1 /*-
2 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE.
25 */
26
27 //__FBSDID("$FreeBSD: src/lib/msun/src/s_fmal.c,v 1.7 2011/10/21 06:30:43 das Exp $");
28
29
30 /*
31 * A struct dd represents a floating-point number with twice the precision
32 * of a long double. We maintain the invariant that "hi" stores the high-order
33 * bits of the result.
34 */
35 struct dd {
36 long double hi;
37 long double lo;
38 };
39
40 /*
41 * Compute a+b exactly, returning the exact result in a struct dd. We assume
42 * that both a and b are finite, but make no assumptions about their relative
43 * magnitudes.
44 */
45 static inline struct dd
dd_add(long double a,long double b)46 dd_add(long double a, long double b)
47 {
48 struct dd ret;
49 long double s;
50
51 ret.hi = a + b;
52 s = ret.hi - a;
53 ret.lo = (a - (ret.hi - s)) + (b - s);
54 return (ret);
55 }
56
57 /*
58 * Compute a+b, with a small tweak: The least significant bit of the
59 * result is adjusted into a sticky bit summarizing all the bits that
60 * were lost to rounding. This adjustment negates the effects of double
61 * rounding when the result is added to another number with a higher
62 * exponent. For an explanation of round and sticky bits, see any reference
63 * on FPU design, e.g.,
64 *
65 * J. Coonen. An Implementation Guide to a Proposed Standard for
66 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
67 */
68 static inline long double
add_adjusted(long double a,long double b)69 add_adjusted(long double a, long double b)
70 {
71 struct dd sum;
72 union IEEEl2bits u;
73
74 sum = dd_add(a, b);
75 if (sum.lo != 0) {
76 u.e = sum.hi;
77 if ((u.bits.manl & 1) == 0)
78 sum.hi = nextafterl(sum.hi, (long double)INFINITY * sum.lo);
79 }
80 return (sum.hi);
81 }
82
83 /*
84 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
85 * that the result will be subnormal, and care is taken to ensure that
86 * double rounding does not occur.
87 */
88 static inline long double
add_and_denormalize(long double a,long double b,int scale)89 add_and_denormalize(long double a, long double b, int scale)
90 {
91 struct dd sum;
92 int bits_lost;
93 union IEEEl2bits u;
94
95 sum = dd_add(a, b);
96
97 /*
98 * If we are losing at least two bits of accuracy to denormalization,
99 * then the first lost bit becomes a round bit, and we adjust the
100 * lowest bit of sum.hi to make it a sticky bit summarizing all the
101 * bits in sum.lo. With the sticky bit adjusted, the hardware will
102 * break any ties in the correct direction.
103 *
104 * If we are losing only one bit to denormalization, however, we must
105 * break the ties manually.
106 */
107 if (sum.lo != 0) {
108 u.e = sum.hi;
109 bits_lost = -u.bits.exp - scale + 1;
110 if ((bits_lost != 1) ^ (int)(u.bits.manl & 1))
111 sum.hi = nextafterl(sum.hi, (long double)INFINITY * sum.lo);
112 }
113 return (ldexpl(sum.hi, scale));
114 }
115
116 /*
117 * Compute a*b exactly, returning the exact result in a struct dd. We assume
118 * that both a and b are normalized, so no underflow or overflow will occur.
119 * The current rounding mode must be round-to-nearest.
120 */
121 static inline struct dd
dd_mul(long double a,long double b)122 dd_mul(long double a, long double b)
123 {
124 #if LDBL_MANT_DIG == 64
125 static const long double split = 0x1p32L + 1.0L;
126 #elif LDBL_MANT_DIG == 113
127 static const long double split = 0x1p57L + 1.0L;
128 #endif
129 struct dd ret;
130 long double ha, hb, la, lb, p, q;
131
132 p = a * split;
133 ha = a - p;
134 ha += p;
135 la = a - ha;
136
137 p = b * split;
138 hb = b - p;
139 hb += p;
140 lb = b - hb;
141
142 p = ha * hb;
143 q = ha * lb + la * hb;
144
145 ret.hi = p + q;
146 ret.lo = p - ret.hi + q + la * lb;
147 return (ret);
148 }
149
150 #ifdef _WANT_MATH_ERRNO
151 static long double
_scalbnl_no_errno(long double x,int n)152 _scalbnl_no_errno(long double x, int n)
153 {
154 int save_errno = errno;
155 x = scalbnl(x, n);
156 errno = save_errno;
157 return x;
158 }
159 #else
160 #define _scalbnl_no_errno(a,b) scalbnl(a,b)
161 #endif
162
163 #ifdef __clang__
164 #pragma STDC FP_CONTRACT OFF
165 #endif
166
167 /*
168 * Fused multiply-add: Compute x * y + z with a single rounding error.
169 *
170 * We use scaling to avoid overflow/underflow, along with the
171 * canonical precision-doubling technique adapted from:
172 *
173 * Dekker, T. A Floating-Point Technique for Extending the
174 * Available Precision. Numer. Math. 18, 224-242 (1971).
175 */
176 long double
fmal(long double x,long double y,long double z)177 fmal(long double x, long double y, long double z)
178 {
179 long double xs, ys, zs, adj;
180 struct dd xy, r;
181 int oround;
182 int ex, ey, ez;
183 int spread;
184
185 /*
186 * Handle special cases. The order of operations and the particular
187 * return values here are crucial in handling special cases involving
188 * infinities, NaNs, overflows, and signed zeroes correctly.
189 */
190 if (x == 0.0L || y == 0.0L)
191 return (x * y + z);
192 if (z == 0.0L)
193 return (x * y);
194 if (!isfinite(x) || !isfinite(y))
195 return (x * y + z);
196 if (!isfinite(z))
197 return (z);
198
199 xs = frexpl(x, &ex);
200 ys = frexpl(y, &ey);
201 zs = frexpl(z, &ez);
202 oround = fegetround();
203 spread = ex + ey - ez;
204
205 /*
206 * If x * y and z are many orders of magnitude apart, the scaling
207 * will overflow, so we handle these cases specially. Rounding
208 * modes other than FE_TONEAREST are painful.
209 */
210 if (spread < -LDBL_MANT_DIG) {
211 #ifdef FE_INEXACT
212 feraiseexcept(FE_INEXACT);
213 #endif
214 #ifdef FE_UNDERFLOW
215 if (!isnormal(z))
216 feraiseexcept(FE_UNDERFLOW);
217 #endif
218 switch (oround) {
219 default:
220 return (z);
221 #ifdef FE_TOWARDZERO
222 case FE_TOWARDZERO:
223 if ((x > 0.0L) ^ (y < 0.0L) ^ (z < 0.0L))
224 return (z);
225 else
226 return (nextafterl(z, 0));
227 #endif
228 #ifdef FE_DOWNWARD
229 case FE_DOWNWARD:
230 if ((x > 0.0L) ^ (y < 0.0L))
231 return (z);
232 else
233 return (nextafterl(z, -(long double)INFINITY));
234 #endif
235 #ifdef FE_UPWARD
236 case FE_UPWARD:
237 if ((x > 0.0L) ^ (y < 0.0L))
238 return (nextafterl(z, (long double)INFINITY));
239 else
240 return (z);
241 #endif
242 }
243 }
244 if (spread <= LDBL_MANT_DIG * 2)
245 zs = _scalbnl_no_errno(zs, -spread);
246 else
247 zs = copysignl(LDBL_MIN, zs);
248
249 fesetround(FE_TONEAREST);
250
251 /*
252 * Basic approach for round-to-nearest:
253 *
254 * (xy.hi, xy.lo) = x * y (exact)
255 * (r.hi, r.lo) = xy.hi + z (exact)
256 * adj = xy.lo + r.lo (inexact; low bit is sticky)
257 * result = r.hi + adj (correctly rounded)
258 */
259 xy = dd_mul(xs, ys);
260 r = dd_add(xy.hi, zs);
261
262 spread = ex + ey;
263
264 if (r.hi == 0.0L) {
265 /*
266 * When the addends cancel to 0, ensure that the result has
267 * the correct sign.
268 */
269 fesetround(oround);
270 volatile long double vzs = zs; /* XXX gcc CSE bug workaround */
271 return (xy.hi + vzs + _scalbnl_no_errno(xy.lo, spread));
272 }
273
274 if (oround != FE_TONEAREST) {
275 /*
276 * There is no need to worry about double rounding in directed
277 * rounding modes.
278 */
279 fesetround(oround);
280 adj = r.lo + xy.lo;
281 return (_scalbnl_no_errno(r.hi + adj, spread));
282 }
283
284 adj = add_adjusted(r.lo, xy.lo);
285 if (spread + ilogbl(r.hi) > -16383)
286 return (_scalbnl_no_errno(r.hi + adj, spread));
287 else
288 return (add_and_denormalize(r.hi, adj, spread));
289 }
290
