2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
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.
9 * Redistribution and use in source and binary forms, with or without
10 * modification, are permitted provided that the following conditions
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.
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
33 * @(#)qdivrem.c 8.1 (Berkeley) 6/4/93
34 * $FreeBSD: src/lib/libc/quad/qdivrem.c,v 1.4 2007/01/09 00:28:03 imp Exp $
35 * $DragonFly: src/lib/libc/quad/qdivrem.c,v 1.5 2005/11/20 09:18:37 swildner Exp $
39 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed),
40 * section 4.3.1, pp. 257--259.
45 #define B (1 << HALF_BITS) /* digit base */
47 /* Combine two `digits' to make a single two-digit number. */
48 #define COMBINE(a, b) (((u_long)(a) << HALF_BITS) | (b))
50 /* select a type for digits in base B: use unsigned short if they fit */
51 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff
52 typedef unsigned short digit
;
58 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
59 * `fall out' the left (there never will be any such anyway).
60 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS.
63 shl(digit
*p
, int len
, int sh
)
67 for (i
= 0; i
< len
; i
++)
68 p
[i
] = LHALF(p
[i
] << sh
) | (p
[i
+ 1] >> (HALF_BITS
- sh
));
69 p
[i
] = LHALF(p
[i
] << sh
);
73 * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
75 * We do this in base 2-sup-HALF_BITS, so that all intermediate products
76 * fit within u_long. As a consequence, the maximum length dividend and
77 * divisor are 4 `digits' in this base (they are shorter if they have
81 __qdivrem(u_quad_t uq
, u_quad_t vq
, u_quad_t
*arq
)
88 digit uspace
[5], vspace
[5], qspace
[5];
91 * Take care of special cases: divide by zero, and u < v.
95 static volatile const unsigned int zero
= 0;
97 tmp
.ul
[H
] = tmp
.ul
[L
] = 1 / zero
;
112 * Break dividend and divisor into digits in base B, then
113 * count leading zeros to determine m and n. When done, we
115 * u = (u[1]u[2]...u[m+n]) sub B
116 * v = (v[1]v[2]...v[n]) sub B
118 * 1 < n <= 4 (if n = 1, we use a different division algorithm)
119 * m >= 0 (otherwise u < v, which we already checked)
126 u
[1] = HHALF(tmp
.ul
[H
]);
127 u
[2] = LHALF(tmp
.ul
[H
]);
128 u
[3] = HHALF(tmp
.ul
[L
]);
129 u
[4] = LHALF(tmp
.ul
[L
]);
131 v
[1] = HHALF(tmp
.ul
[H
]);
132 v
[2] = LHALF(tmp
.ul
[H
]);
133 v
[3] = HHALF(tmp
.ul
[L
]);
134 v
[4] = LHALF(tmp
.ul
[L
]);
135 for (n
= 4; v
[1] == 0; v
++) {
137 u_long rbj
; /* r*B+u[j] (not root boy jim) */
138 digit q1
, q2
, q3
, q4
;
141 * Change of plan, per exercise 16.
144 * q[j] = floor((r*B + u[j]) / v),
145 * r = (r*B + u[j]) % v;
146 * We unroll this completely here.
148 t
= v
[2]; /* nonzero, by definition */
150 rbj
= COMBINE(u
[1] % t
, u
[2]);
152 rbj
= COMBINE(rbj
% t
, u
[3]);
154 rbj
= COMBINE(rbj
% t
, u
[4]);
158 tmp
.ul
[H
] = COMBINE(q1
, q2
);
159 tmp
.ul
[L
] = COMBINE(q3
, q4
);
165 * By adjusting q once we determine m, we can guarantee that
166 * there is a complete four-digit quotient at &qspace[1] when
169 for (m
= 4 - n
; u
[1] == 0; u
++)
171 for (i
= 4 - m
; --i
>= 0;)
176 * Here we run Program D, translated from MIX to C and acquiring
177 * a few minor changes.
179 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
182 for (t
= v
[1]; t
< B
/ 2; t
<<= 1)
185 shl(&u
[0], m
+ n
, d
); /* u <<= d */
186 shl(&v
[1], n
- 1, d
); /* v <<= d */
192 v1
= v
[1]; /* for D3 -- note that v[1..n] are constant */
193 v2
= v
[2]; /* for D3 */
198 * D3: Calculate qhat (\^q, in TeX notation).
199 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
200 * let rhat = (u[j]*B + u[j+1]) mod v[1].
201 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
202 * decrement qhat and increase rhat correspondingly.
203 * Note that if rhat >= B, v[2]*qhat < rhat*B.
205 uj0
= u
[j
+ 0]; /* for D3 only -- note that u[j+...] change */
206 uj1
= u
[j
+ 1]; /* for D3 only */
207 uj2
= u
[j
+ 2]; /* for D3 only */
213 u_long n1
= COMBINE(uj0
, uj1
);
217 while (v2
* qhat
> COMBINE(rhat
, uj2
)) {
220 if ((rhat
+= v1
) >= B
)
224 * D4: Multiply and subtract.
225 * The variable `t' holds any borrows across the loop.
226 * We split this up so that we do not require v[0] = 0,
227 * and to eliminate a final special case.
229 for (t
= 0, i
= n
; i
> 0; i
--) {
230 t
= u
[i
+ j
] - v
[i
] * qhat
- t
;
232 t
= (B
- HHALF(t
)) & (B
- 1);
237 * D5: test remainder.
238 * There is a borrow if and only if HHALF(t) is nonzero;
239 * in that (rare) case, qhat was too large (by exactly 1).
240 * Fix it by adding v[1..n] to u[j..j+n].
244 for (t
= 0, i
= n
; i
> 0; i
--) { /* D6: add back. */
245 t
+= u
[i
+ j
] + v
[i
];
249 u
[j
] = LHALF(u
[j
] + t
);
252 } while (++j
<= m
); /* D7: loop on j. */
255 * If caller wants the remainder, we have to calculate it as
256 * u[m..m+n] >> d (this is at most n digits and thus fits in
257 * u[m+1..m+n], but we may need more source digits).
261 for (i
= m
+ n
; i
> m
; --i
)
263 LHALF(u
[i
- 1] << (HALF_BITS
- d
));
266 tmp
.ul
[H
] = COMBINE(uspace
[1], uspace
[2]);
267 tmp
.ul
[L
] = COMBINE(uspace
[3], uspace
[4]);
271 tmp
.ul
[H
] = COMBINE(qspace
[1], qspace
[2]);
272 tmp
.ul
[L
] = COMBINE(qspace
[3], qspace
[4]);