1 files changed, 102 insertions, 0 deletions
diff --git a/src/math/trig_reduce.go b/src/math/trig_reduce.go
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+++ b/src/math/trig_reduce.go
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+// Copyright 2018 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"
+)
+
+// reduceThreshold is the maximum value of x where the reduction using Pi/4
+// in 3 float64 parts still gives accurate results. This threshold
+// is set by y*C being representable as a float64 without error
+// where y is given by y = floor(x * (4 / Pi)) and C is the leading partial
+// terms of 4/Pi. Since the leading terms (PI4A and PI4B in sin.go) have 30
+// and 32 trailing zero bits, y should have less than 30 significant bits.
+//
+//	y < 1<<30  -> floor(x*4/Pi) < 1<<30 -> x < (1<<30 - 1) * Pi/4
+//
+// So, conservatively we can take x < 1<<29.
+// Above this threshold Payne-Hanek range reduction must be used.
+const reduceThreshold = 1 << 29
+
+// trigReduce implements Payne-Hanek range reduction by Pi/4
+// for x > 0. It returns the integer part mod 8 (j) and
+// the fractional part (z) of x / (Pi/4).
+// The implementation is based on:
+// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
+// K. C. Ng et al, March 24, 1992
+// The simulated multi-precision calculation of x*B uses 64-bit integer arithmetic.
+func trigReduce(x float64) (j uint64, z float64) {
+	const PI4 = Pi / 4
+	if x < PI4 {
+		return 0, x
+	}
+	// Extract out the integer and exponent such that,
+	// x = ix * 2 ** exp.
+	ix := Float64bits(x)
+	exp := int(ix>>shift&mask) - bias - shift
+	ix &^= mask << shift
+	ix |= 1 << shift
+	// Use the exponent to extract the 3 appropriate uint64 digits from mPi4,
+	// B ~ (z0, z1, z2), such that the product leading digit has the exponent -61.
+	// Note, exp >= -53 since x >= PI4 and exp < 971 for maximum float64.
+	digit, bitshift := uint(exp+61)/64, uint(exp+61)%64
+	z0 := (mPi4[digit] << bitshift) | (mPi4[digit+1] >> (64 - bitshift))
+	z1 := (mPi4[digit+1] << bitshift) | (mPi4[digit+2] >> (64 - bitshift))
+	z2 := (mPi4[digit+2] << bitshift) | (mPi4[digit+3] >> (64 - bitshift))
+	// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
+	z2hi, _ := bits.Mul64(z2, ix)
+	z1hi, z1lo := bits.Mul64(z1, ix)
+	z0lo := z0 * ix
+	lo, c := bits.Add64(z1lo, z2hi, 0)
+	hi, _ := bits.Add64(z0lo, z1hi, c)
+	// The top 3 bits are j.
+	j = hi >> 61
+	// Extract the fraction and find its magnitude.
+	hi = hi<<3 | lo>>61
+	lz := uint(bits.LeadingZeros64(hi))
+	e := uint64(bias - (lz + 1))
+	// Clear implicit mantissa bit and shift into place.
+	hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
+	hi >>= 64 - shift
+	// Include the exponent and convert to a float.
+	hi |= e << shift
+	z = Float64frombits(hi)
+	// Map zeros to origin.
+	if j&1 == 1 {
+		j++
+		j &= 7
+		z--
+	}
+	// Multiply the fractional part by pi/4.
+	return j, z * PI4
+}
+
+// mPi4 is the binary digits of 4/pi as a uint64 array,
+// that is, 4/pi = Sum mPi4[i]*2^(-64*i)
+// 19 64-bit digits and the leading one bit give 1217 bits
+// of precision to handle the largest possible float64 exponent.
+var mPi4 = [...]uint64{
+	0x0000000000000001,
+	0x45f306dc9c882a53,
+	0xf84eafa3ea69bb81,
+	0xb6c52b3278872083,
+	0xfca2c757bd778ac3,
+	0x6e48dc74849ba5c0,
+	0x0c925dd413a32439,
+	0xfc3bd63962534e7d,
+	0xd1046bea5d768909,
+	0xd338e04d68befc82,
+	0x7323ac7306a673e9,
+	0x3908bf177bf25076,
+	0x3ff12fffbc0b301f,
+	0xde5e2316b414da3e,
+	0xda6cfd9e4f96136e,
+	0x9e8c7ecd3cbfd45a,
+	0xea4f758fd7cbe2f6,
+	0x7a0e73ef14a525d4,
+	0xd7f6bf623f1aba10,
+	0xac06608df8f6d757,
+}