1 /*
2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * This software was developed by the Computer Systems Engineering group
6 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and
7 * contributed to Berkeley.
8 *
9 * Redistribution and use in source and binary forms, with or without
10 * modification, are permitted provided that the following conditions
11 * are met:
12 * 1. Redistributions of source code must retain the above copyright
13 * notice, this list of conditions and the following disclaimer.
14 * 2. Redistributions in binary form must reproduce the above copyright
15 * notice, this list of conditions and the following disclaimer in the
16 * documentation and/or other materials provided with the distribution.
17 * 3. All advertising materials mentioning features or use of this software
18 * must display the following acknowledgement:
19 * This product includes software developed by the University of
20 * California, Berkeley and its contributors.
21 * 4. Neither the name of the University nor the names of its contributors
22 * may be used to endorse or promote products derived from this software
23 * without specific prior written permission.
24 *
25 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
26 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
27 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
28 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
29 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
30 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
31 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
32 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
33 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
34 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
35 * SUCH DAMAGE.
36 */
37
38 #pragma ident "%Z%%M% %I% %E% SMI"
39
40 /*
41 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed),
42 * section 4.3.1, pp. 257--259.
43 */
44
45 #include "quadint.h"
46
47 #define B (1 << HALF_BITS) /* digit base */
48
49 /* Combine two `digits' to make a single two-digit number. */
50 #define COMBINE(a, b) (((ulong_t)(a) << HALF_BITS) | (b))
51
52 /* select a type for digits in base B: use unsigned short if they fit */
53 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff
54 typedef unsigned short digit;
55 #else
56 typedef ulong_t digit;
57 #endif
58
59 /*
60 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
61 * `fall out' the left (there never will be any such anyway).
62 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS.
63 */
64 static void
shl(digit * p,int len,int sh)65 shl(digit *p, int len, int sh)
66 {
67 int i;
68
69 for (i = 0; i < len; i++)
70 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh));
71 p[i] = LHALF(p[i] << sh);
72 }
73
74 /*
75 * ___qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
76 *
77 * We do this in base 2-sup-HALF_BITS, so that all intermediate products
78 * fit within ulong_t. As a consequence, the maximum length dividend and
79 * divisor are 4 `digits' in this base (they are shorter if they have
80 * leading zeros).
81 */
82 u_longlong_t
___qdivrem(u_longlong_t uq,u_longlong_t vq,u_longlong_t * arq)83 ___qdivrem(u_longlong_t uq, u_longlong_t vq, u_longlong_t *arq)
84 {
85 union uu tmp;
86 digit *u, *v, *q;
87 digit v1, v2;
88 ulong_t qhat, rhat, t;
89 int m, n, d, j, i;
90 digit uspace[5], vspace[5], qspace[5];
91
92 /*
93 * Take care of special cases: divide by zero, and u < v.
94 */
95 if (vq == 0) {
96 /* divide by zero. */
97 static volatile const unsigned int zero = 0;
98
99 tmp.ul[H] = tmp.ul[L] = 1 / zero;
100 if (arq)
101 *arq = uq;
102 return (tmp.q);
103 }
104 if (uq < vq) {
105 if (arq)
106 *arq = uq;
107 return (0);
108 }
109 u = &uspace[0];
110 v = &vspace[0];
111 q = &qspace[0];
112
113 /*
114 * Break dividend and divisor into digits in base B, then
115 * count leading zeros to determine m and n. When done, we
116 * will have:
117 * u = (u[1]u[2]...u[m+n]) sub B
118 * v = (v[1]v[2]...v[n]) sub B
119 * v[1] != 0
120 * 1 < n <= 4 (if n = 1, we use a different division algorithm)
121 * m >= 0 (otherwise u < v, which we already checked)
122 * m + n = 4
123 * and thus
124 * m = 4 - n <= 2
125 */
126 tmp.uq = uq;
127 u[0] = 0;
128 u[1] = HHALF(tmp.ul[H]);
129 u[2] = LHALF(tmp.ul[H]);
130 u[3] = HHALF(tmp.ul[L]);
131 u[4] = LHALF(tmp.ul[L]);
132 tmp.uq = vq;
133 v[1] = HHALF(tmp.ul[H]);
134 v[2] = LHALF(tmp.ul[H]);
135 v[3] = HHALF(tmp.ul[L]);
136 v[4] = LHALF(tmp.ul[L]);
137 for (n = 4; v[1] == 0; v++) {
138 if (--n == 1) {
139 ulong_t rbj; /* r*B+u[j] (not root boy jim) */
140 digit q1, q2, q3, q4;
141
142 /*
143 * Change of plan, per exercise 16.
144 * r = 0;
145 * for j = 1..4:
146 * q[j] = floor((r*B + u[j]) / v),
147 * r = (r*B + u[j]) % v;
148 * We unroll this completely here.
149 */
150 t = v[2]; /* nonzero, by definition */
151 q1 = u[1] / t;
152 rbj = COMBINE(u[1] % t, u[2]);
153 q2 = rbj / t;
154 rbj = COMBINE(rbj % t, u[3]);
155 q3 = rbj / t;
156 rbj = COMBINE(rbj % t, u[4]);
157 q4 = rbj / t;
158 if (arq)
159 *arq = rbj % t;
160 tmp.ul[H] = COMBINE(q1, q2);
161 tmp.ul[L] = COMBINE(q3, q4);
162 return (tmp.q);
163 }
164 }
165
166 /*
167 * By adjusting q once we determine m, we can guarantee that
168 * there is a complete four-digit quotient at &qspace[1] when
169 * we finally stop.
170 */
171 for (m = 4 - n; u[1] == 0; u++)
172 m--;
173 for (i = 4 - m; --i >= 0; )
174 q[i] = 0;
175 q += 4 - m;
176
177 /*
178 * Here we run Program D, translated from MIX to C and acquiring
179 * a few minor changes.
180 *
181 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
182 */
183 d = 0;
184 for (t = v[1]; t < B / 2; t <<= 1)
185 d++;
186 if (d > 0) {
187 shl(&u[0], m + n, d); /* u <<= d */
188 shl(&v[1], n - 1, d); /* v <<= d */
189 }
190 /*
191 * D2: j = 0.
192 */
193 j = 0;
194 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */
195 v2 = v[2]; /* for D3 */
196 do {
197 digit uj0, uj1, uj2;
198
199 /*
200 * D3: Calculate qhat (\^q, in TeX notation).
201 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
202 * let rhat = (u[j]*B + u[j+1]) mod v[1].
203 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
204 * decrement qhat and increase rhat correspondingly.
205 * Note that if rhat >= B, v[2]*qhat < rhat*B.
206 */
207 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */
208 uj1 = u[j + 1]; /* for D3 only */
209 uj2 = u[j + 2]; /* for D3 only */
210 if (uj0 == v1) {
211 qhat = B;
212 rhat = uj1;
213 goto qhat_too_big;
214 } else {
215 ulong_t n = COMBINE(uj0, uj1);
216 qhat = n / v1;
217 rhat = n % v1;
218 }
219 while (v2 * qhat > COMBINE(rhat, uj2)) {
220 qhat_too_big:
221 qhat--;
222 if ((rhat += v1) >= B)
223 break;
224 }
225 /*
226 * D4: Multiply and subtract.
227 * The variable `t' holds any borrows across the loop.
228 * We split this up so that we do not require v[0] = 0,
229 * and to eliminate a final special case.
230 */
231 for (t = 0, i = n; i > 0; i--) {
232 t = u[i + j] - v[i] * qhat - t;
233 u[i + j] = LHALF(t);
234 t = (B - HHALF(t)) & (B - 1);
235 }
236 t = u[j] - t;
237 u[j] = LHALF(t);
238 /*
239 * D5: test remainder.
240 * There is a borrow if and only if HHALF(t) is nonzero;
241 * in that (rare) case, qhat was too large (by exactly 1).
242 * Fix it by adding v[1..n] to u[j..j+n].
243 */
244 if (HHALF(t)) {
245 qhat--;
246 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */
247 t += u[i + j] + v[i];
248 u[i + j] = LHALF(t);
249 t = HHALF(t);
250 }
251 u[j] = LHALF(u[j] + t);
252 }
253 q[j] = (digit)qhat;
254 } while (++j <= m); /* D7: loop on j. */
255
256 /*
257 * If caller wants the remainder, we have to calculate it as
258 * u[m..m+n] >> d (this is at most n digits and thus fits in
259 * u[m+1..m+n], but we may need more source digits).
260 */
261 if (arq) {
262 if (d) {
263 for (i = m + n; i > m; --i)
264 u[i] = (u[i] >> d) |
265 LHALF(u[i - 1] << (HALF_BITS - d));
266 u[i] = 0;
267 }
268 tmp.ul[H] = COMBINE(uspace[1], uspace[2]);
269 tmp.ul[L] = COMBINE(uspace[3], uspace[4]);
270 *arq = tmp.q;
271 }
272
273 tmp.ul[H] = COMBINE(qspace[1], qspace[2]);
274 tmp.ul[L] = COMBINE(qspace[3], qspace[4]);
275 return (tmp.q);
276 }
277