Source: trig_reduce.go in package math


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

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))

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)

The top 3 bits are j.

Extract the fraction and find its magnitude.

	hi = hi<<3 | lo>>61
	lz := uint(bits.LeadingZeros64(hi))

Clear implicit mantissa bit and shift into place.

	hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))

Include the exponent and convert to a float.

	hi |= e << shift

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,