src/math/trig_reduce.go

// 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,
}