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
215
|
// Special Cases:
//
// - exp(+inf) = +inf
// - exp(nan) = nan
const std = @import("../index.zig");
const math = std.math;
const assert = std.debug.assert;
pub fn exp(x: var) @typeOf(x) {
const T = @typeOf(x);
return switch (T) {
f32 => exp32(x),
f64 => exp64(x),
else => @compileError("exp not implemented for " ++ @typeName(T)),
};
}
fn exp32(x_: f32) f32 {
const half = []f32 { 0.5, -0.5 };
const ln2hi = 6.9314575195e-1;
const ln2lo = 1.4286067653e-6;
const invln2 = 1.4426950216e+0;
const P1 = 1.6666625440e-1;
const P2 = -2.7667332906e-3;
var x = x_;
var hx = @bitCast(u32, x);
const sign = i32(hx >> 31);
hx &= 0x7FFFFFFF;
if (math.isNan(x)) {
return x;
}
// |x| >= -87.33655 or nan
if (hx >= 0x42AEAC50) {
// nan
if (hx > 0x7F800000) {
return x;
}
// x >= 88.722839
if (hx >= 0x42b17218 and sign == 0) {
return x * 0x1.0p127;
}
if (sign != 0) {
math.forceEval(-0x1.0p-149 / x); // overflow
// x <= -103.972084
if (hx >= 0x42CFF1B5) {
return 0;
}
}
}
var k: i32 = undefined;
var hi: f32 = undefined;
var lo: f32 = undefined;
// |x| > 0.5 * ln2
if (hx > 0x3EB17218) {
// |x| > 1.5 * ln2
if (hx > 0x3F851592) {
k = i32(invln2 * x + half[usize(sign)]);
}
else {
k = 1 - sign - sign;
}
const fk = f32(k);
hi = x - fk * ln2hi;
lo = fk * ln2lo;
x = hi - lo;
}
// |x| > 2^(-14)
else if (hx > 0x39000000) {
k = 0;
hi = x;
lo = 0;
}
else {
math.forceEval(0x1.0p127 + x); // inexact
return 1 + x;
}
const xx = x * x;
const c = x - xx * (P1 + xx * P2);
const y = 1 + (x * c / (2 - c) - lo + hi);
if (k == 0) {
return y;
} else {
return math.scalbn(y, k);
}
}
fn exp64(x_: f64) f64 {
const half = []const f64 { 0.5, -0.5 };
const ln2hi: f64 = 6.93147180369123816490e-01;
const ln2lo: f64 = 1.90821492927058770002e-10;
const invln2: f64 = 1.44269504088896338700e+00;
const P1: f64 = 1.66666666666666019037e-01;
const P2: f64 = -2.77777777770155933842e-03;
const P3: f64 = 6.61375632143793436117e-05;
const P4: f64 = -1.65339022054652515390e-06;
const P5: f64 = 4.13813679705723846039e-08;
var x = x_;
var ux = @bitCast(u64, x);
var hx = ux >> 32;
const sign = i32(hx >> 31);
hx &= 0x7FFFFFFF;
if (math.isNan(x)) {
return x;
}
// |x| >= 708.39 or nan
if (hx >= 0x4086232B) {
// nan
if (hx > 0x7FF00000) {
return x;
}
if (x > 709.782712893383973096) {
// overflow if x != inf
if (!math.isInf(x)) {
math.raiseOverflow();
}
return math.inf(f64);
}
if (x < -708.39641853226410622) {
// underflow if x != -inf
// math.forceEval(f32(-0x1.0p-149 / x));
if (x < -745.13321910194110842) {
return 0;
}
}
}
// argument reduction
var k: i32 = undefined;
var hi: f64 = undefined;
var lo: f64 = undefined;
// |x| > 0.5 * ln2
if (hx > 0x3EB17218) {
// |x| >= 1.5 * ln2
if (hx > 0x3FF0A2B2) {
k = i32(invln2 * x + half[usize(sign)]);
}
else {
k = 1 - sign - sign;
}
const dk = f64(k);
hi = x - dk * ln2hi;
lo = dk * ln2lo;
x = hi - lo;
}
// |x| > 2^(-28)
else if (hx > 0x3E300000) {
k = 0;
hi = x;
lo = 0;
}
else {
// inexact if x != 0
// math.forceEval(0x1.0p1023 + x);
return 1 + x;
}
const xx = x * x;
const c = x - xx * (P1 + xx * (P2 + xx * (P3 + xx * (P4 + xx * P5))));
const y = 1 + (x * c / (2 - c) - lo + hi);
if (k == 0) {
return y;
} else {
return math.scalbn(y, k);
}
}
test "math.exp" {
assert(exp(f32(0.0)) == exp32(0.0));
assert(exp(f64(0.0)) == exp64(0.0));
}
test "math.exp32" {
const epsilon = 0.000001;
assert(exp32(0.0) == 1.0);
assert(math.approxEq(f32, exp32(0.0), 1.0, epsilon));
assert(math.approxEq(f32, exp32(0.2), 1.221403, epsilon));
assert(math.approxEq(f32, exp32(0.8923), 2.440737, epsilon));
assert(math.approxEq(f32, exp32(1.5), 4.481689, epsilon));
}
test "math.exp64" {
const epsilon = 0.000001;
assert(exp64(0.0) == 1.0);
assert(math.approxEq(f64, exp64(0.0), 1.0, epsilon));
assert(math.approxEq(f64, exp64(0.2), 1.221403, epsilon));
assert(math.approxEq(f64, exp64(0.8923), 2.440737, epsilon));
assert(math.approxEq(f64, exp64(1.5), 4.481689, epsilon));
}
test "math.exp32.special" {
assert(math.isPositiveInf(exp32(math.inf(f32))));
assert(math.isNan(exp32(math.nan(f32))));
}
test "math.exp64.special" {
assert(math.isPositiveInf(exp64(math.inf(f64))));
assert(math.isNan(exp64(math.nan(f64))));
}
|
