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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11 * This product includes software developed by the University of
12 * California, Lawrence Berkeley Laboratory.
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38 * @(#)fpu_div.c 8.1 (Berkeley) 6/11/93
39 * $NetBSD: fpu_div.c,v 1.2 1994/11/20 20:52:38 deraadt Exp $
42 #include <sys/cdefs.h>
43 __FBSDID("$FreeBSD$");
46 * Perform an FPU divide (return x / y).
49 #include <sys/types.h>
51 #include <machine/frame.h>
52 #include <machine/fp.h>
53 #include <machine/fsr.h>
55 #include "fpu_arith.h"
57 #include "fpu_extern.h"
60 * Division of normal numbers is done as follows:
62 * x and y are floating point numbers, i.e., in the form 1.bbbb * 2^e.
63 * If X and Y are the mantissas (1.bbbb's), the quotient is then:
65 * q = (X / Y) * 2^((x exponent) - (y exponent))
67 * Since X and Y are both in [1.0,2.0), the quotient's mantissa (X / Y)
68 * will be in [0.5,2.0). Moreover, it will be less than 1.0 if and only
69 * if X < Y. In that case, it will have to be shifted left one bit to
70 * become a normal number, and the exponent decremented. Thus, the
71 * desired exponent is:
73 * left_shift = x->fp_mant < y->fp_mant;
74 * result_exp = x->fp_exp - y->fp_exp - left_shift;
76 * The quotient mantissa X/Y can then be computed one bit at a time
77 * using the following algorithm:
79 * Q = 0; -- Initial quotient.
80 * R = X; -- Initial remainder,
81 * if (left_shift) -- but fixed up in advance.
83 * for (bit = FP_NMANT; --bit >= 0; R *= 2) {
90 * The subtraction R -= Y always removes the uppermost bit from R (and
91 * can sometimes remove additional lower-order 1 bits); this proof is
94 * This loop correctly calculates the guard and round bits since they are
95 * included in the expanded internal representation. The sticky bit
96 * is to be set if and only if any other bits beyond guard and round
97 * would be set. From the above it is obvious that this is true if and
98 * only if the remainder R is nonzero when the loop terminates.
100 * Examining the loop above, we can see that the quotient Q is built
101 * one bit at a time ``from the top down''. This means that we can
102 * dispense with the multi-word arithmetic and just build it one word
103 * at a time, writing each result word when it is done.
105 * Furthermore, since X and Y are both in [1.0,2.0), we know that,
106 * initially, R >= Y. (Recall that, if X < Y, R is set to X * 2 and
107 * is therefore at in [2.0,4.0).) Thus Q is sure to have bit FP_NMANT-1
108 * set, and R can be set initially to either X - Y (when X >= Y) or
109 * 2X - Y (when X < Y). In addition, comparing R and Y is difficult,
110 * so we will simply calculate R - Y and see if that underflows.
111 * This leads to the following revised version of the algorithm:
117 * result_exp = x->fp_exp - y->fp_exp;
122 * result_exp = x->fp_exp - y->fp_exp - 1;
133 * } while ((bit >>= 1) != 0);
135 * for (i = 1; i < 4; i++) {
136 * q = 0, bit = 1 << 31;
144 * } while ((bit >>= 1) != 0);
148 * This can be refined just a bit further by moving the `R <<= 1'
149 * calculations to the front of the do-loops and eliding the first one.
150 * The process can be terminated immediately whenever R becomes 0, but
151 * this is relatively rare, and we do not bother.
158 struct fpn *x = &fe->fe_f1, *y = &fe->fe_f2;
160 u_int r0, r1, r2, r3, d0, d1, d2, d3, y0, y1, y2, y3;
164 * Since divide is not commutative, we cannot just use ORDER.
165 * Check either operand for NaN first; if there is at least one,
166 * order the signalling one (if only one) onto the right, then
167 * return it. Otherwise we have the following cases:
169 * Inf / Inf = NaN, plus NV exception
170 * Inf / num = Inf [i.e., return x #]
171 * Inf / 0 = Inf [i.e., return x #]
172 * 0 / Inf = 0 [i.e., return x #]
173 * 0 / num = 0 [i.e., return x #]
174 * 0 / 0 = NaN, plus NV exception
176 * num / num = num (do the divide)
177 * num / 0 = Inf #, plus DZ exception
179 * # Sign of result is XOR of operand signs.
181 if (ISNAN(x) || ISNAN(y)) {
185 if (ISINF(x) || ISZERO(x)) {
186 if (x->fp_class == y->fp_class)
187 return (__fpu_newnan(fe));
188 x->fp_sign ^= y->fp_sign;
192 x->fp_sign ^= y->fp_sign;
194 x->fp_class = FPC_ZERO;
199 x->fp_class = FPC_INF;
204 * Macros for the divide. See comments at top for algorithm.
205 * Note that we expand R, D, and Y here.
208 #define SUBTRACT /* D = R - Y */ \
209 FPU_SUBS(d3, r3, y3); FPU_SUBCS(d2, r2, y2); \
210 FPU_SUBCS(d1, r1, y1); FPU_SUBC(d0, r0, y0)
212 #define NONNEGATIVE /* D >= 0 */ \
215 #ifdef FPU_SHL1_BY_ADD
216 #define SHL1 /* R <<= 1 */ \
217 FPU_ADDS(r3, r3, r3); FPU_ADDCS(r2, r2, r2); \
218 FPU_ADDCS(r1, r1, r1); FPU_ADDC(r0, r0, r0)
221 r0 = (r0 << 1) | (r1 >> 31), r1 = (r1 << 1) | (r2 >> 31), \
222 r2 = (r2 << 1) | (r3 >> 31), r3 <<= 1
225 #define LOOP /* do ... while (bit >>= 1) */ \
231 r0 = d0, r1 = d1, r2 = d2, r3 = d3; \
233 } while ((bit >>= 1) != 0)
235 #define WORD(r, i) /* calculate r->fp_mant[i] */ \
241 /* Setup. Note that we put our result in x. */
254 x->fp_exp -= y->fp_exp;
255 r0 = d0, r1 = d1, r2 = d2, r3 = d3;
259 x->fp_exp -= y->fp_exp + 1;
267 x->fp_sticky = r0 | r1 | r2 | r3;