/*- * Copyright (c) 1992, 1993 * The Regents of the University of California. All rights reserved. * * This software was developed by the Computer Systems Engineering group * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and * contributed to Berkeley. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * 3. Neither the name of the University nor the names of its contributors * may be used to endorse or promote products derived from this software * without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF * SUCH DAMAGE. * * From: * @(#)muldi3.c 8.1 (Berkeley) 6/4/93 * NetBSD: muldi3.c,v 1.1 2005/12/20 19:28:51 christos Exp */ #include "longlong.h" /* * Multiply two long longs. * * Our algorithm is based on the following. Split incoming long long * values u and v (where u,v >= 0) into * * u = 2^n u1 * u0 (n = number of bits in `unsigned int', usu. 32) * * and * * v = 2^n v1 * v0 * * Then * * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 * * Now add 2^n u1 v1 to the first term and subtract it from the middle, * and add 2^n u0 v0 to the last term and subtract it from the middle. * This gives: * * uv = (2^2n + 2^n) (u1 v1) + * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + * (2^n + 1) (u0 v0) * * Factoring the middle a bit gives us: * * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] * (2^n + 1) (u0 v0) [u0v0 = low] * * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done * in just half the precision of the original. (Note that either or both * of (u1 - u0) or (v0 - v1) may be negative.) * * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. * * Since C does not give us a `int * int = long long' operator, we split * our input long longs into two ints, then split the two ints into two * shorts. We can then calculate `short * short = int' in native * arithmetic. * * Our product should, strictly speaking, be a `long long long', with * 128 bits, but we are going to discard the upper 64. In other words, * we are not interested in uv, but rather in (uv mod 2^2n). This * makes some of the terms above vanish, and we get: * * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) * * or * * (2^n)(high + mid + low) + low * * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor * of 2^n in either one will also vanish. Only `low' need be computed * mod 2^2n, and only because of the final term above. */ static long long __lmulq(unsigned int, unsigned int); long long __muldi3(long long a, long long b) { union uu u, v, low, prod; unsigned int high, mid, udiff, vdiff; int negall, negmid; #define u1 u.ui[H] #define u0 u.ui[L] #define v1 v.ui[H] #define v0 v.ui[L] /* * Get u and v such that u, v >= 0. When this is finished, * u1, u0, v1, and v0 will be directly accessible through the * int fields. */ if (a >= 0) u.ll = a, negall = 0; else u.ll = -a, negall = 1; if (b >= 0) v.ll = b; else v.ll = -b, negall ^= 1; if (u1 == 0 && v1 == 0) { /* * An (I hope) important optimization occurs when u1 and v1 * are both 0. This should be common since most numbers * are small. Here the product is just u0*v0. */ prod.ll = __lmulq(u0, v0); } else { /* * Compute the three intermediate products, remembering * whether the middle term is negative. We can discard * any upper bits in high and mid, so we can use native * unsigned int * unsigned int => unsigned int arithmetic. */ low.ll = __lmulq(u0, v0); if (u1 >= u0) negmid = 0, udiff = u1 - u0; else negmid = 1, udiff = u0 - u1; if (v0 >= v1) vdiff = v0 - v1; else vdiff = v1 - v0, negmid ^= 1; mid = udiff * vdiff; high = u1 * v1; /* * Assemble the final product. */ prod.ui[H] = high + (negmid ? -mid : mid) + low.ui[L] + low.ui[H]; prod.ui[L] = low.ui[L]; } return (negall ? -prod.ll : prod.ll); #undef u1 #undef u0 #undef v1 #undef v0 } /* * Multiply two 2N-bit ints to produce a 4N-bit long long, where N is * half the number of bits in an int (whatever that is---the code * below does not care as long as the header file does its part of the * bargain---but typically N==16). * * We use the same algorithm from Knuth, but this time the modulo refinement * does not apply. On the other hand, since N is half the size of an int, * we can get away with native multiplication---none of our input terms * exceeds (UINT_MAX >> 1). * * Note that, for unsigned int l, the quad-precision (long long) result * * l << N * * splits into high and low ints as HHALF(l) and LHUP(l) respectively. */ static long long __lmulq(unsigned int u, unsigned int v) { unsigned int u1, u0, v1, v0, udiff, vdiff, high, mid, low; unsigned int prodh, prodl, was; union uu prod; int neg; u1 = HHALF(u); u0 = LHALF(u); v1 = HHALF(v); v0 = LHALF(v); low = u0 * v0; /* This is the same small-number optimization as before. */ if (u1 == 0 && v1 == 0) return (low); if (u1 >= u0) udiff = u1 - u0, neg = 0; else udiff = u0 - u1, neg = 1; if (v0 >= v1) vdiff = v0 - v1; else vdiff = v1 - v0, neg ^= 1; mid = udiff * vdiff; high = u1 * v1; /* prod = (high << 2N) + (high << N); */ prodh = high + HHALF(high); prodl = LHUP(high); /* if (neg) prod -= mid << N; else prod += mid << N; */ if (neg) { was = prodl; prodl -= LHUP(mid); prodh -= HHALF(mid) + (prodl > was); } else { was = prodl; prodl += LHUP(mid); prodh += HHALF(mid) + (prodl < was); } /* prod += low << N */ was = prodl; prodl += LHUP(low); prodh += HHALF(low) + (prodl < was); /* ... + low; */ if ((prodl += low) < low) prodh++; /* return 4N-bit product */ prod.ui[H] = prodh; prod.ui[L] = prodl; return (prod.ll); }