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5 * This software was developed by the Computer Systems Engineering group
6 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
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33 * $FreeBSD: src/sys/libkern/muldi3.c,v 1.6 1999/08/28 00:46:34 peter Exp $
34 * $DragonFly: src/sys/libkern/muldi3.c,v 1.4 2004/01/26 11:09:44 joerg Exp $
42 * Our algorithm is based on the following. Split incoming quad values
43 * u and v (where u,v >= 0) into
45 * u = 2^n u1 * u0 (n = number of bits in `u_long', usu. 32)
53 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0
54 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0
56 * Now add 2^n u1 v1 to the first term and subtract it from the middle,
57 * and add 2^n u0 v0 to the last term and subtract it from the middle.
60 * uv = (2^2n + 2^n) (u1 v1) +
61 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) +
64 * Factoring the middle a bit gives us:
66 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high]
67 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid]
68 * (2^n + 1) (u0 v0) [u0v0 = low]
70 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done
71 * in just half the precision of the original. (Note that either or both
72 * of (u1 - u0) or (v0 - v1) may be negative.)
74 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278.
76 * Since C does not give us a `long * long = quad' operator, we split
77 * our input quads into two longs, then split the two longs into two
78 * shorts. We can then calculate `short * short = long' in native
81 * Our product should, strictly speaking, be a `long quad', with 128
82 * bits, but we are going to discard the upper 64. In other words,
83 * we are not interested in uv, but rather in (uv mod 2^2n). This
84 * makes some of the terms above vanish, and we get:
86 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low)
90 * (2^n)(high + mid + low) + low
92 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor
93 * of 2^n in either one will also vanish. Only `low' need be computed
94 * mod 2^2n, and only because of the final term above.
96 static quad_t __lmulq(u_long u, u_long v);
99 __muldi3(quad_t a, quad_t b)
101 union uu u, v, low, prod;
102 u_long high, mid, udiff, vdiff;
110 * Get u and v such that u, v >= 0. When this is finished,
111 * u1, u0, v1, and v0 will be directly accessible through the
117 u.q = -a, negall = 1;
121 v.q = -b, negall ^= 1;
123 if (u1 == 0 && v1 == 0) {
125 * An (I hope) important optimization occurs when u1 and v1
126 * are both 0. This should be common since most numbers
127 * are small. Here the product is just u0*v0.
129 prod.q = __lmulq(u0, v0);
132 * Compute the three intermediate products, remembering
133 * whether the middle term is negative. We can discard
134 * any upper bits in high and mid, so we can use native
135 * u_long * u_long => u_long arithmetic.
137 low.q = __lmulq(u0, v0);
140 negmid = 0, udiff = u1 - u0;
142 negmid = 1, udiff = u0 - u1;
146 vdiff = v1 - v0, negmid ^= 1;
152 * Assemble the final product.
154 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] +
156 prod.ul[L] = low.ul[L];
158 return (negall ? -prod.q : prod.q);
166 * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half
167 * the number of bits in a long (whatever that is---the code below
168 * does not care as long as quad.h does its part of the bargain---but
171 * We use the same algorithm from Knuth, but this time the modulo refinement
172 * does not apply. On the other hand, since N is half the size of a long,
173 * we can get away with native multiplication---none of our input terms
174 * exceeds (ULONG_MAX >> 1).
176 * Note that, for u_long l, the quad-precision result
180 * splits into high and low longs as HHALF(l) and LHUP(l) respectively.
183 __lmulq(u_long u, u_long v)
185 u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low;
186 u_long prodh, prodl, was;
197 /* This is the same small-number optimization as before. */
198 if (u1 == 0 && v1 == 0)
202 udiff = u1 - u0, neg = 0;
204 udiff = u0 - u1, neg = 1;
208 vdiff = v1 - v0, neg ^= 1;
213 /* prod = (high << 2N) + (high << N); */
214 prodh = high + HHALF(high);
217 /* if (neg) prod -= mid << N; else prod += mid << N; */
221 prodh -= HHALF(mid) + (prodl > was);
225 prodh += HHALF(mid) + (prodl < was);
228 /* prod += low << N */
231 prodh += HHALF(low) + (prodl < was);
233 if ((prodl += low) < low)
236 /* return 4N-bit product */