# Source file src/math/fma.go

```     1  // Copyright 2019 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
3  // license that can be found in the LICENSE file.
4
5  package math
6
7  import "math/bits"
8
9  func zero(x uint64) uint64 {
10  	if x == 0 {
11  		return 1
12  	}
13  	return 0
14  	// branchless:
15  	// return ((x>>1 | x&1) - 1) >> 63
16  }
17
18  func nonzero(x uint64) uint64 {
19  	if x != 0 {
20  		return 1
21  	}
22  	return 0
23  	// branchless:
24  	// return 1 - ((x>>1|x&1)-1)>>63
25  }
26
27  func shl(u1, u2 uint64, n uint) (r1, r2 uint64) {
28  	r1 = u1<<n | u2>>(64-n) | u2<<(n-64)
29  	r2 = u2 << n
30  	return
31  }
32
33  func shr(u1, u2 uint64, n uint) (r1, r2 uint64) {
34  	r2 = u2>>n | u1<<(64-n) | u1>>(n-64)
35  	r1 = u1 >> n
36  	return
37  }
38
39  // shrcompress compresses the bottom n+1 bits of the two-word
40  // value into a single bit. the result is equal to the value
41  // shifted to the right by n, except the result's 0th bit is
42  // set to the bitwise OR of the bottom n+1 bits.
43  func shrcompress(u1, u2 uint64, n uint) (r1, r2 uint64) {
44  	// TODO: Performance here is really sensitive to the
45  	// order/placement of these branches. n == 0 is common
46  	// enough to be in the fast path. Perhaps more measurement
47  	// needs to be done to find the optimal order/placement?
48  	switch {
49  	case n == 0:
50  		return u1, u2
51  	case n == 64:
52  		return 0, u1 | nonzero(u2)
53  	case n >= 128:
54  		return 0, nonzero(u1 | u2)
55  	case n < 64:
56  		r1, r2 = shr(u1, u2, n)
57  		r2 |= nonzero(u2 & (1<<n - 1))
58  	case n < 128:
59  		r1, r2 = shr(u1, u2, n)
60  		r2 |= nonzero(u1&(1<<(n-64)-1) | u2)
61  	}
62  	return
63  }
64
65  func lz(u1, u2 uint64) (l int32) {
66  	l = int32(bits.LeadingZeros64(u1))
67  	if l == 64 {
68  		l += int32(bits.LeadingZeros64(u2))
69  	}
70  	return l
71  }
72
73  // split splits b into sign, biased exponent, and mantissa.
74  // It adds the implicit 1 bit to the mantissa for normal values,
75  // and normalizes subnormal values.
76  func split(b uint64) (sign uint32, exp int32, mantissa uint64) {
77  	sign = uint32(b >> 63)
78  	exp = int32(b>>52) & mask
79  	mantissa = b & fracMask
80
81  	if exp == 0 {
82  		// Normalize value if subnormal.
83  		shift := uint(bits.LeadingZeros64(mantissa) - 11)
84  		mantissa <<= shift
85  		exp = 1 - int32(shift)
86  	} else {
87  		// Add implicit 1 bit
88  		mantissa |= 1 << 52
89  	}
90  	return
91  }
92
93  // FMA returns x * y + z, computed with only one rounding.
94  // (That is, FMA returns the fused multiply-add of x, y, and z.)
95  func FMA(x, y, z float64) float64 {
96  	bx, by, bz := Float64bits(x), Float64bits(y), Float64bits(z)
97
98  	// Inf or NaN or zero involved. At most one rounding will occur.
99  	if x == 0.0 || y == 0.0 || z == 0.0 || bx&uvinf == uvinf || by&uvinf == uvinf {
100  		return x*y + z
101  	}
102  	// Handle non-finite z separately. Evaluating x*y+z where
103  	// x and y are finite, but z is infinite, should always result in z.
104  	if bz&uvinf == uvinf {
105  		return z
106  	}
107
108  	// Inputs are (sub)normal.
109  	// Split x, y, z into sign, exponent, mantissa.
110  	xs, xe, xm := split(bx)
111  	ys, ye, ym := split(by)
112  	zs, ze, zm := split(bz)
113
114  	// Compute product p = x*y as sign, exponent, two-word mantissa.
115  	// Start with exponent. "is normal" bit isn't subtracted yet.
116  	pe := xe + ye - bias + 1
117
118  	// pm1:pm2 is the double-word mantissa for the product p.
119  	// Shift left to leave top bit in product. Effectively
120  	// shifts the 106-bit product to the left by 21.
121  	pm1, pm2 := bits.Mul64(xm<<10, ym<<11)
122  	zm1, zm2 := zm<<10, uint64(0)
123  	ps := xs ^ ys // product sign
124
125  	// normalize to 62nd bit
126  	is62zero := uint((^pm1 >> 62) & 1)
127  	pm1, pm2 = shl(pm1, pm2, is62zero)
128  	pe -= int32(is62zero)
129
130  	// Swap addition operands so |p| >= |z|
131  	if pe < ze || pe == ze && pm1 < zm1 {
132  		ps, pe, pm1, pm2, zs, ze, zm1, zm2 = zs, ze, zm1, zm2, ps, pe, pm1, pm2
133  	}
134
135  	// Align significands
136  	zm1, zm2 = shrcompress(zm1, zm2, uint(pe-ze))
137
138  	// Compute resulting significands, normalizing if necessary.
139  	var m, c uint64
140  	if ps == zs {
141  		// Adding (pm1:pm2) + (zm1:zm2)
142  		pm2, c = bits.Add64(pm2, zm2, 0)
143  		pm1, _ = bits.Add64(pm1, zm1, c)
144  		pe -= int32(^pm1 >> 63)
145  		pm1, m = shrcompress(pm1, pm2, uint(64+pm1>>63))
146  	} else {
147  		// Subtracting (pm1:pm2) - (zm1:zm2)
148  		// TODO: should we special-case cancellation?
149  		pm2, c = bits.Sub64(pm2, zm2, 0)
150  		pm1, _ = bits.Sub64(pm1, zm1, c)
151  		nz := lz(pm1, pm2)
152  		pe -= nz
153  		m, pm2 = shl(pm1, pm2, uint(nz-1))
154  		m |= nonzero(pm2)
155  	}
156
157  	// Round and break ties to even
158  	if pe > 1022+bias || pe == 1022+bias && (m+1<<9)>>63 == 1 {
159  		// rounded value overflows exponent range
160  		return Float64frombits(uint64(ps)<<63 | uvinf)
161  	}
162  	if pe < 0 {
163  		n := uint(-pe)
164  		m = m>>n | nonzero(m&(1<<n-1))
165  		pe = 0
166  	}
167  	m = ((m + 1<<9) >> 10) & ^zero((m&(1<<10-1))^1<<9)
168  	pe &= -int32(nonzero(m))
169  	return Float64frombits(uint64(ps)<<63 + uint64(pe)<<52 + m)
170  }
171
```

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