1129210Scognet/* $NetBSD: muldi3.c,v 1.8 2003/08/07 16:32:09 agc Exp $ */ 2129210Scognet 3129210Scognet/*- 4129210Scognet * Copyright (c) 1992, 1993 5129210Scognet * The Regents of the University of California. All rights reserved. 6129210Scognet * 7129210Scognet * This software was developed by the Computer Systems Engineering group 8129210Scognet * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and 9129210Scognet * contributed to Berkeley. 10129210Scognet * 11129210Scognet * Redistribution and use in source and binary forms, with or without 12129210Scognet * modification, are permitted provided that the following conditions 13129210Scognet * are met: 14129210Scognet * 1. Redistributions of source code must retain the above copyright 15129210Scognet * notice, this list of conditions and the following disclaimer. 16129210Scognet * 2. Redistributions in binary form must reproduce the above copyright 17129210Scognet * notice, this list of conditions and the following disclaimer in the 18129210Scognet * documentation and/or other materials provided with the distribution. 19129210Scognet * 3. Neither the name of the University nor the names of its contributors 20129210Scognet * may be used to endorse or promote products derived from this software 21129210Scognet * without specific prior written permission. 22129210Scognet * 23129210Scognet * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 24129210Scognet * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 25129210Scognet * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 26129210Scognet * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 27129210Scognet * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 28129210Scognet * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 29129210Scognet * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 30129210Scognet * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 31129210Scognet * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 32129210Scognet * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 33129210Scognet * SUCH DAMAGE. 34129210Scognet */ 35129210Scognet 36129210Scognet#include <sys/cdefs.h> 37129210Scognet#if defined(LIBC_SCCS) && !defined(lint) 38129210Scognet#if 0 39129210Scognetstatic char sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93"; 40129210Scognet#else 41129210Scognet__FBSDID("$FreeBSD$"); 42129210Scognet#endif 43129210Scognet#endif /* LIBC_SCCS and not lint */ 44129210Scognet 45129210Scognet#include <libkern/quad.h> 46129210Scognet 47129210Scognet/* 48129210Scognet * Multiply two quads. 49129210Scognet * 50129210Scognet * Our algorithm is based on the following. Split incoming quad values 51129210Scognet * u and v (where u,v >= 0) into 52129210Scognet * 53129210Scognet * u = 2^n u1 * u0 (n = number of bits in `u_int', usu. 32) 54129210Scognet * 55129210Scognet * and 56129210Scognet * 57129210Scognet * v = 2^n v1 * v0 58129210Scognet * 59129210Scognet * Then 60129210Scognet * 61129210Scognet * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 62129210Scognet * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 63129210Scognet * 64129210Scognet * Now add 2^n u1 v1 to the first term and subtract it from the middle, 65129210Scognet * and add 2^n u0 v0 to the last term and subtract it from the middle. 66129210Scognet * This gives: 67129210Scognet * 68129210Scognet * uv = (2^2n + 2^n) (u1 v1) + 69129210Scognet * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 70129210Scognet * (2^n + 1) (u0 v0) 71129210Scognet * 72129210Scognet * Factoring the middle a bit gives us: 73129210Scognet * 74129210Scognet * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 75129210Scognet * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 76129210Scognet * (2^n + 1) (u0 v0) [u0v0 = low] 77129210Scognet * 78129210Scognet * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 79129210Scognet * in just half the precision of the original. (Note that either or both 80129210Scognet * of (u1 - u0) or (v0 - v1) may be negative.) 81129210Scognet * 82129210Scognet * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 83129210Scognet * 84129210Scognet * Since C does not give us a `int * int = quad' operator, we split 85129210Scognet * our input quads into two ints, then split the two ints into two 86129210Scognet * shorts. We can then calculate `short * short = int' in native 87129210Scognet * arithmetic. 88129210Scognet * 89129210Scognet * Our product should, strictly speaking, be a `long quad', with 128 90129210Scognet * bits, but we are going to discard the upper 64. In other words, 91129210Scognet * we are not interested in uv, but rather in (uv mod 2^2n). This 92129210Scognet * makes some of the terms above vanish, and we get: 93129210Scognet * 94129210Scognet * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 95129210Scognet * 96129210Scognet * or 97129210Scognet * 98129210Scognet * (2^n)(high + mid + low) + low 99129210Scognet * 100129210Scognet * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 101129210Scognet * of 2^n in either one will also vanish. Only `low' need be computed 102129210Scognet * mod 2^2n, and only because of the final term above. 103129210Scognet */ 104129210Scognetstatic quad_t __lmulq(u_int, u_int); 105129210Scognet 106129210Scognetquad_t __muldi3(quad_t, quad_t); 107129210Scognetquad_t 108129210Scognet__muldi3(quad_t a, quad_t b) 109129210Scognet{ 110129210Scognet union uu u, v, low, prod; 111129210Scognet u_int high, mid, udiff, vdiff; 112129210Scognet int negall, negmid; 113129210Scognet#define u1 u.ul[H] 114129210Scognet#define u0 u.ul[L] 115129210Scognet#define v1 v.ul[H] 116129210Scognet#define v0 v.ul[L] 117129210Scognet 118129210Scognet /* 119129210Scognet * Get u and v such that u, v >= 0. When this is finished, 120129210Scognet * u1, u0, v1, and v0 will be directly accessible through the 121129210Scognet * int fields. 122129210Scognet */ 123129210Scognet if (a >= 0) 124129210Scognet u.q = a, negall = 0; 125129210Scognet else 126129210Scognet u.q = -a, negall = 1; 127129210Scognet if (b >= 0) 128129210Scognet v.q = b; 129129210Scognet else 130129210Scognet v.q = -b, negall ^= 1; 131129210Scognet 132129210Scognet if (u1 == 0 && v1 == 0) { 133129210Scognet /* 134129210Scognet * An (I hope) important optimization occurs when u1 and v1 135129210Scognet * are both 0. This should be common since most numbers 136129210Scognet * are small. Here the product is just u0*v0. 137129210Scognet */ 138129210Scognet prod.q = __lmulq(u0, v0); 139129210Scognet } else { 140129210Scognet /* 141129210Scognet * Compute the three intermediate products, remembering 142129210Scognet * whether the middle term is negative. We can discard 143129210Scognet * any upper bits in high and mid, so we can use native 144129210Scognet * u_int * u_int => u_int arithmetic. 145129210Scognet */ 146129210Scognet low.q = __lmulq(u0, v0); 147129210Scognet 148129210Scognet if (u1 >= u0) 149129210Scognet negmid = 0, udiff = u1 - u0; 150129210Scognet else 151129210Scognet negmid = 1, udiff = u0 - u1; 152129210Scognet if (v0 >= v1) 153129210Scognet vdiff = v0 - v1; 154129210Scognet else 155129210Scognet vdiff = v1 - v0, negmid ^= 1; 156129210Scognet mid = udiff * vdiff; 157129210Scognet 158129210Scognet high = u1 * v1; 159129210Scognet 160129210Scognet /* 161129210Scognet * Assemble the final product. 162129210Scognet */ 163129210Scognet prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 164129210Scognet low.ul[H]; 165129210Scognet prod.ul[L] = low.ul[L]; 166129210Scognet } 167129210Scognet return (negall ? -prod.q : prod.q); 168129210Scognet#undef u1 169129210Scognet#undef u0 170129210Scognet#undef v1 171129210Scognet#undef v0 172129210Scognet} 173129210Scognet 174129210Scognet/* 175129210Scognet * Multiply two 2N-bit ints to produce a 4N-bit quad, where N is half 176129210Scognet * the number of bits in an int (whatever that is---the code below 177129210Scognet * does not care as long as quad.h does its part of the bargain---but 178129210Scognet * typically N==16). 179129210Scognet * 180129210Scognet * We use the same algorithm from Knuth, but this time the modulo refinement 181129210Scognet * does not apply. On the other hand, since N is half the size of an int, 182129210Scognet * we can get away with native multiplication---none of our input terms 183129210Scognet * exceeds (UINT_MAX >> 1). 184129210Scognet * 185129210Scognet * Note that, for u_int l, the quad-precision result 186129210Scognet * 187129210Scognet * l << N 188129210Scognet * 189129210Scognet * splits into high and low ints as HHALF(l) and LHUP(l) respectively. 190129210Scognet */ 191129210Scognetstatic quad_t 192129210Scognet__lmulq(u_int u, u_int v) 193129210Scognet{ 194129210Scognet u_int u1, u0, v1, v0, udiff, vdiff, high, mid, low; 195129210Scognet u_int prodh, prodl, was; 196129210Scognet union uu prod; 197129210Scognet int neg; 198129210Scognet 199129210Scognet u1 = HHALF(u); 200129210Scognet u0 = LHALF(u); 201129210Scognet v1 = HHALF(v); 202129210Scognet v0 = LHALF(v); 203129210Scognet 204129210Scognet low = u0 * v0; 205129210Scognet 206129210Scognet /* This is the same small-number optimization as before. */ 207129210Scognet if (u1 == 0 && v1 == 0) 208129210Scognet return (low); 209129210Scognet 210129210Scognet if (u1 >= u0) 211129210Scognet udiff = u1 - u0, neg = 0; 212129210Scognet else 213129210Scognet udiff = u0 - u1, neg = 1; 214129210Scognet if (v0 >= v1) 215129210Scognet vdiff = v0 - v1; 216129210Scognet else 217129210Scognet vdiff = v1 - v0, neg ^= 1; 218129210Scognet mid = udiff * vdiff; 219129210Scognet 220129210Scognet high = u1 * v1; 221129210Scognet 222129210Scognet /* prod = (high << 2N) + (high << N); */ 223129210Scognet prodh = high + HHALF(high); 224129210Scognet prodl = LHUP(high); 225129210Scognet 226129210Scognet /* if (neg) prod -= mid << N; else prod += mid << N; */ 227129210Scognet if (neg) { 228129210Scognet was = prodl; 229129210Scognet prodl -= LHUP(mid); 230129210Scognet prodh -= HHALF(mid) + (prodl > was); 231129210Scognet } else { 232129210Scognet was = prodl; 233129210Scognet prodl += LHUP(mid); 234129210Scognet prodh += HHALF(mid) + (prodl < was); 235129210Scognet } 236129210Scognet 237129210Scognet /* prod += low << N */ 238129210Scognet was = prodl; 239129210Scognet prodl += LHUP(low); 240129210Scognet prodh += HHALF(low) + (prodl < was); 241129210Scognet /* ... + low; */ 242129210Scognet if ((prodl += low) < low) 243129210Scognet prodh++; 244129210Scognet 245129210Scognet /* return 4N-bit product */ 246129210Scognet prod.ul[H] = prodh; 247129210Scognet prod.ul[L] = prodl; 248129210Scognet return (prod.q); 249129210Scognet} 250