1 /* $NetBSD: qdivrem.c,v 1.4 2012/03/20 16:21:41 matt Exp $ */
2
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 #if defined(LIBC_SCCS) && !defined(lint)
38 #if 0
39 static char sccsid[] = "@(#)qdivrem.c 8.1 (Berkeley) 6/4/93";
40 #else
41 __RCSID("$NetBSD: qdivrem.c,v 1.4 2012/03/20 16:21:41 matt Exp $");
42 #endif
43 #endif /* LIBC_SCCS and not lint */
44
45 /*
46 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed),
47 * section 4.3.1, pp. 257--259.
48 */
49
50 #include "quad.h"
51
52 #define B ((int)1 << (unsigned int)HALF_BITS) /* digit base */
53
54 /* Combine two `digits' to make a single two-digit number. */
55 #define COMBINE(a, b) (((u_int)(a) << (unsigned int)HALF_BITS) | (b))
56
57 /* select a type for digits in base B: use unsigned short if they fit */
58 #if UINT_MAX == 0xffffffffU && USHRT_MAX >= 0xffff
59 typedef unsigned short digit;
60 #else
61 typedef u_int digit;
62 #endif
63
64 static void shl(digit *p, int len, int sh);
65
66 /*
67 * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
68 *
69 * We do this in base 2-sup-HALF_BITS, so that all intermediate products
70 * fit within u_int. As a consequence, the maximum length dividend and
71 * divisor are 4 `digits' in this base (they are shorter if they have
72 * leading zeros).
73 */
74 u_quad_t
__qdivrem(u_quad_t uq,u_quad_t vq,u_quad_t * arq)75 __qdivrem(u_quad_t uq, u_quad_t vq, u_quad_t *arq)
76 {
77 union uu tmp;
78 digit *u, *v, *q;
79 digit v1, v2;
80 u_int qhat, rhat, t;
81 int m, n, d, j, i;
82 digit uspace[5], vspace[5], qspace[5];
83
84 /*
85 * Take care of special cases: divide by zero, and u < v.
86 */
87 if (vq == 0) {
88 /* divide by zero. */
89 static volatile const unsigned int zero = 0;
90
91 tmp.ul[H] = tmp.ul[L] = 1 / zero;
92 if (arq)
93 *arq = uq;
94 return (tmp.q);
95 }
96 if (uq < vq) {
97 if (arq)
98 *arq = uq;
99 return (0);
100 }
101 u = &uspace[0];
102 v = &vspace[0];
103 q = &qspace[0];
104
105 /*
106 * Break dividend and divisor into digits in base B, then
107 * count leading zeros to determine m and n. When done, we
108 * will have:
109 * u = (u[1]u[2]...u[m+n]) sub B
110 * v = (v[1]v[2]...v[n]) sub B
111 * v[1] != 0
112 * 1 < n <= 4 (if n = 1, we use a different division algorithm)
113 * m >= 0 (otherwise u < v, which we already checked)
114 * m + n = 4
115 * and thus
116 * m = 4 - n <= 2
117 */
118 tmp.uq = uq;
119 u[0] = 0;
120 u[1] = (digit)HHALF(tmp.ul[H]);
121 u[2] = (digit)LHALF(tmp.ul[H]);
122 u[3] = (digit)HHALF(tmp.ul[L]);
123 u[4] = (digit)LHALF(tmp.ul[L]);
124 tmp.uq = vq;
125 v[1] = (digit)HHALF(tmp.ul[H]);
126 v[2] = (digit)LHALF(tmp.ul[H]);
127 v[3] = (digit)HHALF(tmp.ul[L]);
128 v[4] = (digit)LHALF(tmp.ul[L]);
129 for (n = 4; v[1] == 0; v++) {
130 if (--n == 1) {
131 u_int rbj; /* r*B+u[j] (not root boy jim) */
132 digit q1, q2, q3, q4;
133
134 /*
135 * Change of plan, per exercise 16.
136 * r = 0;
137 * for j = 1..4:
138 * q[j] = floor((r*B + u[j]) / v),
139 * r = (r*B + u[j]) % v;
140 * We unroll this completely here.
141 */
142 t = v[2]; /* nonzero, by definition */
143 q1 = (digit)(u[1] / t);
144 rbj = COMBINE(u[1] % t, u[2]);
145 q2 = (digit)(rbj / t);
146 rbj = COMBINE(rbj % t, u[3]);
147 q3 = (digit)(rbj / t);
148 rbj = COMBINE(rbj % t, u[4]);
149 q4 = (digit)(rbj / t);
150 if (arq)
151 *arq = rbj % t;
152 tmp.ul[H] = COMBINE(q1, q2);
153 tmp.ul[L] = COMBINE(q3, q4);
154 return (tmp.q);
155 }
156 }
157
158 /*
159 * By adjusting q once we determine m, we can guarantee that
160 * there is a complete four-digit quotient at &qspace[1] when
161 * we finally stop.
162 */
163 for (m = 4 - n; u[1] == 0; u++)
164 m--;
165 for (i = 4 - m; --i >= 0;)
166 q[i] = 0;
167 q += 4 - m;
168
169 /*
170 * Here we run Program D, translated from MIX to C and acquiring
171 * a few minor changes.
172 *
173 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
174 */
175 d = 0;
176 for (t = v[1]; t < B / 2; t <<= (unsigned int)1)
177 d++;
178 if (d > 0) {
179 shl(&u[0], m + n, d); /* u <<= d */
180 shl(&v[1], n - 1, d); /* v <<= d */
181 }
182 /*
183 * D2: j = 0.
184 */
185 j = 0;
186 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */
187 v2 = v[2]; /* for D3 */
188 do {
189 digit uj0, uj1, uj2;
190
191 /*
192 * D3: Calculate qhat (\^q, in TeX notation).
193 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
194 * let rhat = (u[j]*B + u[j+1]) mod v[1].
195 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
196 * decrement qhat and increase rhat correspondingly.
197 * Note that if rhat >= B, v[2]*qhat < rhat*B.
198 */
199 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */
200 uj1 = u[j + 1]; /* for D3 only */
201 uj2 = u[j + 2]; /* for D3 only */
202 if (uj0 == v1) {
203 qhat = B;
204 rhat = uj1;
205 goto qhat_too_big;
206 } else {
207 u_int nn = COMBINE(uj0, uj1);
208 qhat = nn / v1;
209 rhat = nn % v1;
210 }
211 while (v2 * qhat > COMBINE(rhat, uj2)) {
212 qhat_too_big:
213 qhat--;
214 if ((rhat += v1) >= B)
215 break;
216 }
217 /*
218 * D4: Multiply and subtract.
219 * The variable `t' holds any borrows across the loop.
220 * We split this up so that we do not require v[0] = 0,
221 * and to eliminate a final special case.
222 */
223 for (t = 0, i = n; i > 0; i--) {
224 t = u[i + j] - v[i] * qhat - t;
225 u[i + j] = (digit)LHALF(t);
226 t = (B - HHALF(t)) & (B - 1);
227 }
228 t = u[j] - t;
229 u[j] = (digit)LHALF(t);
230 /*
231 * D5: test remainder.
232 * There is a borrow if and only if HHALF(t) is nonzero;
233 * in that (rare) case, qhat was too large (by exactly 1).
234 * Fix it by adding v[1..n] to u[j..j+n].
235 */
236 if (HHALF(t)) {
237 qhat--;
238 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */
239 t += u[i + j] + v[i];
240 u[i + j] = (digit)LHALF(t);
241 t = HHALF(t);
242 }
243 u[j] = (digit)LHALF(u[j] + t);
244 }
245 q[j] = (digit)qhat;
246 } while (++j <= m); /* D7: loop on j. */
247
248 /*
249 * If caller wants the remainder, we have to calculate it as
250 * u[m..m+n] >> d (this is at most n digits and thus fits in
251 * u[m+1..m+n], but we may need more source digits).
252 */
253 if (arq) {
254 if (d) {
255 for (i = m + n; i > m; --i)
256 u[i] = (digit)(((u_int)u[i] >> d) |
257 LHALF((u_int)u[i - 1] << (unsigned int)(HALF_BITS - d)));
258 u[i] = 0;
259 }
260 tmp.ul[H] = COMBINE(uspace[1], uspace[2]);
261 tmp.ul[L] = COMBINE(uspace[3], uspace[4]);
262 *arq = tmp.q;
263 }
264
265 tmp.ul[H] = COMBINE(qspace[1], qspace[2]);
266 tmp.ul[L] = COMBINE(qspace[3], qspace[4]);
267 return (tmp.q);
268 }
269
270 /*
271 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
272 * `fall out' the left (there never will be any such anyway).
273 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS.
274 */
275 static void
shl(digit * p,int len,int sh)276 shl(digit *p, int len, int sh)
277 {
278 int i;
279
280 for (i = 0; i < len; i++)
281 p[i] = (digit)(LHALF((u_int)p[i] << sh) |
282 ((u_int)p[i + 1] >> (HALF_BITS - sh)));
283 p[i] = (digit)(LHALF((u_int)p[i] << sh));
284 }
285