about summary refs log tree commit diff
path: root/src/math/fma.c
blob: 5fb9540601c520dac6ce9a519e8c9abe91aed7d7 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
#include <fenv.h>
#include "libm.h"

#if LDBL_MANT_DIG==64 && LDBL_MAX_EXP==16384
union ld80 {
	long double x;
	struct {
		uint64_t m;
		uint16_t e : 15;
		uint16_t s : 1;
		uint16_t pad;
	} bits;
};

/* exact add, assumes exponent_x >= exponent_y */
static void add(long double *hi, long double *lo, long double x, long double y)
{
	long double r;

	r = x + y;
	*hi = r;
	r -= x;
	*lo = y - r;
}

/* exact mul, assumes no over/underflow */
static void mul(long double *hi, long double *lo, long double x, long double y)
{
	static const long double c = 1.0 + 0x1p32L;
	long double cx, xh, xl, cy, yh, yl;

	cx = c*x;
	xh = (x - cx) + cx;
	xl = x - xh;
	cy = c*y;
	yh = (y - cy) + cy;
	yl = y - yh;
	*hi = x*y;
	*lo = (xh*yh - *hi) + xh*yl + xl*yh + xl*yl;
}

/*
assume (long double)(hi+lo) == hi
return an adjusted hi so that rounding it to double is correct
*/
static long double adjust(long double hi, long double lo)
{
	union ld80 uhi, ulo;

	if (lo == 0)
		return hi;
	uhi.x = hi;
	if (uhi.bits.m & 0x3ff)
		return hi;
	ulo.x = lo;
	if (uhi.bits.s == ulo.bits.s)
		uhi.bits.m++;
	else
		uhi.bits.m--;
	return uhi.x;
}

static long double dadd(long double x, long double y)
{
	add(&x, &y, x, y);
	return adjust(x, y);
}

static long double dmul(long double x, long double y)
{
	mul(&x, &y, x, y);
	return adjust(x, y);
}

static int getexp(long double x)
{
	union ld80 u;
	u.x = x;
	return u.bits.e;
}

double fma(double x, double y, double z)
{
	long double hi, lo1, lo2, xy;
	int round, ez, exy;

	/* handle +-inf,nan */
	if (!isfinite(x) || !isfinite(y))
		return x*y + z;
	if (!isfinite(z))
		return z;
	/* handle +-0 */
	if (x == 0.0 || y == 0.0)
		return x*y + z;
	round = fegetround();
	if (z == 0.0) {
		if (round == FE_TONEAREST)
			return dmul(x, y);
		return x*y;
	}

	/* exact mul and add require nearest rounding */
	/* spurious inexact exceptions may be raised */
	fesetround(FE_TONEAREST);
	mul(&xy, &lo1, x, y);
	exy = getexp(xy);
	ez = getexp(z);
	if (ez > exy) {
		add(&hi, &lo2, z, xy);
	} else if (ez > exy - 12) {
		add(&hi, &lo2, xy, z);
		if (hi == 0) {
			fesetround(round);
			/* make sure that the sign of 0 is correct */
			return (xy + z) + lo1;
		}
	} else {
		/*
		ez <= exy - 12
		the 12 extra bits (1guard, 11round+sticky) are needed so with
			lo = dadd(lo1, lo2)
		elo <= ehi - 11, and we use the last 10 bits in adjust so
			dadd(hi, lo)
		gives correct result when rounded to double
		*/
		hi = xy;
		lo2 = z;
	}
	fesetround(round);
	if (round == FE_TONEAREST)
		return dadd(hi, dadd(lo1, lo2));
	return hi + (lo1 + lo2);
}
#else
/* origin: FreeBSD /usr/src/lib/msun/src/s_fma.c */
/*-
 * Copyright (c) 2005-2011 David Schultz <das@FreeBSD.ORG>
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 */

/*
 * A struct dd represents a floating-point number with twice the precision
 * of a double.  We maintain the invariant that "hi" stores the 53 high-order
 * bits of the result.
 */
struct dd {
	double hi;
	double lo;
};

/*
 * Compute a+b exactly, returning the exact result in a struct dd.  We assume
 * that both a and b are finite, but make no assumptions about their relative
 * magnitudes.
 */
static inline struct dd dd_add(double a, double b)
{
	struct dd ret;
	double s;

	ret.hi = a + b;
	s = ret.hi - a;
	ret.lo = (a - (ret.hi - s)) + (b - s);
	return (ret);
}

/*
 * Compute a+b, with a small tweak:  The least significant bit of the
 * result is adjusted into a sticky bit summarizing all the bits that
 * were lost to rounding.  This adjustment negates the effects of double
 * rounding when the result is added to another number with a higher
 * exponent.  For an explanation of round and sticky bits, see any reference
 * on FPU design, e.g.,
 *
 *     J. Coonen.  An Implementation Guide to a Proposed Standard for
 *     Floating-Point Arithmetic.  Computer, vol. 13, no. 1, Jan 1980.
 */
static inline double add_adjusted(double a, double b)
{
	struct dd sum;
	uint64_t hibits, lobits;

	sum = dd_add(a, b);
	if (sum.lo != 0) {
		EXTRACT_WORD64(hibits, sum.hi);
		if ((hibits & 1) == 0) {
			/* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
			EXTRACT_WORD64(lobits, sum.lo);
			hibits += 1 - ((hibits ^ lobits) >> 62);
			INSERT_WORD64(sum.hi, hibits);
		}
	}
	return (sum.hi);
}

/*
 * Compute ldexp(a+b, scale) with a single rounding error. It is assumed
 * that the result will be subnormal, and care is taken to ensure that
 * double rounding does not occur.
 */
static inline double add_and_denormalize(double a, double b, int scale)
{
	struct dd sum;
	uint64_t hibits, lobits;
	int bits_lost;

	sum = dd_add(a, b);

	/*
	 * If we are losing at least two bits of accuracy to denormalization,
	 * then the first lost bit becomes a round bit, and we adjust the
	 * lowest bit of sum.hi to make it a sticky bit summarizing all the
	 * bits in sum.lo. With the sticky bit adjusted, the hardware will
	 * break any ties in the correct direction.
	 *
	 * If we are losing only one bit to denormalization, however, we must
	 * break the ties manually.
	 */
	if (sum.lo != 0) {
		EXTRACT_WORD64(hibits, sum.hi);
		bits_lost = -((int)(hibits >> 52) & 0x7ff) - scale + 1;
		if (bits_lost != 1 ^ (int)(hibits & 1)) {
			/* hibits += (int)copysign(1.0, sum.hi * sum.lo) */
			EXTRACT_WORD64(lobits, sum.lo);
			hibits += 1 - (((hibits ^ lobits) >> 62) & 2);
			INSERT_WORD64(sum.hi, hibits);
		}
	}
	return scalbn(sum.hi, scale);
}

/*
 * Compute a*b exactly, returning the exact result in a struct dd.  We assume
 * that both a and b are normalized, so no underflow or overflow will occur.
 * The current rounding mode must be round-to-nearest.
 */
static inline struct dd dd_mul(double a, double b)
{
	static const double split = 0x1p27 + 1.0;
	struct dd ret;
	double ha, hb, la, lb, p, q;

	p = a * split;
	ha = a - p;
	ha += p;
	la = a - ha;

	p = b * split;
	hb = b - p;
	hb += p;
	lb = b - hb;

	p = ha * hb;
	q = ha * lb + la * hb;

	ret.hi = p + q;
	ret.lo = p - ret.hi + q + la * lb;
	return (ret);
}

/*
 * Fused multiply-add: Compute x * y + z with a single rounding error.
 *
 * We use scaling to avoid overflow/underflow, along with the
 * canonical precision-doubling technique adapted from:
 *
 *      Dekker, T.  A Floating-Point Technique for Extending the
 *      Available Precision.  Numer. Math. 18, 224-242 (1971).
 *
 * This algorithm is sensitive to the rounding precision.  FPUs such
 * as the i387 must be set in double-precision mode if variables are
 * to be stored in FP registers in order to avoid incorrect results.
 * This is the default on FreeBSD, but not on many other systems.
 *
 * Hardware instructions should be used on architectures that support it,
 * since this implementation will likely be several times slower.
 */
double fma(double x, double y, double z)
{
	double xs, ys, zs, adj;
	struct dd xy, r;
	int oround;
	int ex, ey, ez;
	int spread;

	/*
	 * Handle special cases. The order of operations and the particular
	 * return values here are crucial in handling special cases involving
	 * infinities, NaNs, overflows, and signed zeroes correctly.
	 */
	if (!isfinite(x) || !isfinite(y))
		return (x * y + z);
	if (!isfinite(z))
		return (z);
	if (x == 0.0 || y == 0.0)
		return (x * y + z);
	if (z == 0.0)
		return (x * y);

	xs = frexp(x, &ex);
	ys = frexp(y, &ey);
	zs = frexp(z, &ez);
	oround = fegetround();
	spread = ex + ey - ez;

	/*
	 * If x * y and z are many orders of magnitude apart, the scaling
	 * will overflow, so we handle these cases specially.  Rounding
	 * modes other than FE_TONEAREST are painful.
	 */
	if (spread < -DBL_MANT_DIG) {
#ifdef FE_INEXACT
		feraiseexcept(FE_INEXACT);
#endif
#ifdef FE_UNDERFLOW
		if (!isnormal(z))
			feraiseexcept(FE_UNDERFLOW);
#endif
		switch (oround) {
		default: /* FE_TONEAREST */
			return (z);
#ifdef FE_TOWARDZERO
		case FE_TOWARDZERO:
			if (x > 0.0 ^ y < 0.0 ^ z < 0.0)
				return (z);
			else
				return (nextafter(z, 0));
#endif
#ifdef FE_DOWNWARD
		case FE_DOWNWARD:
			if (x > 0.0 ^ y < 0.0)
				return (z);
			else
				return (nextafter(z, -INFINITY));
#endif
#ifdef FE_UPWARD
		case FE_UPWARD:
			if (x > 0.0 ^ y < 0.0)
				return (nextafter(z, INFINITY));
			else
				return (z);
#endif
		}
	}
	if (spread <= DBL_MANT_DIG * 2)
		zs = scalbn(zs, -spread);
	else
		zs = copysign(DBL_MIN, zs);

	fesetround(FE_TONEAREST);

	/*
	 * Basic approach for round-to-nearest:
	 *
	 *     (xy.hi, xy.lo) = x * y           (exact)
	 *     (r.hi, r.lo)   = xy.hi + z       (exact)
	 *     adj = xy.lo + r.lo               (inexact; low bit is sticky)
	 *     result = r.hi + adj              (correctly rounded)
	 */
	xy = dd_mul(xs, ys);
	r = dd_add(xy.hi, zs);

	spread = ex + ey;

	if (r.hi == 0.0) {
		/*
		 * When the addends cancel to 0, ensure that the result has
		 * the correct sign.
		 */
		fesetround(oround);
		volatile double vzs = zs; /* XXX gcc CSE bug workaround */
		return xy.hi + vzs + scalbn(xy.lo, spread);
	}

	if (oround != FE_TONEAREST) {
		/*
		 * There is no need to worry about double rounding in directed
		 * rounding modes.
		 */
		fesetround(oround);
		adj = r.lo + xy.lo;
		return scalbn(r.hi + adj, spread);
	}

	adj = add_adjusted(r.lo, xy.lo);
	if (spread + ilogb(r.hi) > -1023)
		return scalbn(r.hi + adj, spread);
	else
		return add_and_denormalize(r.hi, adj, spread);
}
#endif