// Copyright 2018 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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import (
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"math/bits"
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)
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// reduceThreshold is the maximum value where the reduction using Pi/4
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// in 3 float64 parts still gives accurate results. Above this
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// threshold Payne-Hanek range reduction must be used.
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const reduceThreshold = (1 << 52) / (4 / Pi)
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// trigReduce implements Payne-Hanek range reduction by Pi/4
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// for x > 0. It returns the integer part mod 8 (j) and
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// the fractional part (z) of x / (Pi/4).
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// The implementation is based on:
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// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
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// K. C. Ng et al, March 24, 1992
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// The simulated multi-precision calculation of x*B uses 64-bit integer arithmetic.
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func trigReduce(x float64) (j uint64, z float64) {
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const PI4 = Pi / 4
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if x < PI4 {
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return 0, x
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}
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// Extract out the integer and exponent such that,
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// x = ix * 2 ** exp.
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ix := Float64bits(x)
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exp := int(ix>>shift&mask) - bias - shift
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ix &^= mask << shift
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ix |= 1 << shift
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// Use the exponent to extract the 3 appropriate uint64 digits from mPi4,
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// B ~ (z0, z1, z2), such that the product leading digit has the exponent -61.
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// Note, exp >= -53 since x >= PI4 and exp < 971 for maximum float64.
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digit, bitshift := uint(exp+61)/64, uint(exp+61)%64
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z0 := (mPi4[digit] << bitshift) | (mPi4[digit+1] >> (64 - bitshift))
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z1 := (mPi4[digit+1] << bitshift) | (mPi4[digit+2] >> (64 - bitshift))
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z2 := (mPi4[digit+2] << bitshift) | (mPi4[digit+3] >> (64 - bitshift))
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// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
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z2hi, _ := bits.Mul64(z2, ix)
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z1hi, z1lo := bits.Mul64(z1, ix)
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z0lo := z0 * ix
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lo, c := bits.Add64(z1lo, z2hi, 0)
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hi, _ := bits.Add64(z0lo, z1hi, c)
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// The top 3 bits are j.
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j = hi >> 61
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// Extract the fraction and find its magnitude.
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hi = hi<<3 | lo>>61
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lz := uint(bits.LeadingZeros64(hi))
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e := uint64(bias - (lz + 1))
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// Clear implicit mantissa bit and shift into place.
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hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
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hi >>= 64 - shift
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// Include the exponent and convert to a float.
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hi |= e << shift
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z = Float64frombits(hi)
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// Map zeros to origin.
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if j&1 == 1 {
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j++
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j &= 7
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z--
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}
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// Multiply the fractional part by pi/4.
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return j, z * PI4
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}
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// mPi4 is the binary digits of 4/pi as a uint64 array,
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// that is, 4/pi = Sum mPi4[i]*2^(-64*i)
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// 19 64-bit digits and the leading one bit give 1217 bits
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// of precision to handle the largest possible float64 exponent.
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var mPi4 = [...]uint64{
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0x0000000000000001,
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0x45f306dc9c882a53,
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0xf84eafa3ea69bb81,
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0xb6c52b3278872083,
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0xfca2c757bd778ac3,
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0x6e48dc74849ba5c0,
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0x0c925dd413a32439,
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0xfc3bd63962534e7d,
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0xd1046bea5d768909,
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0xd338e04d68befc82,
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0x7323ac7306a673e9,
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0x3908bf177bf25076,
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0x3ff12fffbc0b301f,
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0xde5e2316b414da3e,
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0xda6cfd9e4f96136e,
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0x9e8c7ecd3cbfd45a,
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0xea4f758fd7cbe2f6,
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0x7a0e73ef14a525d4,
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0xd7f6bf623f1aba10,
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0xac06608df8f6d757,
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}
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