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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 : : */
26 : :
27 : : #include "cdefs-compat.h"
28 : : //__FBSDID("$FreeBSD: src/lib/msun/src/s_fma.c,v 1.8 2011/10/21 06:30:43 das Exp $");
29 : :
30 : : #include <float.h>
31 : : #include <openlibm_fenv.h>
32 : : #include <openlibm_math.h>
33 : :
34 : : #include "math_private.h"
35 : :
36 : : /*
37 : : * A struct dd represents a floating-point number with twice the precision
38 : : * of a double. We maintain the invariant that "hi" stores the 53 high-order
39 : : * bits of the result.
40 : : */
41 : : struct dd {
42 : : double hi;
43 : : double lo;
44 : : };
45 : :
46 : : /*
47 : : * Compute a+b exactly, returning the exact result in a struct dd. We assume
48 : : * that both a and b are finite, but make no assumptions about their relative
49 : : * magnitudes.
50 : : */
51 : : static inline struct dd
52 : 2 : dd_add(double a, double b)
53 : : {
54 : : struct dd ret;
55 : : double s;
56 : :
57 : 2 : ret.hi = a + b;
58 : 2 : s = ret.hi - a;
59 : 2 : ret.lo = (a - (ret.hi - s)) + (b - s);
60 : 2 : return (ret);
61 : : }
62 : :
63 : : /*
64 : : * Compute a+b, with a small tweak: The least significant bit of the
65 : : * result is adjusted into a sticky bit summarizing all the bits that
66 : : * were lost to rounding. This adjustment negates the effects of double
67 : : * rounding when the result is added to another number with a higher
68 : : * exponent. For an explanation of round and sticky bits, see any reference
69 : : * on FPU design, e.g.,
70 : : *
71 : : * J. Coonen. An Implementation Guide to a Proposed Standard for
72 : : * Floating-Point Arithmetic. Computer, vol. 13, no. 1, Jan 1980.
73 : : */
74 : : static inline double
75 : 1 : add_adjusted(double a, double b)
76 : : {
77 : : struct dd sum;
78 : : u_int64_t hibits, lobits;
79 : :
80 : 1 : sum = dd_add(a, b);
81 [ - + ]: 1 : if (sum.lo != 0) {
82 : 0 : EXTRACT_WORD64(hibits, sum.hi);
83 [ # # ]: 0 : if ((hibits & 1) == 0) {
84 : : /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
85 : 0 : EXTRACT_WORD64(lobits, sum.lo);
86 : 0 : hibits += 1 - ((hibits ^ lobits) >> 62);
87 : 0 : INSERT_WORD64(sum.hi, hibits);
88 : : }
89 : : }
90 : 1 : return (sum.hi);
91 : : }
92 : :
93 : : /*
94 : : * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
95 : : * that the result will be subnormal, and care is taken to ensure that
96 : : * double rounding does not occur.
97 : : */
98 : : static inline double
99 : 0 : add_and_denormalize(double a, double b, int scale)
100 : : {
101 : : struct dd sum;
102 : : u_int64_t hibits, lobits;
103 : : int bits_lost;
104 : :
105 : 0 : sum = dd_add(a, b);
106 : :
107 : : /*
108 : : * If we are losing at least two bits of accuracy to denormalization,
109 : : * then the first lost bit becomes a round bit, and we adjust the
110 : : * lowest bit of sum.hi to make it a sticky bit summarizing all the
111 : : * bits in sum.lo. With the sticky bit adjusted, the hardware will
112 : : * break any ties in the correct direction.
113 : : *
114 : : * If we are losing only one bit to denormalization, however, we must
115 : : * break the ties manually.
116 : : */
117 [ # # ]: 0 : if (sum.lo != 0) {
118 : 0 : EXTRACT_WORD64(hibits, sum.hi);
119 : 0 : bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
120 [ # # ]: 0 : if ((bits_lost != 1) ^ (int)(hibits & 1)) {
121 : : /* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
122 : 0 : EXTRACT_WORD64(lobits, sum.lo);
123 : 0 : hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
124 : 0 : INSERT_WORD64(sum.hi, hibits);
125 : : }
126 : : }
127 : 0 : return (ldexp(sum.hi, scale));
128 : : }
129 : :
130 : : /*
131 : : * Compute a*b exactly, returning the exact result in a struct dd. We assume
132 : : * that both a and b are normalized, so no underflow or overflow will occur.
133 : : * The current rounding mode must be round-to-nearest.
134 : : */
135 : : static inline struct dd
136 : 1 : dd_mul(double a, double b)
137 : : {
138 : : static const double split = 0x1p27 + 1.0;
139 : : struct dd ret;
140 : : double ha, hb, la, lb, p, q;
141 : :
142 : 1 : p = a * split;
143 : 1 : ha = a - p;
144 : 1 : ha += p;
145 : 1 : la = a - ha;
146 : :
147 : 1 : p = b * split;
148 : 1 : hb = b - p;
149 : 1 : hb += p;
150 : 1 : lb = b - hb;
151 : :
152 : 1 : p = ha * hb;
153 : 1 : q = ha * lb + la * hb;
154 : :
155 : 1 : ret.hi = p + q;
156 : 1 : ret.lo = p - ret.hi + q + la * lb;
157 : 1 : return (ret);
158 : : }
159 : :
160 : : /*
161 : : * Fused multiply-add: Compute x * y + z with a single rounding error.
162 : : *
163 : : * We use scaling to avoid overflow/underflow, along with the
164 : : * canonical precision-doubling technique adapted from:
165 : : *
166 : : * Dekker, T. A Floating-Point Technique for Extending the
167 : : * Available Precision. Numer. Math. 18, 224-242 (1971).
168 : : *
169 : : * This algorithm is sensitive to the rounding precision. FPUs such
170 : : * as the i387 must be set in double-precision mode if variables are
171 : : * to be stored in FP registers in order to avoid incorrect results.
172 : : * This is the default on FreeBSD, but not on many other systems.
173 : : *
174 : : * Hardware instructions should be used on architectures that support it,
175 : : * since this implementation will likely be several times slower.
176 : : */
177 : : OLM_DLLEXPORT double
178 : 16 : fma(double x, double y, double z)
179 : : {
180 : : double xs, ys, zs, adj;
181 : : struct dd xy, r;
182 : : int oround;
183 : : int ex, ey, ez;
184 : : int spread;
185 : :
186 : : /*
187 : : * Handle special cases. The order of operations and the particular
188 : : * return values here are crucial in handling special cases involving
189 : : * infinities, NaNs, overflows, and signed zeroes correctly.
190 : : */
191 [ + + + + ]: 16 : if (x == 0.0 || y == 0.0)
192 : 8 : return (x * y + z);
193 [ - + ]: 8 : if (z == 0.0)
194 : 0 : return (x * y);
195 [ + + + + ]: 8 : if (!isfinite(x) || !isfinite(y))
196 : 6 : return (x * y + z);
197 [ + + ]: 2 : if (!isfinite(z))
198 : 1 : return (z);
199 : :
200 : 1 : xs = frexp(x, &ex);
201 : 1 : ys = frexp(y, &ey);
202 : 1 : zs = frexp(z, &ez);
203 : 1 : oround = fegetround();
204 : 1 : spread = ex + ey - ez;
205 : :
206 : : /*
207 : : * If x * y and z are many orders of magnitude apart, the scaling
208 : : * will overflow, so we handle these cases specially. Rounding
209 : : * modes other than FE_TONEAREST are painful.
210 : : */
211 [ - + ]: 1 : if (spread < -DBL_MANT_DIG) {
212 : 0 : feraiseexcept(FE_INEXACT);
213 [ # # ]: 0 : if (!isnormal(z))
214 : 0 : feraiseexcept(FE_UNDERFLOW);
215 [ # # # # ]: 0 : switch (oround) {
216 : 0 : case FE_TONEAREST:
217 : 0 : return (z);
218 : 0 : case FE_TOWARDZERO:
219 [ # # ]: 0 : if ((x > 0.0) ^ (y < 0.0) ^ (z < 0.0))
220 : 0 : return (z);
221 : : else
222 : 0 : return (nextafter(z, 0));
223 : 0 : case FE_DOWNWARD:
224 [ # # ]: 0 : if ((x > 0.0) ^ (y < 0.0))
225 : 0 : return (z);
226 : : else
227 : 0 : return (nextafter(z, -INFINITY));
228 : 0 : default: /* FE_UPWARD */
229 [ # # ]: 0 : if ((x > 0.0) ^ (y < 0.0))
230 : 0 : return (nextafter(z, INFINITY));
231 : : else
232 : 0 : return (z);
233 : : }
234 : : }
235 [ + - ]: 1 : if (spread <= DBL_MANT_DIG * 2)
236 : 1 : zs = ldexp(zs, -spread);
237 : : else
238 : 0 : zs = copysign(DBL_MIN, zs);
239 : :
240 : 1 : fesetround(FE_TONEAREST);
241 : :
242 : : /*
243 : : * Basic approach for round-to-nearest:
244 : : *
245 : : * (xy.hi, xy.lo) = x * y (exact)
246 : : * (r.hi, r.lo) = xy.hi + z (exact)
247 : : * adj = xy.lo + r.lo (inexact; low bit is sticky)
248 : : * result = r.hi + adj (correctly rounded)
249 : : */
250 : 1 : xy = dd_mul(xs, ys);
251 : 1 : r = dd_add(xy.hi, zs);
252 : :
253 : 1 : spread = ex + ey;
254 : :
255 [ - + ]: 1 : if (r.hi == 0.0) {
256 : : /*
257 : : * When the addends cancel to 0, ensure that the result has
258 : : * the correct sign.
259 : : */
260 : 0 : fesetround(oround);
261 : 0 : volatile double vzs = zs; /* XXX gcc CSE bug workaround */
262 : 0 : return (xy.hi + vzs + ldexp(xy.lo, spread));
263 : : }
264 : :
265 [ - + ]: 1 : if (oround != FE_TONEAREST) {
266 : : /*
267 : : * There is no need to worry about double rounding in directed
268 : : * rounding modes.
269 : : */
270 : 0 : fesetround(oround);
271 : 0 : adj = r.lo + xy.lo;
272 : 0 : return (ldexp(r.hi + adj, spread));
273 : : }
274 : :
275 : 1 : adj = add_adjusted(r.lo, xy.lo);
276 [ + - ]: 1 : if (spread + ilogb(r.hi) > -1023)
277 : 1 : return (ldexp(r.hi + adj, spread));
278 : : else
279 : 0 : return (add_and_denormalize(r.hi, adj, spread));
280 : : }
281 : :
282 : : #if (LDBL_MANT_DIG == 53)
283 : : openlibm_weak_reference(fma, fmal);
284 : : #endif
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