2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
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/lib/libstand/qdivrem.c,v 1.2 1999/08/28 00:05:33 peter Exp $
34 * From: Id: qdivrem.c,v 1.7 1997/11/07 09:20:40 phk Exp
38 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed),
39 * section 4.3.1, pp. 257--259.
44 #define B (1 << HALF_BITS) /* digit base */
46 /* Combine two `digits' to make a single two-digit number. */
47 #define COMBINE(a, b) (((u_int)(a) << HALF_BITS) | (b))
49 _Static_assert(sizeof(int) / 2 == sizeof(short),
50 "Bitwise functions in libstand are broken on this architecture");
52 /* select a type for digits in base B: use unsigned short if they fit */
53 typedef unsigned short digit;
56 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
57 * `fall out' the left (there never will be any such anyway).
58 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS.
61 shl(digit *p, int len, int sh)
65 for (i = 0; i < len; i++)
66 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh));
67 p[i] = LHALF(p[i] << sh);
71 * __udivmoddi4(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
73 * We do this in base 2-sup-HALF_BITS, so that all intermediate products
74 * fit within u_int. As a consequence, the maximum length dividend and
75 * divisor are 4 `digits' in this base (they are shorter if they have
79 __udivmoddi4(u_quad_t uq, u_quad_t vq, u_quad_t *arq)
86 digit uspace[5], vspace[5], qspace[5];
89 * Take care of special cases: divide by zero, and u < v.
93 static volatile const unsigned int zero = 0;
95 tmp.ul[H] = tmp.ul[L] = 1 / zero;
110 * Break dividend and divisor into digits in base B, then
111 * count leading zeros to determine m and n. When done, we
113 * u = (u[1]u[2]...u[m+n]) sub B
114 * v = (v[1]v[2]...v[n]) sub B
116 * 1 < n <= 4 (if n = 1, we use a different division algorithm)
117 * m >= 0 (otherwise u < v, which we already checked)
124 u[1] = HHALF(tmp.ul[H]);
125 u[2] = LHALF(tmp.ul[H]);
126 u[3] = HHALF(tmp.ul[L]);
127 u[4] = LHALF(tmp.ul[L]);
129 v[1] = HHALF(tmp.ul[H]);
130 v[2] = LHALF(tmp.ul[H]);
131 v[3] = HHALF(tmp.ul[L]);
132 v[4] = LHALF(tmp.ul[L]);
133 for (n = 4; v[1] == 0; v++) {
135 u_int rbj; /* r*B+u[j] (not root boy jim) */
136 digit q1, q2, q3, q4;
139 * Change of plan, per exercise 16.
142 * q[j] = floor((r*B + u[j]) / v),
143 * r = (r*B + u[j]) % v;
144 * We unroll this completely here.
146 t = v[2]; /* nonzero, by definition */
148 rbj = COMBINE(u[1] % t, u[2]);
150 rbj = COMBINE(rbj % t, u[3]);
152 rbj = COMBINE(rbj % t, u[4]);
156 tmp.ul[H] = COMBINE(q1, q2);
157 tmp.ul[L] = COMBINE(q3, q4);
163 * By adjusting q once we determine m, we can guarantee that
164 * there is a complete four-digit quotient at &qspace[1] when
167 for (m = 4 - n; u[1] == 0; u++)
169 for (i = 4 - m; --i >= 0;)
174 * Here we run Program D, translated from MIX to C and acquiring
175 * a few minor changes.
177 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
180 for (t = v[1]; t < B / 2; t <<= 1)
183 shl(&u[0], m + n, d); /* u <<= d */
184 shl(&v[1], n - 1, d); /* v <<= d */
190 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */
191 v2 = v[2]; /* for D3 */
196 * D3: Calculate qhat (\^q, in TeX notation).
197 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
198 * let rhat = (u[j]*B + u[j+1]) mod v[1].
199 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
200 * decrement qhat and increase rhat correspondingly.
201 * Note that if rhat >= B, v[2]*qhat < rhat*B.
203 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */
204 uj1 = u[j + 1]; /* for D3 only */
205 uj2 = u[j + 2]; /* for D3 only */
211 u_int nn = COMBINE(uj0, uj1);
215 while (v2 * qhat > COMBINE(rhat, uj2)) {
218 if ((rhat += v1) >= B)
222 * D4: Multiply and subtract.
223 * The variable `t' holds any borrows across the loop.
224 * We split this up so that we do not require v[0] = 0,
225 * and to eliminate a final special case.
227 for (t = 0, i = n; i > 0; i--) {
228 t = u[i + j] - v[i] * qhat - t;
230 t = (B - HHALF(t)) & (B - 1);
235 * D5: test remainder.
236 * There is a borrow if and only if HHALF(t) is nonzero;
237 * in that (rare) case, qhat was too large (by exactly 1).
238 * Fix it by adding v[1..n] to u[j..j+n].
242 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */
243 t += u[i + j] + v[i];
247 u[j] = LHALF(u[j] + t);
250 } while (++j <= m); /* D7: loop on j. */
253 * If caller wants the remainder, we have to calculate it as
254 * u[m..m+n] >> d (this is at most n digits and thus fits in
255 * u[m+1..m+n], but we may need more source digits).
259 for (i = m + n; i > m; --i)
261 LHALF(u[i - 1] << (HALF_BITS - d));
264 tmp.ul[H] = COMBINE(uspace[1], uspace[2]);
265 tmp.ul[L] = COMBINE(uspace[3], uspace[4]);
269 tmp.ul[H] = COMBINE(qspace[1], qspace[2]);
270 tmp.ul[L] = COMBINE(qspace[3], qspace[4]);
275 * Divide two unsigned quads.
279 __udivdi3(u_quad_t a, u_quad_t b)
282 return (__udivmoddi4(a, b, (u_quad_t *)0));
286 * Return remainder after dividing two unsigned quads.
289 __umoddi3(u_quad_t a, u_quad_t b)
293 (void)__udivmoddi4(a, b, &r);