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 * 4. 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
34#if defined(LIBC_SCCS) && !defined(lint)
35static char sccsid[] = "@(#)qdivrem.c	8.1 (Berkeley) 6/4/93";
36#endif /* LIBC_SCCS and not lint */
37#include <sys/cdefs.h>
38__FBSDID("$FreeBSD$");
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 "quad.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) (((u_long)(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
54typedef unsigned short digit;
55#else
56typedef u_long 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 */
64static void
65shl(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 u_long.  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 */
82u_quad_t
83__qdivrem(uq, vq, arq)
84	u_quad_t uq, vq, *arq;
85{
86	union uu tmp;
87	digit *u, *v, *q;
88	digit v1, v2;
89	u_long qhat, rhat, t;
90	int m, n, d, j, i;
91	digit uspace[5], vspace[5], qspace[5];
92
93	/*
94	 * Take care of special cases: divide by zero, and u < v.
95	 */
96	if (vq == 0) {
97		/* divide by zero. */
98		static volatile const unsigned int zero = 0;
99
100		tmp.ul[H] = tmp.ul[L] = 1 / zero;
101		if (arq)
102			*arq = uq;
103		return (tmp.q);
104	}
105	if (uq < vq) {
106		if (arq)
107			*arq = uq;
108		return (0);
109	}
110	u = &uspace[0];
111	v = &vspace[0];
112	q = &qspace[0];
113
114	/*
115	 * Break dividend and divisor into digits in base B, then
116	 * count leading zeros to determine m and n.  When done, we
117	 * will have:
118	 *	u = (u[1]u[2]...u[m+n]) sub B
119	 *	v = (v[1]v[2]...v[n]) sub B
120	 *	v[1] != 0
121	 *	1 < n <= 4 (if n = 1, we use a different division algorithm)
122	 *	m >= 0 (otherwise u < v, which we already checked)
123	 *	m + n = 4
124	 * and thus
125	 *	m = 4 - n <= 2
126	 */
127	tmp.uq = uq;
128	u[0] = 0;
129	u[1] = HHALF(tmp.ul[H]);
130	u[2] = LHALF(tmp.ul[H]);
131	u[3] = HHALF(tmp.ul[L]);
132	u[4] = LHALF(tmp.ul[L]);
133	tmp.uq = vq;
134	v[1] = HHALF(tmp.ul[H]);
135	v[2] = LHALF(tmp.ul[H]);
136	v[3] = HHALF(tmp.ul[L]);
137	v[4] = LHALF(tmp.ul[L]);
138	for (n = 4; v[1] == 0; v++) {
139		if (--n == 1) {
140			u_long rbj;	/* r*B+u[j] (not root boy jim) */
141			digit q1, q2, q3, q4;
142
143			/*
144			 * Change of plan, per exercise 16.
145			 *	r = 0;
146			 *	for j = 1..4:
147			 *		q[j] = floor((r*B + u[j]) / v),
148			 *		r = (r*B + u[j]) % v;
149			 * We unroll this completely here.
150			 */
151			t = v[2];	/* nonzero, by definition */
152			q1 = u[1] / t;
153			rbj = COMBINE(u[1] % t, u[2]);
154			q2 = rbj / t;
155			rbj = COMBINE(rbj % t, u[3]);
156			q3 = rbj / t;
157			rbj = COMBINE(rbj % t, u[4]);
158			q4 = rbj / t;
159			if (arq)
160				*arq = rbj % t;
161			tmp.ul[H] = COMBINE(q1, q2);
162			tmp.ul[L] = COMBINE(q3, q4);
163			return (tmp.q);
164		}
165	}
166
167	/*
168	 * By adjusting q once we determine m, we can guarantee that
169	 * there is a complete four-digit quotient at &qspace[1] when
170	 * we finally stop.
171	 */
172	for (m = 4 - n; u[1] == 0; u++)
173		m--;
174	for (i = 4 - m; --i >= 0;)
175		q[i] = 0;
176	q += 4 - m;
177
178	/*
179	 * Here we run Program D, translated from MIX to C and acquiring
180	 * a few minor changes.
181	 *
182	 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
183	 */
184	d = 0;
185	for (t = v[1]; t < B / 2; t <<= 1)
186		d++;
187	if (d > 0) {
188		shl(&u[0], m + n, d);		/* u <<= d */
189		shl(&v[1], n - 1, d);		/* v <<= d */
190	}
191	/*
192	 * D2: j = 0.
193	 */
194	j = 0;
195	v1 = v[1];	/* for D3 -- note that v[1..n] are constant */
196	v2 = v[2];	/* for D3 */
197	do {
198		digit uj0, uj1, uj2;
199
200		/*
201		 * D3: Calculate qhat (\^q, in TeX notation).
202		 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
203		 * let rhat = (u[j]*B + u[j+1]) mod v[1].
204		 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
205		 * decrement qhat and increase rhat correspondingly.
206		 * Note that if rhat >= B, v[2]*qhat < rhat*B.
207		 */
208		uj0 = u[j + 0];	/* for D3 only -- note that u[j+...] change */
209		uj1 = u[j + 1];	/* for D3 only */
210		uj2 = u[j + 2];	/* for D3 only */
211		if (uj0 == v1) {
212			qhat = B;
213			rhat = uj1;
214			goto qhat_too_big;
215		} else {
216			u_long n = COMBINE(uj0, uj1);
217			qhat = n / v1;
218			rhat = n % v1;
219		}
220		while (v2 * qhat > COMBINE(rhat, uj2)) {
221	qhat_too_big:
222			qhat--;
223			if ((rhat += v1) >= B)
224				break;
225		}
226		/*
227		 * D4: Multiply and subtract.
228		 * The variable `t' holds any borrows across the loop.
229		 * We split this up so that we do not require v[0] = 0,
230		 * and to eliminate a final special case.
231		 */
232		for (t = 0, i = n; i > 0; i--) {
233			t = u[i + j] - v[i] * qhat - t;
234			u[i + j] = LHALF(t);
235			t = (B - HHALF(t)) & (B - 1);
236		}
237		t = u[j] - t;
238		u[j] = LHALF(t);
239		/*
240		 * D5: test remainder.
241		 * There is a borrow if and only if HHALF(t) is nonzero;
242		 * in that (rare) case, qhat was too large (by exactly 1).
243		 * Fix it by adding v[1..n] to u[j..j+n].
244		 */
245		if (HHALF(t)) {
246			qhat--;
247			for (t = 0, i = n; i > 0; i--) { /* D6: add back. */
248				t += u[i + j] + v[i];
249				u[i + j] = LHALF(t);
250				t = HHALF(t);
251			}
252			u[j] = LHALF(u[j] + t);
253		}
254		q[j] = qhat;
255	} while (++j <= m);		/* D7: loop on j. */
256
257	/*
258	 * If caller wants the remainder, we have to calculate it as
259	 * u[m..m+n] >> d (this is at most n digits and thus fits in
260	 * u[m+1..m+n], but we may need more source digits).
261	 */
262	if (arq) {
263		if (d) {
264			for (i = m + n; i > m; --i)
265				u[i] = (u[i] >> d) |
266				    LHALF(u[i - 1] << (HALF_BITS - d));
267			u[i] = 0;
268		}
269		tmp.ul[H] = COMBINE(uspace[1], uspace[2]);
270		tmp.ul[L] = COMBINE(uspace[3], uspace[4]);
271		*arq = tmp.q;
272	}
273
274	tmp.ul[H] = COMBINE(qspace[1], qspace[2]);
275	tmp.ul[L] = COMBINE(qspace[3], qspace[4]);
276	return (tmp.q);
277}
278