https://www.cnblogs.com/kukri/p/8484992.html

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

// Copyright 2009 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

// The original C code and the long comment below are
// from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
// came with this notice. The go code is a simplified
// version of the original C.
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// __ieee754_sqrt(x)
// Return correctly rounded sqrt.
// -----------------------------------------
// | Use the hardware sqrt if you have one |
// -----------------------------------------
// Method:
// Bit by bit method using integer arithmetic. (Slow, but portable)
// 1. Normalization
// Scale x to y in [1,4) with even powers of 2:
// find an integer k such that 1 <= (y=x*2**(2k)) < 4, then
// sqrt(x) = 2**k * sqrt(y)
// 2. Bit by bit computation
// Let q = sqrt(y) truncated to i bit after binary point (q = 1),
// i 0
// i+1 2
// s = 2*q , and y = 2 * ( y - q ). (1)
// i i i i
//
// To compute q from q , one checks whether
// i+1 i
//
// -(i+1) 2
// (q + 2 ) <= y. (2)
// i
// -(i+1)
// If (2) is false, then q = q ; otherwise q = q + 2 .
// i+1 i i+1 i
//
// With some algebraic manipulation, it is not difficult to see
// that (2) is equivalent to
// -(i+1)
// s + 2 <= y (3)
// i i
//
// The advantage of (3) is that s and y can be computed by
// i i
// the following recurrence formula:
// if (3) is false
//
// s = s , y = y ; (4)
// i+1 i i+1 i
//
// otherwise,
// -i -(i+1)
// s = s + 2 , y = y - s - 2 (5)
// i+1 i i+1 i i
//
// One may easily use induction to prove (4) and (5).
// Note. Since the left hand side of (3) contain only i+2 bits,
// it does not necessary to do a full (53-bit) comparison
// in (3).
// 3. Final rounding
// After generating the 53 bits result, we compute one more bit.
// Together with the remainder, we can decide whether the
// result is exact, bigger than 1/2ulp, or less than 1/2ulp
// (it will never equal to 1/2ulp).
// The rounding mode can be detected by checking whether
// huge + tiny is equal to huge, and whether huge - tiny is
// equal to huge for some floating point number "huge" and "tiny".
//
//
// Notes: Rounding mode detection omitted. The constants "mask", "shift",
// and "bias" are found in src/math/bits.go

// Sqrt returns the square root of x.
//
// Special cases are:
// Sqrt(+Inf) = +Inf
// Sqrt(±0) = ±0
// Sqrt(x < 0) = NaN
// Sqrt(NaN) = NaN
func Sqrt(x float64) float64

// Note: Sqrt is implemented in assembly on some systems.
// Others have assembly stubs that jump to func sqrt below.
// On systems where Sqrt is a single instruction, the compiler
// may turn a direct call into a direct use of that instruction instead.

func sqrt(x float64) float64 {
// special cases
switch {
case x == 0 || IsNaN(x) || IsInf(x, 1):
return x
case x < 0:
return NaN()
}
ix := Float64bits(x)
// normalize x
exp := int((ix >> shift) & mask)
if exp == 0 { // subnormal x
for ix&(1<<shift) == 0 {
ix <<= 1
exp--
}
exp++
}
exp -= bias // unbias exponent
ix &^= mask << shift
ix |= 1 << shift
if exp&1 == 1 { // odd exp, double x to make it even
ix <<= 1
}
exp >>= 1 // exp = exp/2, exponent of square root
// generate sqrt(x) bit by bit
ix <<= 1
var q, s uint64 // q = sqrt(x)
r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
for r != 0 {
t := s + r
if t <= ix {
s = t + r
ix -= t
q += r
}
ix <<= 1
r >>= 1
}
// final rounding
if ix != 0 { // remainder, result not exact
q += q & 1 // round according to extra bit
}
ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
return Float64frombits(ix)
}

IEEE二进制浮点数算术标准(IEEE 754)