summaryrefslogtreecommitdiff
path: root/src/math/fma.c
blob: 0c6f90c9cfd14e7ba584c3828dd01077171ca420 (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
#include <stdint.h>
#include <float.h>
#include <math.h>
#include "atomic.h"

#define ASUINT64(x) ((union {double f; uint64_t i;}){x}).i
#define ZEROINFNAN (0x7ff-0x3ff-52-1)

struct num { uint64_t m; int e; int sign; };

static struct num normalize(double x)
{
	uint64_t ix = ASUINT64(x);
	int e = ix>>52;
	int sign = e & 0x800;
	e &= 0x7ff;
	if (!e) {
		ix = ASUINT64(x*0x1p63);
		e = ix>>52 & 0x7ff;
		e = e ? e-63 : 0x800;
	}
	ix &= (1ull<<52)-1;
	ix |= 1ull<<52;
	ix <<= 1;
	e -= 0x3ff + 52 + 1;
	return (struct num){ix,e,sign};
}

static void mul(uint64_t *hi, uint64_t *lo, uint64_t x, uint64_t y)
{
	uint64_t t1,t2,t3;
	uint64_t xlo = (uint32_t)x, xhi = x>>32;
	uint64_t ylo = (uint32_t)y, yhi = y>>32;

	t1 = xlo*ylo;
	t2 = xlo*yhi + xhi*ylo;
	t3 = xhi*yhi;
	*lo = t1 + (t2<<32);
	*hi = t3 + (t2>>32) + (t1 > *lo);
}

double fma(double x, double y, double z)
{
	#pragma STDC FENV_ACCESS ON

	/* normalize so top 10bits and last bit are 0 */
	struct num nx, ny, nz;
	nx = normalize(x);
	ny = normalize(y);
	nz = normalize(z);

	if (nx.e >= ZEROINFNAN || ny.e >= ZEROINFNAN)
		return x*y + z;
	if (nz.e >= ZEROINFNAN) {
		if (nz.e > ZEROINFNAN) /* z==0 */
			return x*y + z;
		return z;
	}

	/* mul: r = x*y */
	uint64_t rhi, rlo, zhi, zlo;
	mul(&rhi, &rlo, nx.m, ny.m);
	/* either top 20 or 21 bits of rhi and last 2 bits of rlo are 0 */

	/* align exponents */
	int e = nx.e + ny.e;
	int d = nz.e - e;
	/* shift bits z<<=kz, r>>=kr, so kz+kr == d, set e = e+kr (== ez-kz) */
	if (d > 0) {
		if (d < 64) {
			zlo = nz.m<<d;
			zhi = nz.m>>64-d;
		} else {
			zlo = 0;
			zhi = nz.m;
			e = nz.e - 64;
			d -= 64;
			if (d == 0) {
			} else if (d < 64) {
				rlo = rhi<<64-d | rlo>>d | !!(rlo<<64-d);
				rhi = rhi>>d;
			} else {
				rlo = 1;
				rhi = 0;
			}
		}
	} else {
		zhi = 0;
		d = -d;
		if (d == 0) {
			zlo = nz.m;
		} else if (d < 64) {
			zlo = nz.m>>d | !!(nz.m<<64-d);
		} else {
			zlo = 1;
		}
	}

	/* add */
	int sign = nx.sign^ny.sign;
	int samesign = !(sign^nz.sign);
	int nonzero = 1;
	if (samesign) {
		/* r += z */
		rlo += zlo;
		rhi += zhi + (rlo < zlo);
	} else {
		/* r -= z */
		uint64_t t = rlo;
		rlo -= zlo;
		rhi = rhi - zhi - (t < rlo);
		if (rhi>>63) {
			rlo = -rlo;
			rhi = -rhi-!!rlo;
			sign = !sign;
		}
		nonzero = !!rhi;
	}

	/* set rhi to top 63bit of the result (last bit is sticky) */
	if (nonzero) {
		e += 64;
		d = a_clz_64(rhi)-1;
		/* note: d > 0 */
		rhi = rhi<<d | rlo>>64-d | !!(rlo<<d);
	} else if (rlo) {
		d = a_clz_64(rlo)-1;
		if (d < 0)
			rhi = rlo>>1 | (rlo&1);
		else
			rhi = rlo<<d;
	} else {
		/* exact +-0 */
		return x*y + z;
	}
	e -= d;

	/* convert to double */
	int64_t i = rhi; /* i is in [1<<62,(1<<63)-1] */
	if (sign)
		i = -i;
	double r = i; /* |r| is in [0x1p62,0x1p63] */

	if (e < -1022-62) {
		/* result is subnormal before rounding */
		if (e == -1022-63) {
			double c = 0x1p63;
			if (sign)
				c = -c;
			if (r == c) {
				/* min normal after rounding, underflow depends
				   on arch behaviour which can be imitated by
				   a double to float conversion */
				float fltmin = 0x0.ffffff8p-63*FLT_MIN * r;
				return DBL_MIN/FLT_MIN * fltmin;
			}
			/* one bit is lost when scaled, add another top bit to
			   only round once at conversion if it is inexact */
			if (rhi << 53) {
				i = rhi>>1 | (rhi&1) | 1ull<<62;
				if (sign)
					i = -i;
				r = i;
				r = 2*r - c; /* remove top bit */

				/* raise underflow portably, such that it
				   cannot be optimized away */
				{
					double_t tiny = DBL_MIN/FLT_MIN * r;
					r += (double)(tiny*tiny) * (r-r);
				}
			}
		} else {
			/* only round once when scaled */
			d = 10;
			i = ( rhi>>d | !!(rhi<<64-d) ) << d;
			if (sign)
				i = -i;
			r = i;
		}
	}
	return scalbn(r, e);
}