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
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
|
module math
import math.internal
const (
f64_max_exp = f64(1024)
f64_min_exp = f64(-1021)
threshold = 7.09782712893383973096e+02 // 0x40862E42FEFA39EF
ln2_x56 = 3.88162421113569373274e+01 // 0x4043687a9f1af2b1
ln2_halfx3 = 1.03972077083991796413e+00 // 0x3ff0a2b23f3bab73
ln2_half = 3.46573590279972654709e-01 // 0x3fd62e42fefa39ef
ln2hi = 6.93147180369123816490e-01 // 0x3fe62e42fee00000
ln2lo = 1.90821492927058770002e-10 // 0x3dea39ef35793c76
inv_ln2 = 1.44269504088896338700e+00 // 0x3ff71547652b82fe
// scaled coefficients related to expm1
expm1_q1 = -3.33333333333331316428e-02 // 0xBFA11111111110F4
expm1_q2 = 1.58730158725481460165e-03 // 0x3F5A01A019FE5585
expm1_q3 = -7.93650757867487942473e-05 // 0xBF14CE199EAADBB7
expm1_q4 = 4.00821782732936239552e-06 // 0x3ED0CFCA86E65239
expm1_q5 = -2.01099218183624371326e-07 // 0xBE8AFDB76E09C32D
)
// exp returns e**x, the base-e exponential of x.
//
// special cases are:
// exp(+inf) = +inf
// exp(nan) = nan
// Very large values overflow to 0 or +inf.
// Very small values underflow to 1.
pub fn exp(x f64) f64 {
log2e := 1.44269504088896338700e+00
overflow := 7.09782712893383973096e+02
underflow := -7.45133219101941108420e+02
near_zero := 1.0 / (1 << 28) // 2**-28
// special cases
if is_nan(x) || is_inf(x, 1) {
return x
}
if is_inf(x, -1) {
return 0.0
}
if x > overflow {
return inf(1)
}
if x < underflow {
return 0.0
}
if -near_zero < x && x < near_zero {
return 1.0 + x
}
// reduce; computed as r = hi - lo for extra precision.
mut k := 0
if x < 0 {
k = int(log2e * x - 0.5)
}
if x > 0 {
k = int(log2e * x + 0.5)
}
hi := x - f64(k) * math.ln2hi
lo := f64(k) * math.ln2lo
// compute
return expmulti(hi, lo, k)
}
// exp2 returns 2**x, the base-2 exponential of x.
//
// special cases are the same as exp.
pub fn exp2(x f64) f64 {
overflow := 1.0239999999999999e+03
underflow := -1.0740e+03
if is_nan(x) || is_inf(x, 1) {
return x
}
if is_inf(x, -1) {
return 0
}
if x > overflow {
return inf(1)
}
if x < underflow {
return 0
}
// argument reduction; x = r×lg(e) + k with |r| ≤ ln(2)/2.
// computed as r = hi - lo for extra precision.
mut k := 0
if x > 0 {
k = int(x + 0.5)
}
if x < 0 {
k = int(x - 0.5)
}
mut t := x - f64(k)
hi := t * math.ln2hi
lo := -t * math.ln2lo
// compute
return expmulti(hi, lo, k)
}
pub fn ldexp(x f64, e int) f64 {
if x == 0.0 {
return x
} else {
mut y, ex := frexp(x)
mut e2 := f64(e + ex)
if e2 >= math.f64_max_exp {
y *= pow(2.0, e2 - math.f64_max_exp + 1.0)
e2 = math.f64_max_exp - 1.0
} else if e2 <= math.f64_min_exp {
y *= pow(2.0, e2 - math.f64_min_exp - 1.0)
e2 = math.f64_min_exp + 1.0
}
return y * pow(2.0, e2)
}
}
// frexp breaks f into a normalized fraction
// and an integral power of two.
// It returns frac and exp satisfying f == frac × 2**exp,
// with the absolute value of frac in the interval [½, 1).
//
// special cases are:
// frexp(±0) = ±0, 0
// frexp(±inf) = ±inf, 0
// frexp(nan) = nan, 0
// pub fn frexp(f f64) (f64, int) {
// // special cases
// if f == 0.0 {
// return f, 0 // correctly return -0
// }
// if is_inf(f, 0) || is_nan(f) {
// return f, 0
// }
// f_norm, mut exp := normalize(f)
// mut x := f64_bits(f_norm)
// exp += int((x>>shift)&mask) - bias + 1
// x &= ~(mask << shift)
// x |= (-1 + bias) << shift
// return f64_from_bits(x), exp
pub fn frexp(x f64) (f64, int) {
if x == 0.0 {
return 0.0, 0
} else if !is_finite(x) {
return x, 0
} else if abs(x) >= 0.5 && abs(x) < 1 { // Handle the common case
return x, 0
} else {
ex := ceil(log(abs(x)) / ln2)
mut ei := int(ex) // Prevent underflow and overflow of 2**(-ei)
if ei < int(math.f64_min_exp) {
ei = int(math.f64_min_exp)
}
if ei > -int(math.f64_min_exp) {
ei = -int(math.f64_min_exp)
}
mut f := x * pow(2.0, -ei)
if !is_finite(f) { // This should not happen
return f, 0
}
for abs(f) >= 1.0 {
ei++
f /= 2.0
}
for abs(f) > 0 && abs(f) < 0.5 {
ei--
f *= 2.0
}
return f, ei
}
}
// special cases are:
// expm1(+inf) = +inf
// expm1(-inf) = -1
// expm1(nan) = nan
pub fn expm1(x f64) f64 {
if is_inf(x, 1) || is_nan(x) {
return x
}
if is_inf(x, -1) {
return f64(-1)
}
// FIXME: this should be improved
if abs(x) < ln2 { // Compute the taylor series S = x + (1/2!) x^2 + (1/3!) x^3 + ...
mut i := 1.0
mut sum := x
mut term := x / 1.0
i++
term *= x / f64(i)
sum += term
for abs(term) > abs(sum) * internal.f64_epsilon {
i++
term *= x / f64(i)
sum += term
}
return sum
} else {
return exp(x) - 1
}
}
// exp1 returns e**r × 2**k where r = hi - lo and |r| ≤ ln(2)/2.
fn expmulti(hi f64, lo f64, k int) f64 {
exp_p1 := 1.66666666666666657415e-01 // 0x3FC55555; 0x55555555
exp_p2 := -2.77777777770155933842e-03 // 0xBF66C16C; 0x16BEBD93
exp_p3 := 6.61375632143793436117e-05 // 0x3F11566A; 0xAF25DE2C
exp_p4 := -1.65339022054652515390e-06 // 0xBEBBBD41; 0xC5D26BF1
exp_p5 := 4.13813679705723846039e-08 // 0x3E663769; 0x72BEA4D0
r := hi - lo
t := r * r
c := r - t * (exp_p1 + t * (exp_p2 + t * (exp_p3 + t * (exp_p4 + t * exp_p5))))
y := 1 - ((lo - (r * c) / (2 - c)) - hi)
// TODO(rsc): make sure ldexp can handle boundary k
return ldexp(y, k)
}
|
