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
54typedef unsigned short digit;
55#else
56typedef 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 */
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 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 */
82u_longlong_t
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