blob: 5cdf4fa01379e4589ec4827b5015896bbf20425a [file] [log] [blame]
 // 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, }