1 /* origin: FreeBSD /usr/src/lib/msun/src/s_fmal.c */
2 /*-
3 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
4 * All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
25 * SUCH DAMAGE.
26 */
27 28 29 #include "libm.h"
30 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
31 long double fmal(long double x, long double y, long double z)
32 {
33 return fma(x, y, z);
34 }
35 #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
36 #include <fenv.h>
37 #if LDBL_MANT_DIG == 64
38 #define LASTBIT(u) (u.i.m & 1)
39 #define SPLIT (0x1p32L + 1)
40 #elif LDBL_MANT_DIG == 113
41 #define LASTBIT(u) (u.i.lo & 1)
42 #define SPLIT (0x1p57L + 1)
43 #endif
44 45 /*
46 * A struct dd represents a floating-point number with twice the precision
47 * of a long double. We maintain the invariant that "hi" stores the high-order
48 * bits of the result.
49 */
50 struct dd {
51 long double hi;
52 long double lo;
53 };
54 55 /*
56 * Compute a+b exactly, returning the exact result in a struct dd. We assume
57 * that both a and b are finite, but make no assumptions about their relative
58 * magnitudes.
59 */
60 static inline struct dd dd_add(long double a, long double b)
61 {
62 struct dd ret;
63 long double s;
64 65 ret.hi = a + b;
66 s = ret.hi - a;
67 ret.lo = (a - (ret.hi - s)) + (b - s);
68 return (ret);
69 }
70 71 /*
72 * Compute a+b, with a small tweak: The least significant bit of the
73 * result is adjusted into a sticky bit summarizing all the bits that
74 * were lost to rounding. This adjustment negates the effects of double
75 * rounding when the result is added to another number with a higher
76 * exponent. For an explanation of round and sticky bits, see any reference
77 * on FPU design, e.g.,
78 *
79 * J. Coonen. An Implementation Guide to a Proposed Standard for
80 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
81 */
82 static inline long double add_adjusted(long double a, long double b)
83 {
84 struct dd sum;
85 union ldshape u;
86 87 sum = dd_add(a, b);
88 if (sum.lo != 0) {
89 u.f = sum.hi;
90 if (!LASTBIT(u))
91 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
92 }
93 return (sum.hi);
94 }
95 96 /*
97 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
98 * that the result will be subnormal, and care is taken to ensure that
99 * double rounding does not occur.
100 */
101 static inline long double add_and_denormalize(long double a, long double b, int scale)
102 {
103 struct dd sum;
104 int bits_lost;
105 union ldshape u;
106 107 sum = dd_add(a, b);
108 109 /*
110 * If we are losing at least two bits of accuracy to denormalization,
111 * then the first lost bit becomes a round bit, and we adjust the
112 * lowest bit of sum.hi to make it a sticky bit summarizing all the
113 * bits in sum.lo. With the sticky bit adjusted, the hardware will
114 * break any ties in the correct direction.
115 *
116 * If we are losing only one bit to denormalization, however, we must
117 * break the ties manually.
118 */
119 if (sum.lo != 0) {
120 u.f = sum.hi;
121 bits_lost = -u.i.se - scale + 1;
122 if ((bits_lost != 1) ^ LASTBIT(u))
123 sum.hi = nextafterl(sum.hi, INFINITY * sum.lo);
124 }
125 return scalbnl(sum.hi, scale);
126 }
127 128 /*
129 * Compute a*b exactly, returning the exact result in a struct dd. We assume
130 * that both a and b are normalized, so no underflow or overflow will occur.
131 * The current rounding mode must be round-to-nearest.
132 */
133 static inline struct dd dd_mul(long double a, long double b)
134 {
135 struct dd ret;
136 long double ha, hb, la, lb, p, q;
137 138 p = a * SPLIT;
139 ha = a - p;
140 ha += p;
141 la = a - ha;
142 143 p = b * SPLIT;
144 hb = b - p;
145 hb += p;
146 lb = b - hb;
147 148 p = ha * hb;
149 q = ha * lb + la * hb;
150 151 ret.hi = p + q;
152 ret.lo = p - ret.hi + q + la * lb;
153 return (ret);
154 }
155 156 /*
157 * Fused multiply-add: Compute x * y + z with a single rounding error.
158 *
159 * We use scaling to avoid overflow/underflow, along with the
160 * canonical precision-doubling technique adapted from:
161 *
162 * Dekker, T. A Floating-Point Technique for Extending the
163 * Available Precision. Numer. Math. 18, 224-242 (1971).
164 */
165 long double fmal(long double x, long double y, long double z)
166 {
167 #pragma STDC FENV_ACCESS ON
168 long double xs, ys, zs, adj;
169 struct dd xy, r;
170 int oround;
171 int ex, ey, ez;
172 int spread;
173 174 /*
175 * Handle special cases. The order of operations and the particular
176 * return values here are crucial in handling special cases involving
177 * infinities, NaNs, overflows, and signed zeroes correctly.
178 */
179 if (!isfinite(x) || !isfinite(y))
180 return (x * y + z);
181 if (!isfinite(z))
182 return (z);
183 if (x == 0.0 || y == 0.0)
184 return (x * y + z);
185 if (z == 0.0)
186 return (x * y);
187 188 xs = frexpl(x, &ex);
189 ys = frexpl(y, &ey);
190 zs = frexpl(z, &ez);
191 oround = fegetround();
192 spread = ex + ey - ez;
193 194 /*
195 * If x * y and z are many orders of magnitude apart, the scaling
196 * will overflow, so we handle these cases specially. Rounding
197 * modes other than FE_TONEAREST are painful.
198 */
199 if (spread < -LDBL_MANT_DIG) {
200 #ifdef FE_INEXACT
201 feraiseexcept(FE_INEXACT);
202 #endif
203 #ifdef FE_UNDERFLOW
204 if (!isnormal(z))
205 feraiseexcept(FE_UNDERFLOW);
206 #endif
207 switch (oround) {
208 default: /* FE_TONEAREST */
209 return (z);
210 #ifdef FE_TOWARDZERO
211 case FE_TOWARDZERO:
212 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
213 return (z);
214 else
215 return (nextafterl(z, 0));
216 #endif
217 #ifdef FE_DOWNWARD
218 case FE_DOWNWARD:
219 if (x > 0.0 ^ y < 0.0)
220 return (z);
221 else
222 return (nextafterl(z, -INFINITY));
223 #endif
224 #ifdef FE_UPWARD
225 case FE_UPWARD:
226 if (x > 0.0 ^ y < 0.0)
227 return (nextafterl(z, INFINITY));
228 else
229 return (z);
230 #endif
231 }
232 }
233 if (spread <= LDBL_MANT_DIG * 2)
234 zs = scalbnl(zs, -spread);
235 else
236 zs = copysignl(LDBL_MIN, zs);
237 238 fesetround(FE_TONEAREST);
239 240 /*
241 * Basic approach for round-to-nearest:
242 *
243 * (xy.hi, xy.lo) = x * y (exact)
244 * (r.hi, r.lo) = xy.hi + z (exact)
245 * adj = xy.lo + r.lo (inexact; low bit is sticky)
246 * result = r.hi + adj (correctly rounded)
247 */
248 xy = dd_mul(xs, ys);
249 r = dd_add(xy.hi, zs);
250 251 spread = ex + ey;
252 253 if (r.hi == 0.0) {
254 /*
255 * When the addends cancel to 0, ensure that the result has
256 * the correct sign.
257 */
258 fesetround(oround);
259 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
260 return xy.hi + vzs + scalbnl(xy.lo, spread);
261 }
262 263 if (oround != FE_TONEAREST) {
264 /*
265 * There is no need to worry about double rounding in directed
266 * rounding modes.
267 * But underflow may not be raised correctly, example in downward rounding:
268 * fmal(0x1.0000000001p-16000L, 0x1.0000000001p-400L, -0x1p-16440L)
269 */
270 long double ret;
271 #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
272 int e = fetestexcept(FE_INEXACT);
273 feclearexcept(FE_INEXACT);
274 #endif
275 fesetround(oround);
276 adj = r.lo + xy.lo;
277 ret = scalbnl(r.hi + adj, spread);
278 #if defined(FE_INEXACT) && defined(FE_UNDERFLOW)
279 if (ilogbl(ret) < -16382 && fetestexcept(FE_INEXACT))
280 feraiseexcept(FE_UNDERFLOW);
281 else if (e)
282 feraiseexcept(FE_INEXACT);
283 #endif
284 return ret;
285 }
286 287 adj = add_adjusted(r.lo, xy.lo);
288 if (spread + ilogbl(r.hi) > -16383)
289 return scalbnl(r.hi + adj, spread);
290 else
291 return add_and_denormalize(r.hi, adj, spread);
292 }
293 #endif
294