Source File
trig_reduce.go
Belonging Package
math
// 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 (
)
// 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( float64) ( uint64, float64) {
const = Pi / 4
if < {
return 0,
}
// Extract out the integer and exponent such that,
// x = ix * 2 ** exp.
:= Float64bits()
:= int(>>shift&mask) - bias - shift
&^= mask << shift
|= 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.
, := uint(+61)/64, uint(+61)%64
:= (mPi4[] << ) | (mPi4[+1] >> (64 - ))
:= (mPi4[+1] << ) | (mPi4[+2] >> (64 - ))
:= (mPi4[+2] << ) | (mPi4[+3] >> (64 - ))
// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
, := bits.Mul64(, )
, := bits.Mul64(, )
:= *
, := bits.Add64(, , 0)
, := bits.Add64(, , )
// The top 3 bits are j.
= >> 61
// Extract the fraction and find its magnitude.
= <<3 | >>61
:= uint(bits.LeadingZeros64())
:= uint64(bias - ( + 1))
// Clear implicit mantissa bit and shift into place.
= ( << ( + 1)) | ( >> (64 - ( + 1)))
>>= 64 - shift
// Include the exponent and convert to a float.
|= << shift
= Float64frombits()
// Map zeros to origin.
if &1 == 1 {
++
&= 7
--
}
// Multiply the fractional part by pi/4.
return , *
}
// 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,
}
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