barrett Division</a>, Up: <a href="Division-Algorithms.html#Division-Algorithms" accesskey="u" rel="up">Division 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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