| 1 | /* $NetBSD: muldi3.c,v 1.10 2005/12/11 12:24:37 christos Exp $ */
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| 2 |
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| 3 | /*-
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| 4 | * Copyright (c) 1992, 1993
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| 5 | * The Regents of the University of California. All rights reserved.
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| 6 | *
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| 7 | * This software was developed by the Computer Systems Engineering group
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| 8 | * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
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| 9 | * contributed to Berkeley.
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| 10 | *
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| 11 | * Redistribution and use in source and binary forms, with or without
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| 12 | * modification, are permitted provided that the following conditions
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| 13 | * are met:
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| 14 | * 1. Redistributions of source code must retain the above copyright
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| 15 | * notice, this list of conditions and the following disclaimer.
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| 16 | * 2. Redistributions in binary form must reproduce the above copyright
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| 17 | * notice, this list of conditions and the following disclaimer in the
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| 18 | * documentation and/or other materials provided with the distribution.
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| 19 | * 3. Neither the name of the University nor the names of its contributors
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| 20 | * may be used to endorse or promote products derived from this software
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| 21 | * without specific prior written permission.
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| 22 | *
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| 23 | * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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| 24 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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| 25 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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| 26 | * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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| 27 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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| 28 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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| 29 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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| 30 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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| 31 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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| 32 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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| 33 | * SUCH DAMAGE.
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| 34 | */
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| 35 |
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| 36 | /*#include <sys/cdefs.h>
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| 37 | #if defined(LIBC_SCCS) && !defined(lint)
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| 38 | #if 0
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| 39 | static char sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93";
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| 40 | #else
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| 41 | __RCSID("$NetBSD: muldi3.c,v 1.10 2005/12/11 12:24:37 christos Exp $");
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| 42 | #endif
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| 43 | #endif*/ /* LIBC_SCCS and not lint */
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| 44 |
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| 45 | #include "quad.h"
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| 46 |
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| 47 | /*
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| 48 | * Multiply two quads.
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| 49 | *
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| 50 | * Our algorithm is based on the following. Split incoming quad values
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| 51 | * u and v (where u,v >= 0) into
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| 52 | *
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| 53 | * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32)
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| 54 | *
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| 55 | * and
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| 56 | *
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| 57 | * v = 2^n v1 * v0
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| 58 | *
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| 59 | * Then
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| 60 | *
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| 61 | * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
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| 62 | * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
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| 63 | *
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| 64 | * Now add 2^n u1 v1 to the first term and subtract it from the middle,
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| 65 | * and add 2^n u0 v0 to the last term and subtract it from the middle.
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| 66 | * This gives:
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| 67 | *
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| 68 | * uv = (2^2n + 2^n) (u1 v1) +
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| 69 | * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
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| 70 | * (2^n + 1) (u0 v0)
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| 71 | *
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| 72 | * Factoring the middle a bit gives us:
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| 73 | *
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| 74 | * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
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| 75 | * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
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| 76 | * (2^n + 1) (u0 v0) [u0v0 = low]
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| 77 | *
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| 78 | * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
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| 79 | * in just half the precision of the original. (Note that either or both
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| 80 | * of (u1 - u0) or (v0 - v1) may be negative.)
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| 81 | *
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| 82 | * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
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| 83 | *
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| 84 | * Since C does not give us a `int * int = quad' operator, we split
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| 85 | * our input quads into two ints, then split the two ints into two
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| 86 | * shorts. We can then calculate `short * short = int' in native
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| 87 | * arithmetic.
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| 88 | *
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| 89 | * Our product should, strictly speaking, be a `long quad', with 128
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| 90 | * bits, but we are going to discard the upper 64. In other words,
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| 91 | * we are not interested in uv, but rather in (uv mod 2^2n). This
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| 92 | * makes some of the terms above vanish, and we get:
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| 93 | *
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| 94 | * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
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| 95 | *
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| 96 | * or
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| 97 | *
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| 98 | * (2^n)(high + mid + low) + low
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| 99 | *
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| 100 | * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
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| 101 | * of 2^n in either one will also vanish. Only `low' need be computed
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| 102 | * mod 2^2n, and only because of the final term above.
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| 103 | */
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| 104 | static quad_t __lmulq(u_int, u_int);
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| 105 |
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| 106 | quad_t
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| 107 | __muldi3(a, b)
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| 108 | quad_t a, b;
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| 109 | {
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| 110 | union uu u, v, low, prod;
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| 111 | u_int high, mid, udiff, vdiff;
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| 112 | int negall, negmid;
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| 113 | #define u1 u.ul[H]
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| 114 | #define u0 u.ul[L]
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| 115 | #define v1 v.ul[H]
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| 116 | #define v0 v.ul[L]
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| 117 |
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| 118 | /*
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| 119 | * Get u and v such that u, v >= 0. When this is finished,
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| 120 | * u1, u0, v1, and v0 will be directly accessible through the
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| 121 | * int fields.
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| 122 | */
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| 123 | if (a >= 0)
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| 124 | u.q = a, negall = 0;
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| 125 | else
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| 126 | u.q = -a, negall = 1;
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| 127 | if (b >= 0)
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| 128 | v.q = b;
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| 129 | else
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| 130 | v.q = -b, negall ^= 1;
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| 131 |
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| 132 | if (u1 == 0 && v1 == 0) {
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| 133 | /*
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| 134 | * An (I hope) important optimization occurs when u1 and v1
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| 135 | * are both 0. This should be common since most numbers
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| 136 | * are small. Here the product is just u0*v0.
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| 137 | */
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| 138 | prod.q = __lmulq(u0, v0);
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| 139 | } else {
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| 140 | /*
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| 141 | * Compute the three intermediate products, remembering
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| 142 | * whether the middle term is negative. We can discard
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| 143 | * any upper bits in high and mid, so we can use native
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| 144 | * u_int * u_int => u_int arithmetic.
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| 145 | */
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| 146 | low.q = __lmulq(u0, v0);
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| 147 |
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| 148 | if (u1 >= u0)
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| 149 | negmid = 0, udiff = u1 - u0;
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| 150 | else
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| 151 | negmid = 1, udiff = u0 - u1;
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| 152 | if (v0 >= v1)
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| 153 | vdiff = v0 - v1;
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| 154 | else
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| 155 | vdiff = v1 - v0, negmid ^= 1;
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| 156 | mid = udiff * vdiff;
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| 157 |
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| 158 | high = u1 * v1;
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| 159 |
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| 160 | /*
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| 161 | * Assemble the final product.
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| 162 | */
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| 163 | prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
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| 164 | low.ul[H];
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| 165 | prod.ul[L] = low.ul[L];
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| 166 | }
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| 167 | return (negall ? -prod.q : prod.q);
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| 168 | #undef u1
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| 169 | #undef u0
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| 170 | #undef v1
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| 171 | #undef v0
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| 172 | }
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| 173 |
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| 174 | /*
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| 175 | * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
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| 176 | * the number of bits in an int (whatever that is---the code below
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| 177 | * does not care as long as quad.h does its part of the bargain---but
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| 178 | * typically N==16).
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| 179 | *
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| 180 | * We use the same algorithm from Knuth, but this time the modulo refinement
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| 181 | * does not apply. On the other hand, since N is half the size of an int,
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| 182 | * we can get away with native multiplication---none of our input terms
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| 183 | * exceeds (UINT_MAX >> 1).
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| 184 | *
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| 185 | * Note that, for u_int l, the quad-precision result
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| 186 | *
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| 187 | * l << N
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| 188 | *
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| 189 | * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
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| 190 | */
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| 191 | static quad_t
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| 192 | __lmulq(u_int u, u_int v)
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| 193 | {
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| 194 | u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
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| 195 | u_int prodh, prodl, was;
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| 196 | union uu prod;
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| 197 | int neg;
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| 198 |
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| 199 | u1 = HHALF(u);
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| 200 | u0 = LHALF(u);
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| 201 | v1 = HHALF(v);
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| 202 | v0 = LHALF(v);
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| 203 |
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| 204 | low = u0 * v0;
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| 205 |
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| 206 | /* This is the same small-number optimization as before. */
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| 207 | if (u1 == 0 && v1 == 0)
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| 208 | return (low);
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| 209 |
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| 210 | if (u1 >= u0)
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| 211 | udiff = u1 - u0, neg = 0;
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| 212 | else
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| 213 | udiff = u0 - u1, neg = 1;
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| 214 | if (v0 >= v1)
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| 215 | vdiff = v0 - v1;
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| 216 | else
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| 217 | vdiff = v1 - v0, neg ^= 1;
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| 218 | mid = udiff * vdiff;
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| 219 |
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| 220 | high = u1 * v1;
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| 221 |
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| 222 | /* prod = (high << 2N) + (high << N); */
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| 223 | prodh = high + HHALF(high);
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| 224 | prodl = LHUP(high);
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| 225 |
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| 226 | /* if (neg) prod -= mid << N; else prod += mid << N; */
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| 227 | if (neg) {
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| 228 | was = prodl;
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| 229 | prodl -= LHUP(mid);
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| 230 | prodh -= HHALF(mid) + (prodl > was);
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| 231 | } else {
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| 232 | was = prodl;
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| 233 | prodl += LHUP(mid);
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| 234 | prodh += HHALF(mid) + (prodl < was);
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| 235 | }
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| 236 |
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| 237 | /* prod += low << N */
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| 238 | was = prodl;
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| 239 | prodl += LHUP(low);
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| 240 | prodh += HHALF(low) + (prodl < was);
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| 241 | /* ... + low; */
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| 242 | if ((prodl += low) < low)
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| 243 | prodh++;
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| 244 |
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| 245 | /* return 4N-bit product */
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| 246 | prod.ul[H] = prodh;
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| 247 | prod.ul[L] = prodl;
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| 248 | return (prod.q);
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| 249 | }
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