Annotation of sys/lib/libkern/muldi3.c, Revision 1.1
1.1 ! nbrk 1: /*-
! 2: * Copyright (c) 1992, 1993
! 3: * The Regents of the University of California. All rights reserved.
! 4: *
! 5: * This software was developed by the Computer Systems Engineering group
! 6: * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
! 7: * contributed to Berkeley.
! 8: *
! 9: * Redistribution and use in source and binary forms, with or without
! 10: * modification, are permitted provided that the following conditions
! 11: * are met:
! 12: * 1. Redistributions of source code must retain the above copyright
! 13: * notice, this list of conditions and the following disclaimer.
! 14: * 2. Redistributions in binary form must reproduce the above copyright
! 15: * notice, this list of conditions and the following disclaimer in the
! 16: * documentation and/or other materials provided with the distribution.
! 17: * 3. Neither the name of the University nor the names of its contributors
! 18: * may be used to endorse or promote products derived from this software
! 19: * without specific prior written permission.
! 20: *
! 21: * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
! 22: * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
! 23: * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
! 24: * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
! 25: * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
! 26: * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
! 27: * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
! 28: * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
! 29: * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
! 30: * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
! 31: * SUCH DAMAGE.
! 32: */
! 33:
! 34: #if defined(LIBC_SCCS) && !defined(lint)
! 35: static char rcsid[] = "$OpenBSD: muldi3.c,v 1.7 2004/11/28 07:23:41 mickey Exp $";
! 36: #endif /* LIBC_SCCS and not lint */
! 37:
! 38: #include "quad.h"
! 39:
! 40: /*
! 41: * Multiply two quads.
! 42: *
! 43: * Our algorithm is based on the following. Split incoming quad values
! 44: * u and v (where u,v >= 0) into
! 45: *
! 46: * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32)
! 47: *
! 48: * and
! 49: *
! 50: * v = 2^n v1 * v0
! 51: *
! 52: * Then
! 53: *
! 54: * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
! 55: * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
! 56: *
! 57: * Now add 2^n u1 v1 to the first term and subtract it from the middle,
! 58: * and add 2^n u0 v0 to the last term and subtract it from the middle.
! 59: * This gives:
! 60: *
! 61: * uv = (2^2n + 2^n) (u1 v1) +
! 62: * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
! 63: * (2^n + 1) (u0 v0)
! 64: *
! 65: * Factoring the middle a bit gives us:
! 66: *
! 67: * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
! 68: * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
! 69: * (2^n + 1) (u0 v0) [u0v0 = low]
! 70: *
! 71: * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
! 72: * in just half the precision of the original. (Note that either or both
! 73: * of (u1 - u0) or (v0 - v1) may be negative.)
! 74: *
! 75: * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
! 76: *
! 77: * Since C does not give us a `int * int = quad' operator, we split
! 78: * our input quads into two ints, then split the two ints into two
! 79: * shorts. We can then calculate `short * short = int' in native
! 80: * arithmetic.
! 81: *
! 82: * Our product should, strictly speaking, be a `long quad', with 128
! 83: * bits, but we are going to discard the upper 64. In other words,
! 84: * we are not interested in uv, but rather in (uv mod 2^2n). This
! 85: * makes some of the terms above vanish, and we get:
! 86: *
! 87: * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
! 88: *
! 89: * or
! 90: *
! 91: * (2^n)(high + mid + low) + low
! 92: *
! 93: * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
! 94: * of 2^n in either one will also vanish. Only `low' need be computed
! 95: * mod 2^2n, and only because of the final term above.
! 96: */
! 97: static quad_t __lmulq(u_int, u_int);
! 98:
! 99: quad_t
! 100: __muldi3(a, b)
! 101: quad_t a, b;
! 102: {
! 103: union uu u, v, low, prod;
! 104: u_int high, mid, udiff, vdiff;
! 105: int negall, negmid;
! 106: #define u1 u.ul[H]
! 107: #define u0 u.ul[L]
! 108: #define v1 v.ul[H]
! 109: #define v0 v.ul[L]
! 110:
! 111: /*
! 112: * Get u and v such that u, v >= 0. When this is finished,
! 113: * u1, u0, v1, and v0 will be directly accessible through the
! 114: * int fields.
! 115: */
! 116: if (a >= 0)
! 117: u.q = a, negall = 0;
! 118: else
! 119: u.q = -a, negall = 1;
! 120: if (b >= 0)
! 121: v.q = b;
! 122: else
! 123: v.q = -b, negall ^= 1;
! 124:
! 125: if (u1 == 0 && v1 == 0) {
! 126: /*
! 127: * An (I hope) important optimization occurs when u1 and v1
! 128: * are both 0. This should be common since most numbers
! 129: * are small. Here the product is just u0*v0.
! 130: */
! 131: prod.q = __lmulq(u0, v0);
! 132: } else {
! 133: /*
! 134: * Compute the three intermediate products, remembering
! 135: * whether the middle term is negative. We can discard
! 136: * any upper bits in high and mid, so we can use native
! 137: * u_int * u_int => u_int arithmetic.
! 138: */
! 139: low.q = __lmulq(u0, v0);
! 140:
! 141: if (u1 >= u0)
! 142: negmid = 0, udiff = u1 - u0;
! 143: else
! 144: negmid = 1, udiff = u0 - u1;
! 145: if (v0 >= v1)
! 146: vdiff = v0 - v1;
! 147: else
! 148: vdiff = v1 - v0, negmid ^= 1;
! 149: mid = udiff * vdiff;
! 150:
! 151: high = u1 * v1;
! 152:
! 153: /*
! 154: * Assemble the final product.
! 155: */
! 156: prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
! 157: low.ul[H];
! 158: prod.ul[L] = low.ul[L];
! 159: }
! 160: return (negall ? -prod.q : prod.q);
! 161: #undef u1
! 162: #undef u0
! 163: #undef v1
! 164: #undef v0
! 165: }
! 166:
! 167: /*
! 168: * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half
! 169: * the number of bits in an int (whatever that is---the code below
! 170: * does not care as long as quad.h does its part of the bargain---but
! 171: * typically N==16).
! 172: *
! 173: * We use the same algorithm from Knuth, but this time the modulo refinement
! 174: * does not apply. On the other hand, since N is half the size of an int,
! 175: * we can get away with native multiplication---none of our input terms
! 176: * exceeds (UINT_MAX >> 1).
! 177: *
! 178: * Note that, for u_int l, the quad-precision result
! 179: *
! 180: * l << N
! 181: *
! 182: * splits into high and low ints as HHALF(l) and LHUP(l) respectively.
! 183: */
! 184: static quad_t
! 185: __lmulq(u_int u, u_int v)
! 186: {
! 187: u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low;
! 188: u_int prodh, prodl, was;
! 189: union uu prod;
! 190: int neg;
! 191:
! 192: u1 = HHALF(u);
! 193: u0 = LHALF(u);
! 194: v1 = HHALF(v);
! 195: v0 = LHALF(v);
! 196:
! 197: low = u0 * v0;
! 198:
! 199: /* This is the same small-number optimization as before. */
! 200: if (u1 == 0 && v1 == 0)
! 201: return (low);
! 202:
! 203: if (u1 >= u0)
! 204: udiff = u1 - u0, neg = 0;
! 205: else
! 206: udiff = u0 - u1, neg = 1;
! 207: if (v0 >= v1)
! 208: vdiff = v0 - v1;
! 209: else
! 210: vdiff = v1 - v0, neg ^= 1;
! 211: mid = udiff * vdiff;
! 212:
! 213: high = u1 * v1;
! 214:
! 215: /* prod = (high << 2N) + (high << N); */
! 216: prodh = high + HHALF(high);
! 217: prodl = LHUP(high);
! 218:
! 219: /* if (neg) prod -= mid << N; else prod += mid << N; */
! 220: if (neg) {
! 221: was = prodl;
! 222: prodl -= LHUP(mid);
! 223: prodh -= HHALF(mid) + (prodl > was);
! 224: } else {
! 225: was = prodl;
! 226: prodl += LHUP(mid);
! 227: prodh += HHALF(mid) + (prodl < was);
! 228: }
! 229:
! 230: /* prod += low << N */
! 231: was = prodl;
! 232: prodl += LHUP(low);
! 233: prodh += HHALF(low) + (prodl < was);
! 234: /* ... + low; */
! 235: if ((prodl += low) < low)
! 236: prodh++;
! 237:
! 238: /* return 4N-bit product */
! 239: prod.ul[H] = prodh;
! 240: prod.ul[L] = prodl;
! 241: return (prod.q);
! 242: }
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