1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
|
// Copyright 2017 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 big
import "math"
var (
half = NewFloat(0.5)
two = NewFloat(2.0)
three = NewFloat(3.0)
)
// Sqrt sets z to the rounded square root of x, and returns it.
//
// If z's precision is 0, it is changed to x's precision before the
// operation. Rounding is performed according to z's precision and
// rounding mode.
//
// The function panics if z < 0. The value of z is undefined in that
// case.
func (z *Float) Sqrt(x *Float) *Float {
if debugFloat {
x.validate()
}
if z.prec == 0 {
z.prec = x.prec
}
if x.Sign() == -1 {
// following IEEE754-2008 (section 7.2)
panic(ErrNaN{"square root of negative operand"})
}
// handle ±0 and +∞
if x.form != finite {
z.acc = Exact
z.form = x.form
z.neg = x.neg // IEEE754-2008 requires √±0 = ±0
return z
}
// MantExp sets the argument's precision to the receiver's, and
// when z.prec > x.prec this will lower z.prec. Restore it after
// the MantExp call.
prec := z.prec
b := x.MantExp(z)
z.prec = prec
// Compute √(z·2**b) as
// √( z)·2**(½b) if b is even
// √(2z)·2**(⌊½b⌋) if b > 0 is odd
// √(½z)·2**(⌈½b⌉) if b < 0 is odd
switch b % 2 {
case 0:
// nothing to do
case 1:
z.Mul(two, z)
case -1:
z.Mul(half, z)
}
// 0.25 <= z < 2.0
// Solving x² - z = 0 directly requires a Quo call, but it's
// faster for small precisions.
//
// Solving 1/x² - z = 0 avoids the Quo call and is much faster for
// high precisions.
//
// 128bit precision is an empirically chosen threshold.
if z.prec <= 128 {
z.sqrtDirect(z)
} else {
z.sqrtInverse(z)
}
// re-attach halved exponent
return z.SetMantExp(z, b/2)
}
// Compute √x (up to prec 128) by solving
// t² - x = 0
// for t, starting with a 53 bits precision guess from math.Sqrt and
// then using at most two iterations of Newton's method.
func (z *Float) sqrtDirect(x *Float) {
// let
// f(t) = t² - x
// then
// g(t) = f(t)/f'(t) = ½(t² - x)/t
// and the next guess is given by
// t2 = t - g(t) = ½(t² + x)/t
u := new(Float)
ng := func(t *Float) *Float {
u.prec = t.prec
u.Mul(t, t) // u = t²
u.Add(u, x) // = t² + x
u.Mul(half, u) // = ½(t² + x)
return t.Quo(u, t) // = ½(t² + x)/t
}
xf, _ := x.Float64()
sq := NewFloat(math.Sqrt(xf))
switch {
case z.prec > 128:
panic("sqrtDirect: only for z.prec <= 128")
case z.prec > 64:
sq.prec *= 2
sq = ng(sq)
fallthrough
default:
sq.prec *= 2
sq = ng(sq)
}
z.Set(sq)
}
// Compute √x (to z.prec precision) by solving
// 1/t² - x = 0
// for t (using Newton's method), and then inverting.
func (z *Float) sqrtInverse(x *Float) {
// let
// f(t) = 1/t² - x
// then
// g(t) = f(t)/f'(t) = -½t(1 - xt²)
// and the next guess is given by
// t2 = t - g(t) = ½t(3 - xt²)
u := new(Float)
ng := func(t *Float) *Float {
u.prec = t.prec
u.Mul(t, t) // u = t²
u.Mul(x, u) // = xt²
u.Sub(three, u) // = 3 - xt²
u.Mul(t, u) // = t(3 - xt²)
return t.Mul(half, u) // = ½t(3 - xt²)
}
xf, _ := x.Float64()
sqi := NewFloat(1 / math.Sqrt(xf))
for prec := z.prec + 32; sqi.prec < prec; {
sqi.prec *= 2
sqi = ng(sqi)
}
// sqi = 1/√x
// x/√x = √x
z.Mul(x, sqi)
}
|
