src/math/fma.go

// Copyright 2019 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package math

import "math/bits"

func zero(x uint64) uint64 {
	if x == 0 {
		return 1
	}
	return 0
	// branchless:
	// return ((x>>1 | x&1) - 1) >> 63
}

func nonzero(x uint64) uint64 {
	if x != 0 {
		return 1
	}
	return 0
	// branchless:
	// return 1 - ((x>>1|x&1)-1)>>63
}

func shl(u1, u2 uint64, n uint) (r1, r2 uint64) {
	r1 = u1<<n | u2>>(64-n) | u2<<(n-64)
	r2 = u2 << n
	return
}

func shr(u1, u2 uint64, n uint) (r1, r2 uint64) {
	r2 = u2>>n | u1<<(64-n) | u1>>(n-64)
	r1 = u1 >> n
	return
}

// shrcompress compresses the bottom n+1 bits of the two-word
// value into a single bit. the result is equal to the value
// shifted to the right by n, except the result's 0th bit is
// set to the bitwise OR of the bottom n+1 bits.
func shrcompress(u1, u2 uint64, n uint) (r1, r2 uint64) {
	// TODO: Performance here is really sensitive to the
	// order/placement of these branches. n == 0 is common
	// enough to be in the fast path. Perhaps more measurement
	// needs to be done to find the optimal order/placement?
	switch {
	case n == 0:
		return u1, u2
	case n == 64:
		return 0, u1 | nonzero(u2)
	case n >= 128:
		return 0, nonzero(u1 | u2)
	case n < 64:
		r1, r2 = shr(u1, u2, n)
		r2 |= nonzero(u2 & (1<<n - 1))
	case n < 128:
		r1, r2 = shr(u1, u2, n)
		r2 |= nonzero(u1&(1<<(n-64)-1) | u2)
	}
	return
}

func lz(u1, u2 uint64) (l int32) {
	l = int32(bits.LeadingZeros64(u1))
	if l == 64 {
		l += int32(bits.LeadingZeros64(u2))
	}
	return l
}

// split splits b into sign, biased exponent, and mantissa.
// It adds the implicit 1 bit to the mantissa for normal values,
// and normalizes subnormal values.
func split(b uint64) (sign uint32, exp int32, mantissa uint64) {
	sign = uint32(b >> 63)
	exp = int32(b>>52) & mask
	mantissa = b & fracMask

	if exp == 0 {
		// Normalize value if subnormal.
		shift := uint(bits.LeadingZeros64(mantissa) - 11)
		mantissa <<= shift
		exp = 1 - int32(shift)
	} else {
		// Add implicit 1 bit
		mantissa |= 1 << 52
	}
	return
}

// FMA returns x * y + z, computed with only one rounding.
// (That is, FMA returns the fused multiply-add of x, y, and z.)
func FMA(x, y, z float64) float64 {
	bx, by, bz := Float64bits(x), Float64bits(y), Float64bits(z)

	// Inf or NaN or zero involved. At most one rounding will occur.
	if x == 0.0 || y == 0.0 || z == 0.0 || bx&uvinf == uvinf || by&uvinf == uvinf {
		return x*y + z
	}
	// Handle non-finite z separately. Evaluating x*y+z where
	// x and y are finite, but z is infinite, should always result in z.
	if bz&uvinf == uvinf {
		return z
	}

	// Inputs are (sub)normal.
	// Split x, y, z into sign, exponent, mantissa.
	xs, xe, xm := split(bx)
	ys, ye, ym := split(by)
	zs, ze, zm := split(bz)

	// Compute product p = x*y as sign, exponent, two-word mantissa.
	// Start with exponent. "is normal" bit isn't subtracted yet.
	pe := xe + ye - bias + 1

	// pm1:pm2 is the double-word mantissa for the product p.
	// Shift left to leave top bit in product. Effectively
	// shifts the 106-bit product to the left by 21.
	pm1, pm2 := bits.Mul64(xm<<10, ym<<11)
	zm1, zm2 := zm<<10, uint64(0)
	ps := xs ^ ys // product sign

	// normalize to 62nd bit
	is62zero := uint((^pm1 >> 62) & 1)
	pm1, pm2 = shl(pm1, pm2, is62zero)
	pe -= int32(is62zero)

	// Swap addition operands so |p| >= |z|
	if pe < ze || pe == ze && pm1 < zm1 {
		ps, pe, pm1, pm2, zs, ze, zm1, zm2 = zs, ze, zm1, zm2, ps, pe, pm1, pm2
	}

	// Align significands
	zm1, zm2 = shrcompress(zm1, zm2, uint(pe-ze))

	// Compute resulting significands, normalizing if necessary.
	var m, c uint64
	if ps == zs {
		// Adding (pm1:pm2) + (zm1:zm2)
		pm2, c = bits.Add64(pm2, zm2, 0)
		pm1, _ = bits.Add64(pm1, zm1, c)
		pe -= int32(^pm1 >> 63)
		pm1, m = shrcompress(pm1, pm2, uint(64+pm1>>63))
	} else {
		// Subtracting (pm1:pm2) - (zm1:zm2)
		// TODO: should we special-case cancellation?
		pm2, c = bits.Sub64(pm2, zm2, 0)
		pm1, _ = bits.Sub64(pm1, zm1, c)
		nz := lz(pm1, pm2)
		pe -= nz
		m, pm2 = shl(pm1, pm2, uint(nz-1))
		m |= nonzero(pm2)
	}

	// Round and break ties to even
	if pe > 1022+bias || pe == 1022+bias && (m+1<<9)>>63 == 1 {
		// rounded value overflows exponent range
		return Float64frombits(uint64(ps)<<63 | uvinf)
	}
	if pe < 0 {
		n := uint(-pe)
		m = m>>n | nonzero(m&(1<<n-1))
		pe = 0
	}
	m = ((m + 1<<9) >> 10) & ^zero((m&(1<<10-1))^1<<9)
	pe &= -int32(nonzero(m))
	return Float64frombits(uint64(ps)<<63 + uint64(pe)<<52 + m)
}