ipiq: Add simple IPI latency measure sysctls (2)
[dragonfly.git] / lib / libm / src / s_fma.c
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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 *
01429808 26 * $FreeBSD: head/lib/msun/src/s_fma.c 252170 2013-06-24 19:12:17Z eadler $
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27 */
28
29#include <fenv.h>
30#include <float.h>
31#include <math.h>
32
33#include "math_private.h"
34
35/*
36 * A struct dd represents a floating-point number with twice the precision
37 * of a double. We maintain the invariant that "hi" stores the 53 high-order
38 * bits of the result.
39 */
40struct dd {
41 double hi;
42 double lo;
43};
44
45/*
46 * Compute a+b exactly, returning the exact result in a struct dd. We assume
47 * that both a and b are finite, but make no assumptions about their relative
48 * magnitudes.
49 */
50static inline struct dd
51dd_add(double a, double b)
52{
53 struct dd ret;
54 double s;
55
56 ret.hi = a + b;
57 s = ret.hi - a;
58 ret.lo = (a - (ret.hi - s)) + (b - s);
59 return (ret);
60}
61
62/*
63 * Compute a+b, with a small tweak: The least significant bit of the
64 * result is adjusted into a sticky bit summarizing all the bits that
65 * were lost to rounding. This adjustment negates the effects of double
66 * rounding when the result is added to another number with a higher
67 * exponent. For an explanation of round and sticky bits, see any reference
68 * on FPU design, e.g.,
69 *
70 * J. Coonen. An Implementation Guide to a Proposed Standard for
71 * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
72 */
73static inline double
74add_adjusted(double a, double b)
75{
76 struct dd sum;
77 uint64_t hibits, lobits;
78
79 sum = dd_add(a, b);
80 if (sum.lo != 0) {
81 EXTRACT_WORD64(hibits, sum.hi);
82 if ((hibits & 1) == 0) {
83 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
84 EXTRACT_WORD64(lobits, sum.lo);
85 hibits += 1 - ((hibits ^ lobits) >> 62);
86 INSERT_WORD64(sum.hi, hibits);
87 }
88 }
89 return (sum.hi);
90}
91
92/*
93 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
94 * that the result will be subnormal, and care is taken to ensure that
95 * double rounding does not occur.
96 */
97static inline double
98add_and_denormalize(double a, double b, int scale)
99{
100 struct dd sum;
101 uint64_t hibits, lobits;
102 int bits_lost;
103
104 sum = dd_add(a, b);
105
106 /*
107 * If we are losing at least two bits of accuracy to denormalization,
108 * then the first lost bit becomes a round bit, and we adjust the
109 * lowest bit of sum.hi to make it a sticky bit summarizing all the
110 * bits in sum.lo. With the sticky bit adjusted, the hardware will
111 * break any ties in the correct direction.
112 *
113 * If we are losing only one bit to denormalization, however, we must
114 * break the ties manually.
115 */
116 if (sum.lo != 0) {
117 EXTRACT_WORD64(hibits, sum.hi);
118 bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
01429808 119 if ((bits_lost != 1) ^ (int)(hibits & 1)) {
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120 /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
121 EXTRACT_WORD64(lobits, sum.lo);
122 hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
123 INSERT_WORD64(sum.hi, hibits);
124 }
125 }
126 return (ldexp(sum.hi, scale));
127}
128
129/*
130 * Compute a*b exactly, returning the exact result in a struct dd. We assume
131 * that both a and b are normalized, so no underflow or overflow will occur.
132 * The current rounding mode must be round-to-nearest.
133 */
134static inline struct dd
135dd_mul(double a, double b)
136{
137 static const double split = 0x1p27 + 1.0;
138 struct dd ret;
139 double ha, hb, la, lb, p, q;
140
141 p = a * split;
142 ha = a - p;
143 ha += p;
144 la = a - ha;
145
146 p = b * split;
147 hb = b - p;
148 hb += p;
149 lb = b - hb;
150
151 p = ha * hb;
152 q = ha * lb + la * hb;
153
154 ret.hi = p + q;
155 ret.lo = p - ret.hi + q + la * lb;
156 return (ret);
157}
158
159/*
160 * Fused multiply-add: Compute x * y + z with a single rounding error.
161 *
162 * We use scaling to avoid overflow/underflow, along with the
163 * canonical precision-doubling technique adapted from:
164 *
165 * Dekker, T. A Floating-Point Technique for Extending the
166 * Available Precision. Numer. Math. 18, 224-242 (1971).
167 *
168 * This algorithm is sensitive to the rounding precision. FPUs such
169 * as the i387 must be set in double-precision mode if variables are
170 * to be stored in FP registers in order to avoid incorrect results.
171 * This is the default on FreeBSD, but not on many other systems.
172 *
173 * Hardware instructions should be used on architectures that support it,
174 * since this implementation will likely be several times slower.
175 */
176double
177fma(double x, double y, double z)
178{
179 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 == 0.0 || y == 0.0)
191 return (x * y + z);
192 if (z == 0.0)
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 = frexp(x, &ex);
200 ys = frexp(y, &ey);
201 zs = frexp(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 < -DBL_MANT_DIG) {
211 feraiseexcept(FE_INEXACT);
212 if (!isnormal(z))
213 feraiseexcept(FE_UNDERFLOW);
214 switch (oround) {
215 case FE_TONEAREST:
216 return (z);
217 case FE_TOWARDZERO:
6ff43c94 218 if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
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219 return (z);
220 else
221 return (nextafter(z, 0));
222 case FE_DOWNWARD:
6ff43c94 223 if (x > 0.0 ^ y < 0.0)
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224 return (z);
225 else
226 return (nextafter(z, -INFINITY));
227 default: /* FE_UPWARD */
6ff43c94 228 if (x > 0.0 ^ y < 0.0)
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229 return (nextafter(z, INFINITY));
230 else
231 return (z);
232 }
233 }
234 if (spread <= DBL_MANT_DIG * 2)
235 zs = ldexp(zs, -spread);
236 else
237 zs = copysign(DBL_MIN, zs);
238
239 fesetround(FE_TONEAREST);
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240 /* work around clang bug 8100 */
241 volatile double vxs = xs;
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242
243 /*
244 * Basic approach for round-to-nearest:
245 *
246 * (xy.hi, xy.lo) = x * y (exact)
247 * (r.hi, r.lo) = xy.hi + z (exact)
248 * adj = xy.lo + r.lo (inexact; low bit is sticky)
249 * result = r.hi + adj (correctly rounded)
250 */
6ff43c94 251 xy = dd_mul(vxs, ys);
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252 r = dd_add(xy.hi, zs);
253
254 spread = ex + ey;
255
256 if (r.hi == 0.0) {
257 /*
258 * When the addends cancel to 0, ensure that the result has
259 * the correct sign.
260 */
261 fesetround(oround);
262 volatile double vzs = zs; /* XXX gcc CSE bug workaround */
263 return (xy.hi + vzs + ldexp(xy.lo, spread));
264 }
265
266 if (oround != FE_TONEAREST) {
267 /*
268 * There is no need to worry about double rounding in directed
269 * rounding modes.
270 */
271 fesetround(oround);
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272 /* work around clang bug 8100 */
273 volatile double vrlo = r.lo;
274 adj = vrlo + xy.lo;
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275 return (ldexp(r.hi + adj, spread));
276 }
277
278 adj = add_adjusted(r.lo, xy.lo);
279 if (spread + ilogb(r.hi) > -1023)
280 return (ldexp(r.hi + adj, spread));
281 else
282 return (add_and_denormalize(r.hi, adj, spread));
283}
284
285#if (LDBL_MANT_DIG == 53)
286__weak_reference(fma, fmal);
287#endif