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.
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10 * modification, are permitted provided that the following conditions
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13 * notice, this list of conditions and the following disclaimer.
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15 * notice, this list of conditions and the following disclaimer in the
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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.
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33 * $FreeBSD: src/sys/libkern/qdivrem.c,v 1.8 1999/08/28 00:46:35 peter Exp $
34 * $DragonFly: src/sys/libkern/qdivrem.c,v 1.4 2004/01/26 11:09:44 joerg Exp $
38 * Multiprecision divide. This algorithm is from Knuth vol. 2 (2nd ed),
39 * section 4.3.1, pp. 257--259.
42 #include <libkern/quad.h>
44 #define B (1 << HALF_BITS) /* digit base */
46 /* Combine two `digits' to make a single two-digit number. */
47 #define COMBINE(a, b) (((u_long)(a) << HALF_BITS) | (b))
49 /* select a type for digits in base B: use unsigned short if they fit */
50 #if ULONG_MAX == 0xffffffff && USHRT_MAX >= 0xffff
51 typedef unsigned short digit
;
57 * Shift p[0]..p[len] left `sh' bits, ignoring any bits that
58 * `fall out' the left (there never will be any such anyway).
59 * We may assume len >= 0. NOTE THAT THIS WRITES len+1 DIGITS.
62 shl(digit
*p
, int len
, int sh
)
66 for (i
= 0; i
< len
; i
++)
67 p
[i
] = LHALF(p
[i
] << sh
) | (p
[i
+ 1] >> (HALF_BITS
- sh
));
68 p
[i
] = LHALF(p
[i
] << sh
);
72 * __qdivrem(u, v, rem) returns u/v and, optionally, sets *rem to u%v.
74 * We do this in base 2-sup-HALF_BITS, so that all intermediate products
75 * fit within u_long. As a consequence, the maximum length dividend and
76 * divisor are 4 `digits' in this base (they are shorter if they have
80 __qdivrem(u_quad_t uq
, u_quad_t vq
, u_quad_t
*arq
)
87 digit uspace
[5], vspace
[5], qspace
[5];
90 * Take care of special cases: divide by zero, and u < v.
94 static volatile const unsigned int zero
= 0;
96 tmp
.ul
[H
] = tmp
.ul
[L
] = 1 / zero
;
111 * Break dividend and divisor into digits in base B, then
112 * count leading zeros to determine m and n. When done, we
114 * u = (u[1]u[2]...u[m+n]) sub B
115 * v = (v[1]v[2]...v[n]) sub B
117 * 1 < n <= 4 (if n = 1, we use a different division algorithm)
118 * m >= 0 (otherwise u < v, which we already checked)
125 u
[1] = HHALF(tmp
.ul
[H
]);
126 u
[2] = LHALF(tmp
.ul
[H
]);
127 u
[3] = HHALF(tmp
.ul
[L
]);
128 u
[4] = LHALF(tmp
.ul
[L
]);
130 v
[1] = HHALF(tmp
.ul
[H
]);
131 v
[2] = LHALF(tmp
.ul
[H
]);
132 v
[3] = HHALF(tmp
.ul
[L
]);
133 v
[4] = LHALF(tmp
.ul
[L
]);
134 for (n
= 4; v
[1] == 0; v
++) {
136 u_long rbj
; /* r*B+u[j] (not root boy jim) */
137 digit q1
, q2
, q3
, q4
;
140 * Change of plan, per exercise 16.
143 * q[j] = floor((r*B + u[j]) / v),
144 * r = (r*B + u[j]) % v;
145 * We unroll this completely here.
147 t
= v
[2]; /* nonzero, by definition */
149 rbj
= COMBINE(u
[1] % t
, u
[2]);
151 rbj
= COMBINE(rbj
% t
, u
[3]);
153 rbj
= COMBINE(rbj
% t
, u
[4]);
157 tmp
.ul
[H
] = COMBINE(q1
, q2
);
158 tmp
.ul
[L
] = COMBINE(q3
, q4
);
164 * By adjusting q once we determine m, we can guarantee that
165 * there is a complete four-digit quotient at &qspace[1] when
168 for (m
= 4 - n
; u
[1] == 0; u
++)
170 for (i
= 4 - m
; --i
>= 0;)
175 * Here we run Program D, translated from MIX to C and acquiring
176 * a few minor changes.
178 * D1: choose multiplier 1 << d to ensure v[1] >= B/2.
181 for (t
= v
[1]; t
< B
/ 2; t
<<= 1)
184 shl(&u
[0], m
+ n
, d
); /* u <<= d */
185 shl(&v
[1], n
- 1, d
); /* v <<= d */
191 v1
= v
[1]; /* for D3 -- note that v[1..n] are constant */
192 v2
= v
[2]; /* for D3 */
197 * D3: Calculate qhat (\^q, in TeX notation).
198 * Let qhat = min((u[j]*B + u[j+1])/v[1], B-1), and
199 * let rhat = (u[j]*B + u[j+1]) mod v[1].
200 * While rhat < B and v[2]*qhat > rhat*B+u[j+2],
201 * decrement qhat and increase rhat correspondingly.
202 * Note that if rhat >= B, v[2]*qhat < rhat*B.
204 uj0
= u
[j
+ 0]; /* for D3 only -- note that u[j+...] change */
205 uj1
= u
[j
+ 1]; /* for D3 only */
206 uj2
= u
[j
+ 2]; /* for D3 only */
212 u_long nn
= COMBINE(uj0
, uj1
);
216 while (v2
* qhat
> COMBINE(rhat
, uj2
)) {
219 if ((rhat
+= v1
) >= B
)
223 * D4: Multiply and subtract.
224 * The variable `t' holds any borrows across the loop.
225 * We split this up so that we do not require v[0] = 0,
226 * and to eliminate a final special case.
228 for (t
= 0, i
= n
; i
> 0; i
--) {
229 t
= u
[i
+ j
] - v
[i
] * qhat
- t
;
231 t
= (B
- HHALF(t
)) & (B
- 1);
236 * D5: test remainder.
237 * There is a borrow if and only if HHALF(t) is nonzero;
238 * in that (rare) case, qhat was too large (by exactly 1).
239 * Fix it by adding v[1..n] to u[j..j+n].
243 for (t
= 0, i
= n
; i
> 0; i
--) { /* D6: add back. */
244 t
+= u
[i
+ j
] + v
[i
];
248 u
[j
] = LHALF(u
[j
] + t
);
251 } while (++j
<= m
); /* D7: loop on j. */
254 * If caller wants the remainder, we have to calculate it as
255 * u[m..m+n] >> d (this is at most n digits and thus fits in
256 * u[m+1..m+n], but we may need more source digits).
260 for (i
= m
+ n
; i
> m
; --i
)
262 LHALF(u
[i
- 1] << (HALF_BITS
- d
));
265 tmp
.ul
[H
] = COMBINE(uspace
[1], uspace
[2]);
266 tmp
.ul
[L
] = COMBINE(uspace
[3], uspace
[4]);
270 tmp
.ul
[H
] = COMBINE(qspace
[1], qspace
[2]);
271 tmp
.ul
[L
] = COMBINE(qspace
[3], qspace
[4]);