11573Srgrimes/*- 21573Srgrimes * Copyright (c) 1992, 1993 31573Srgrimes * The Regents of the University of California. All rights reserved. 41573Srgrimes * 51573Srgrimes * This software was developed by the Computer Systems Engineering group 61573Srgrimes * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and 71573Srgrimes * contributed to Berkeley. 81573Srgrimes * 91573Srgrimes * Redistribution and use in source and binary forms, with or without 101573Srgrimes * modification, are permitted provided that the following conditions 111573Srgrimes * are met: 121573Srgrimes * 1. Redistributions of source code must retain the above copyright 131573Srgrimes * notice, this list of conditions and the following disclaimer. 141573Srgrimes * 2. Redistributions in binary form must reproduce the above copyright 151573Srgrimes * notice, this list of conditions and the following disclaimer in the 161573Srgrimes * documentation and/or other materials provided with the distribution. 171573Srgrimes * 4. Neither the name of the University nor the names of its contributors 181573Srgrimes * may be used to endorse or promote products derived from this software 191573Srgrimes * without specific prior written permission. 201573Srgrimes * 211573Srgrimes * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 221573Srgrimes * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 231573Srgrimes * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 241573Srgrimes * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 251573Srgrimes * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 261573Srgrimes * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 271573Srgrimes * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 281573Srgrimes * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 291573Srgrimes * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 301573Srgrimes * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 311573Srgrimes * SUCH DAMAGE. 321573Srgrimes */ 331573Srgrimes 341573Srgrimes#if defined(LIBC_SCCS) && !defined(lint) 351573Srgrimesstatic char sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93"; 361573Srgrimes#endif /* LIBC_SCCS and not lint */ 3792889Sobrien#include <sys/cdefs.h> 3892889Sobrien__FBSDID("$FreeBSD$"); 391573Srgrimes 401573Srgrimes#include "quad.h" 411573Srgrimes 421573Srgrimes/* 431573Srgrimes * Multiply two quads. 441573Srgrimes * 451573Srgrimes * Our algorithm is based on the following. Split incoming quad values 461573Srgrimes * u and v (where u,v >= 0) into 471573Srgrimes * 481573Srgrimes * u = 2^n u1 * u0 (n = number of bits in `u_long', usu. 32) 491573Srgrimes * 508870Srgrimes * and 511573Srgrimes * 521573Srgrimes * v = 2^n v1 * v0 531573Srgrimes * 541573Srgrimes * Then 551573Srgrimes * 561573Srgrimes * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 571573Srgrimes * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 581573Srgrimes * 591573Srgrimes * Now add 2^n u1 v1 to the first term and subtract it from the middle, 601573Srgrimes * and add 2^n u0 v0 to the last term and subtract it from the middle. 611573Srgrimes * This gives: 621573Srgrimes * 631573Srgrimes * uv = (2^2n + 2^n) (u1 v1) + 641573Srgrimes * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 651573Srgrimes * (2^n + 1) (u0 v0) 661573Srgrimes * 671573Srgrimes * Factoring the middle a bit gives us: 681573Srgrimes * 691573Srgrimes * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 701573Srgrimes * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 711573Srgrimes * (2^n + 1) (u0 v0) [u0v0 = low] 721573Srgrimes * 731573Srgrimes * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 741573Srgrimes * in just half the precision of the original. (Note that either or both 751573Srgrimes * of (u1 - u0) or (v0 - v1) may be negative.) 761573Srgrimes * 771573Srgrimes * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 781573Srgrimes * 791573Srgrimes * Since C does not give us a `long * long = quad' operator, we split 801573Srgrimes * our input quads into two longs, then split the two longs into two 811573Srgrimes * shorts. We can then calculate `short * short = long' in native 821573Srgrimes * arithmetic. 831573Srgrimes * 841573Srgrimes * Our product should, strictly speaking, be a `long quad', with 128 851573Srgrimes * bits, but we are going to discard the upper 64. In other words, 861573Srgrimes * we are not interested in uv, but rather in (uv mod 2^2n). This 871573Srgrimes * makes some of the terms above vanish, and we get: 881573Srgrimes * 891573Srgrimes * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 901573Srgrimes * 911573Srgrimes * or 921573Srgrimes * 931573Srgrimes * (2^n)(high + mid + low) + low 941573Srgrimes * 951573Srgrimes * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 961573Srgrimes * of 2^n in either one will also vanish. Only `low' need be computed 971573Srgrimes * mod 2^2n, and only because of the final term above. 981573Srgrimes */ 991573Srgrimesstatic quad_t __lmulq(u_long, u_long); 1001573Srgrimes 1011573Srgrimesquad_t 1021573Srgrimes__muldi3(a, b) 1031573Srgrimes quad_t a, b; 1041573Srgrimes{ 1051573Srgrimes union uu u, v, low, prod; 10692889Sobrien u_long high, mid, udiff, vdiff; 10792889Sobrien int negall, negmid; 1081573Srgrimes#define u1 u.ul[H] 1091573Srgrimes#define u0 u.ul[L] 1101573Srgrimes#define v1 v.ul[H] 1111573Srgrimes#define v0 v.ul[L] 1121573Srgrimes 1131573Srgrimes /* 1141573Srgrimes * Get u and v such that u, v >= 0. When this is finished, 1151573Srgrimes * u1, u0, v1, and v0 will be directly accessible through the 1161573Srgrimes * longword fields. 1171573Srgrimes */ 1181573Srgrimes if (a >= 0) 1191573Srgrimes u.q = a, negall = 0; 1201573Srgrimes else 1211573Srgrimes u.q = -a, negall = 1; 1221573Srgrimes if (b >= 0) 1231573Srgrimes v.q = b; 1241573Srgrimes else 1251573Srgrimes v.q = -b, negall ^= 1; 1261573Srgrimes 1271573Srgrimes if (u1 == 0 && v1 == 0) { 1281573Srgrimes /* 1291573Srgrimes * An (I hope) important optimization occurs when u1 and v1 1301573Srgrimes * are both 0. This should be common since most numbers 1311573Srgrimes * are small. Here the product is just u0*v0. 1321573Srgrimes */ 1331573Srgrimes prod.q = __lmulq(u0, v0); 1341573Srgrimes } else { 1351573Srgrimes /* 1361573Srgrimes * Compute the three intermediate products, remembering 1371573Srgrimes * whether the middle term is negative. We can discard 1381573Srgrimes * any upper bits in high and mid, so we can use native 1391573Srgrimes * u_long * u_long => u_long arithmetic. 1401573Srgrimes */ 1411573Srgrimes low.q = __lmulq(u0, v0); 1421573Srgrimes 1431573Srgrimes if (u1 >= u0) 1441573Srgrimes negmid = 0, udiff = u1 - u0; 1451573Srgrimes else 1461573Srgrimes negmid = 1, udiff = u0 - u1; 1471573Srgrimes if (v0 >= v1) 1481573Srgrimes vdiff = v0 - v1; 1491573Srgrimes else 1501573Srgrimes vdiff = v1 - v0, negmid ^= 1; 1511573Srgrimes mid = udiff * vdiff; 1521573Srgrimes 1531573Srgrimes high = u1 * v1; 1541573Srgrimes 1551573Srgrimes /* 1561573Srgrimes * Assemble the final product. 1571573Srgrimes */ 1581573Srgrimes prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 1591573Srgrimes low.ul[H]; 1601573Srgrimes prod.ul[L] = low.ul[L]; 1611573Srgrimes } 1621573Srgrimes return (negall ? -prod.q : prod.q); 1631573Srgrimes#undef u1 1641573Srgrimes#undef u0 1651573Srgrimes#undef v1 1661573Srgrimes#undef v0 1671573Srgrimes} 1681573Srgrimes 1691573Srgrimes/* 1701573Srgrimes * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half 1711573Srgrimes * the number of bits in a long (whatever that is---the code below 1721573Srgrimes * does not care as long as quad.h does its part of the bargain---but 1731573Srgrimes * typically N==16). 1741573Srgrimes * 1751573Srgrimes * We use the same algorithm from Knuth, but this time the modulo refinement 1761573Srgrimes * does not apply. On the other hand, since N is half the size of a long, 1771573Srgrimes * we can get away with native multiplication---none of our input terms 1781573Srgrimes * exceeds (ULONG_MAX >> 1). 1791573Srgrimes * 1801573Srgrimes * Note that, for u_long l, the quad-precision result 1811573Srgrimes * 1821573Srgrimes * l << N 1831573Srgrimes * 1841573Srgrimes * splits into high and low longs as HHALF(l) and LHUP(l) respectively. 1851573Srgrimes */ 1861573Srgrimesstatic quad_t 1871573Srgrimes__lmulq(u_long u, u_long v) 1881573Srgrimes{ 1891573Srgrimes u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low; 1901573Srgrimes u_long prodh, prodl, was; 1911573Srgrimes union uu prod; 1921573Srgrimes int neg; 1931573Srgrimes 1941573Srgrimes u1 = HHALF(u); 1951573Srgrimes u0 = LHALF(u); 1961573Srgrimes v1 = HHALF(v); 1971573Srgrimes v0 = LHALF(v); 1981573Srgrimes 1991573Srgrimes low = u0 * v0; 2001573Srgrimes 2011573Srgrimes /* This is the same small-number optimization as before. */ 2021573Srgrimes if (u1 == 0 && v1 == 0) 2031573Srgrimes return (low); 2041573Srgrimes 2051573Srgrimes if (u1 >= u0) 2061573Srgrimes udiff = u1 - u0, neg = 0; 2071573Srgrimes else 2081573Srgrimes udiff = u0 - u1, neg = 1; 2091573Srgrimes if (v0 >= v1) 2101573Srgrimes vdiff = v0 - v1; 2111573Srgrimes else 2121573Srgrimes vdiff = v1 - v0, neg ^= 1; 2131573Srgrimes mid = udiff * vdiff; 2141573Srgrimes 2151573Srgrimes high = u1 * v1; 2161573Srgrimes 2171573Srgrimes /* prod = (high << 2N) + (high << N); */ 2181573Srgrimes prodh = high + HHALF(high); 2191573Srgrimes prodl = LHUP(high); 2201573Srgrimes 2211573Srgrimes /* if (neg) prod -= mid << N; else prod += mid << N; */ 2221573Srgrimes if (neg) { 2231573Srgrimes was = prodl; 2241573Srgrimes prodl -= LHUP(mid); 2251573Srgrimes prodh -= HHALF(mid) + (prodl > was); 2261573Srgrimes } else { 2271573Srgrimes was = prodl; 2281573Srgrimes prodl += LHUP(mid); 2291573Srgrimes prodh += HHALF(mid) + (prodl < was); 2301573Srgrimes } 2311573Srgrimes 2321573Srgrimes /* prod += low << N */ 2331573Srgrimes was = prodl; 2341573Srgrimes prodl += LHUP(low); 2351573Srgrimes prodh += HHALF(low) + (prodl < was); 2361573Srgrimes /* ... + low; */ 2371573Srgrimes if ((prodl += low) < low) 2381573Srgrimes prodh++; 2391573Srgrimes 2401573Srgrimes /* return 4N-bit product */ 2411573Srgrimes prod.ul[H] = prodh; 2421573Srgrimes prod.ul[L] = prodl; 2431573Srgrimes return (prod.q); 2441573Srgrimes} 245