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. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
20 *
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
31 * SUCH DAMAGE.
32 *
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
35 */
36
37 /*
38 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed),
39 * section 4.3.1, pp. 257--259.
40 */
41
42 #include "quad.h"
43 #include <sys/endian.h> /* _QUAD_HIGHWORD */
44
45 #define B (1 << HALF_BITS) /* digit base */
46
47 /*
48 * Define high and low longwords.
49 */
50 #define H _QUAD_HIGHWORD
51 #define L _QUAD_LOWWORD
52
53 /* Combine two `digits' to make a single two-digit number. */
54 #define COMBINE(a, b) (((u_int)(a) << HALF_BITS) | (b))
55
56 _Static_assert(sizeof(int) / 2 == sizeof(short),
57 "Bitwise functions in libstand are broken on this architecture");
58
59 /* select a type for digits in base B: use unsigned short if they fit */
60 typedef unsigned short digit;
61
62 /*
63 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
64 * `fall out' the left (there never will be any such anyway).
65 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS.
66 */
67 static void
shl(digit * p,int len,int sh)68 shl(digit *p, int len, int sh)
69 {
70 int i;
71
72 for (i = 0; i < len; i++)
73 p[i] = LHALF(p[i] << sh) | (p[i + 1] >> (HALF_BITS - sh));
74 p[i] = LHALF(p[i] << sh);
75 }
76
77 /*
78 * __udivmoddi4(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
79 *
80 * We do this in base 2-sup-HALF_BITS, so that all intermediate products
81 * fit within u_int. As a consequence, the maximum length dividend and
82 * divisor are 4 `digits' in this base (they are shorter if they have
83 * leading zeros).
84 */
85 u_quad_t
__udivmoddi4(u_quad_t uq,u_quad_t vq,u_quad_t * arq)86 __udivmoddi4(u_quad_t uq, u_quad_t vq, u_quad_t *arq)
87 {
88 union uu tmp;
89 digit *u, *v, *q;
90 digit v1, v2;
91 u_int qhat, rhat, t;
92 int m, n, d, j, i;
93 digit uspace[5], vspace[5], qspace[5];
94
95 /*
96 * Take care of special cases: divide by zero, and u < v.
97 */
98 if (vq == 0) {
99 /* divide by zero. */
100 static volatile const unsigned int zero = 0;
101
102 tmp.ul[H] = tmp.ul[L] = 1 / zero;
103 if (arq)
104 *arq = uq;
105 return (tmp.q);
106 }
107 if (uq < vq) {
108 if (arq)
109 *arq = uq;
110 return (0);
111 }
112 u = &uspace[0];
113 v = &vspace[0];
114 q = &qspace[0];
115
116 /*
117 * Break dividend and divisor into digits in base B, then
118 * count leading zeros to determine m and n. When done, we
119 * will have:
120 * u = (u[1]u[2]...u[m+n]) sub B
121 * v = (v[1]v[2]...v[n]) sub B
122 * v[1] != 0
123 * 1 < n <= 4 (if n = 1, we use a different division algorithm)
124 * m >= 0 (otherwise u < v, which we already checked)
125 * m + n = 4
126 * and thus
127 * m = 4 - n <= 2
128 */
129 tmp.uq = uq;
130 u[0] = 0;
131 u[1] = HHALF(tmp.ul[H]);
132 u[2] = LHALF(tmp.ul[H]);
133 u[3] = HHALF(tmp.ul[L]);
134 u[4] = LHALF(tmp.ul[L]);
135 tmp.uq = vq;
136 v[1] = HHALF(tmp.ul[H]);
137 v[2] = LHALF(tmp.ul[H]);
138 v[3] = HHALF(tmp.ul[L]);
139 v[4] = LHALF(tmp.ul[L]);
140 for (n = 4; v[1] == 0; v++) {
141 if (--n == 1) {
142 u_int rbj; /* r*B+u[j] (not root boy jim) */
143 digit q1, q2, q3, q4;
144
145 /*
146 * Change of plan, per exercise 16.
147 * r = 0;
148 * for j = 1..4:
149 * q[j] = floor((r*B + u[j]) / v),
150 * r = (r*B + u[j]) % v;
151 * We unroll this completely here.
152 */
153 t = v[2]; /* nonzero, by definition */
154 q1 = u[1] / t;
155 rbj = COMBINE(u[1] % t, u[2]);
156 q2 = rbj / t;
157 rbj = COMBINE(rbj % t, u[3]);
158 q3 = rbj / t;
159 rbj = COMBINE(rbj % t, u[4]);
160 q4 = rbj / t;
161 if (arq)
162 *arq = rbj % t;
163 tmp.ul[H] = COMBINE(q1, q2);
164 tmp.ul[L] = COMBINE(q3, q4);
165 return (tmp.q);
166 }
167 }
168
169 /*
170 * By adjusting q once we determine m, we can guarantee that
171 * there is a complete four-digit quotient at &qspace[1] when
172 * we finally stop.
173 */
174 for (m = 4 - n; u[1] == 0; u++)
175 m--;
176 for (i = 4 - m; --i >= 0;)
177 q[i] = 0;
178 q += 4 - m;
179
180 /*
181 * Here we run Program D, translated from MIX to C and acquiring
182 * a few minor changes.
183 *
184 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
185 */
186 d = 0;
187 for (t = v[1]; t < B / 2; t <<= 1)
188 d++;
189 if (d > 0) {
190 shl(&u[0], m + n, d); /* u <<= d */
191 shl(&v[1], n - 1, d); /* v <<= d */
192 }
193 /*
194 * D2: j = 0.
195 */
196 j = 0;
197 v1 = v[1]; /* for D3 -- note that v[1..n] are constant */
198 v2 = v[2]; /* for D3 */
199 do {
200 digit uj0, uj1, uj2;
201
202 /*
203 * D3: Calculate qhat (\^q, in TeX notation).
204 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
205 * let rhat = (u[j]*B + u[j+1]) mod v[1].
206 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
207 * decrement qhat and increase rhat correspondingly.
208 * Note that if rhat >= B, v[2]*qhat < rhat*B.
209 */
210 uj0 = u[j + 0]; /* for D3 only -- note that u[j+...] change */
211 uj1 = u[j + 1]; /* for D3 only */
212 uj2 = u[j + 2]; /* for D3 only */
213 if (uj0 == v1) {
214 qhat = B;
215 rhat = uj1;
216 goto qhat_too_big;
217 } else {
218 u_int nn = COMBINE(uj0, uj1);
219 qhat = nn / v1;
220 rhat = nn % v1;
221 }
222 while (v2 * qhat > COMBINE(rhat, uj2)) {
223 qhat_too_big:
224 qhat--;
225 if ((rhat += v1) >= B)
226 break;
227 }
228 /*
229 * D4: Multiply and subtract.
230 * The variable `t' holds any borrows across the loop.
231 * We split this up so that we do not require v[0] = 0,
232 * and to eliminate a final special case.
233 */
234 for (t = 0, i = n; i > 0; i--) {
235 t = u[i + j] - v[i] * qhat - t;
236 u[i + j] = LHALF(t);
237 t = (B - HHALF(t)) & (B - 1);
238 }
239 t = u[j] - t;
240 u[j] = LHALF(t);
241 /*
242 * D5: test remainder.
243 * There is a borrow if and only if HHALF(t) is nonzero;
244 * in that (rare) case, qhat was too large (by exactly 1).
245 * Fix it by adding v[1..n] to u[j..j+n].
246 */
247 if (HHALF(t)) {
248 qhat--;
249 for (t = 0, i = n; i > 0; i--) { /* D6: add back. */
250 t += u[i + j] + v[i];
251 u[i + j] = LHALF(t);
252 t = HHALF(t);
253 }
254 u[j] = LHALF(u[j] + t);
255 }
256 q[j] = qhat;
257 } while (++j <= m); /* D7: loop on j. */
258
259 /*
260 * If caller wants the remainder, we have to calculate it as
261 * u[m..m+n] >> d (this is at most n digits and thus fits in
262 * u[m+1..m+n], but we may need more source digits).
263 */
264 if (arq) {
265 if (d) {
266 for (i = m + n; i > m; --i)
267 u[i] = (u[i] >> d) |
268 LHALF(u[i - 1] << (HALF_BITS - d));
269 u[i] = 0;
270 }
271 tmp.ul[H] = COMBINE(uspace[1], uspace[2]);
272 tmp.ul[L] = COMBINE(uspace[3], uspace[4]);
273 *arq = tmp.q;
274 }
275
276 tmp.ul[H] = COMBINE(qspace[1], qspace[2]);
277 tmp.ul[L] = COMBINE(qspace[3], qspace[4]);
278 return (tmp.q);
279 }
280
281 /*
282 * Divide two unsigned quads.
283 */
284
285 u_quad_t
__udivdi3(u_quad_t a,u_quad_t b)286 __udivdi3(u_quad_t a, u_quad_t b)
287 {
288
289 return (__udivmoddi4(a, b, (u_quad_t *)0));
290 }
291
292 /*
293 * Return remainder after dividing two unsigned quads.
294 */
295 u_quad_t
__umoddi3(u_quad_t a,u_quad_t b)296 __umoddi3(u_quad_t a, u_quad_t b)
297 {
298 u_quad_t r;
299
300 (void)__udivmoddi4(a, b, &r);
301 return (r);
302 }
303