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