1 // Copyright 2018 The Go Authors. All rights reserved.
2 // Use of this source code is governed by a BSD-style
3 // license that can be found in the LICENSE file.
4 5 package math
6 7 import (
8 "math/bits"
9 )
10 11 // reduceThreshold is the maximum value of x where the reduction using Pi/4
12 // in 3 float64 parts still gives accurate results. This threshold
13 // is set by y*C being representable as a float64 without error
14 // where y is given by y = floor(x * (4 / Pi)) and C is the leading partial
15 // terms of 4/Pi. Since the leading terms (PI4A and PI4B in sin.go) have 30
16 // and 32 trailing zero bits, y should have less than 30 significant bits.
17 //
18 // y < 1<<30 -> floor(x*4/Pi) < 1<<30 -> x < (1<<30 - 1) * Pi/4
19 //
20 // So, conservatively we can take x < 1<<29.
21 // Above this threshold Payne-Hanek range reduction must be used.
22 const reduceThreshold = 1 << 29
23 24 // trigReduce implements Payne-Hanek range reduction by Pi/4
25 // for x > 0. It returns the integer part mod 8 (j) and
26 // the fractional part (z) of x / (Pi/4).
27 // The implementation is based on:
28 // "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
29 // K. C. Ng et al, March 24, 1992
30 // The simulated multi-precision calculation of x*B uses 64-bit integer arithmetic.
31 func trigReduce(x float64) (j uint64, z float64) {
32 const PI4 = Pi / 4
33 if x < PI4 {
34 return 0, x
35 }
36 // Extract out the integer and exponent such that,
37 // x = ix * 2 ** exp.
38 ix := Float64bits(x)
39 exp := int(ix>>shift&mask) - bias - shift
40 ix &^= mask << shift
41 ix |= 1 << shift
42 // Use the exponent to extract the 3 appropriate uint64 digits from mPi4,
43 // B ~ (z0, z1, z2), such that the product leading digit has the exponent -61.
44 // Note, exp >= -53 since x >= PI4 and exp < 971 for maximum float64.
45 digit, bitshift := uint(exp+61)/64, uint(exp+61)%64
46 z0 := (mPi4[digit] << bitshift) | (mPi4[digit+1] >> (64 - bitshift))
47 z1 := (mPi4[digit+1] << bitshift) | (mPi4[digit+2] >> (64 - bitshift))
48 z2 := (mPi4[digit+2] << bitshift) | (mPi4[digit+3] >> (64 - bitshift))
49 // Multiply mantissa by the digits and extract the upper two digits (hi, lo).
50 z2hi, _ := bits.Mul64(z2, ix)
51 z1hi, z1lo := bits.Mul64(z1, ix)
52 z0lo := z0 * ix
53 lo, c := bits.Add64(z1lo, z2hi, 0)
54 hi, _ := bits.Add64(z0lo, z1hi, c)
55 // The top 3 bits are j.
56 j = hi >> 61
57 // Extract the fraction and find its magnitude.
58 hi = hi<<3 | lo>>61
59 lz := uint(bits.LeadingZeros64(hi))
60 e := uint64(bias - (lz + 1))
61 // Clear implicit mantissa bit and shift into place.
62 hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
63 hi >>= 64 - shift
64 // Include the exponent and convert to a float.
65 hi |= e << shift
66 z = Float64frombits(hi)
67 // Map zeros to origin.
68 if j&1 == 1 {
69 j++
70 j &= 7
71 z--
72 }
73 // Multiply the fractional part by pi/4.
74 return j, z * PI4
75 }
76 77 // mPi4 is the binary digits of 4/pi as a uint64 array,
78 // that is, 4/pi = Sum mPi4[i]*2^(-64*i)
79 // 19 64-bit digits and the leading one bit give 1217 bits
80 // of precision to handle the largest possible float64 exponent.
81 var mPi4 = [...]uint64{
82 0x0000000000000001,
83 0x45f306dc9c882a53,
84 0xf84eafa3ea69bb81,
85 0xb6c52b3278872083,
86 0xfca2c757bd778ac3,
87 0x6e48dc74849ba5c0,
88 0x0c925dd413a32439,
89 0xfc3bd63962534e7d,
90 0xd1046bea5d768909,
91 0xd338e04d68befc82,
92 0x7323ac7306a673e9,
93 0x3908bf177bf25076,
94 0x3ff12fffbc0b301f,
95 0xde5e2316b414da3e,
96 0xda6cfd9e4f96136e,
97 0x9e8c7ecd3cbfd45a,
98 0xea4f758fd7cbe2f6,
99 0x7a0e73ef14a525d4,
100 0xd7f6bf623f1aba10,
101 0xac06608df8f6d757,
102 }
103