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