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
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
|
use core::{f32, f64};
use super::scalbn;
const ZEROINFNAN: i32 = 0x7ff - 0x3ff - 52 - 1;
struct Num {
m: u64,
e: i32,
sign: i32,
}
fn normalize(x: f64) -> Num {
let x1p63: f64 = f64::from_bits(0x43e0000000000000); // 0x1p63 === 2 ^ 63
let mut ix: u64 = x.to_bits();
let mut e: i32 = (ix >> 52) as i32;
let sign: i32 = e & 0x800;
e &= 0x7ff;
if e == 0 {
ix = (x * x1p63).to_bits();
e = (ix >> 52) as i32 & 0x7ff;
e = if e != 0 { e - 63 } else { 0x800 };
}
ix &= (1 << 52) - 1;
ix |= 1 << 52;
ix <<= 1;
e -= 0x3ff + 52 + 1;
Num { m: ix, e, sign }
}
fn mul(x: u64, y: u64) -> (u64, u64) {
let t1: u64;
let t2: u64;
let t3: u64;
let xlo: u64 = x as u32 as u64;
let xhi: u64 = x >> 32;
let ylo: u64 = y as u32 as u64;
let yhi: u64 = y >> 32;
t1 = xlo * ylo;
t2 = xlo * yhi + xhi * ylo;
t3 = xhi * yhi;
let lo = t1.wrapping_add(t2 << 32);
let hi = t3 + (t2 >> 32) + (t1 > lo) as u64;
(hi, lo)
}
/// Floating multiply add (f64)
///
/// Computes `(x*y)+z`, rounded as one ternary operation:
/// Computes the value (as if) to infinite precision and rounds once to the result format,
/// according to the rounding mode characterized by the value of FLT_ROUNDS.
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn fma(x: f64, y: f64, z: f64) -> f64 {
let x1p63: f64 = f64::from_bits(0x43e0000000000000); // 0x1p63 === 2 ^ 63
let x0_ffffff8p_63 = f64::from_bits(0x3bfffffff0000000); // 0x0.ffffff8p-63
/* normalize so top 10bits and last bit are 0 */
let nx = normalize(x);
let ny = normalize(y);
let nz = normalize(z);
if nx.e >= ZEROINFNAN || ny.e >= ZEROINFNAN {
return x * y + z;
}
if nz.e >= ZEROINFNAN {
if nz.e > ZEROINFNAN {
/* z==0 */
return x * y + z;
}
return z;
}
/* mul: r = x*y */
let zhi: u64;
let zlo: u64;
let (mut rhi, mut rlo) = mul(nx.m, ny.m);
/* either top 20 or 21 bits of rhi and last 2 bits of rlo are 0 */
/* align exponents */
let mut e: i32 = nx.e + ny.e;
let mut d: i32 = nz.e - e;
/* shift bits z<<=kz, r>>=kr, so kz+kr == d, set e = e+kr (== ez-kz) */
if d > 0 {
if d < 64 {
zlo = nz.m << d;
zhi = nz.m >> (64 - d);
} else {
zlo = 0;
zhi = nz.m;
e = nz.e - 64;
d -= 64;
if d == 0 {
} else if d < 64 {
rlo = rhi << (64 - d) | rlo >> d | ((rlo << (64 - d)) != 0) as u64;
rhi = rhi >> d;
} else {
rlo = 1;
rhi = 0;
}
}
} else {
zhi = 0;
d = -d;
if d == 0 {
zlo = nz.m;
} else if d < 64 {
zlo = nz.m >> d | ((nz.m << (64 - d)) != 0) as u64;
} else {
zlo = 1;
}
}
/* add */
let mut sign: i32 = nx.sign ^ ny.sign;
let samesign: bool = (sign ^ nz.sign) == 0;
let mut nonzero: i32 = 1;
if samesign {
/* r += z */
rlo = rlo.wrapping_add(zlo);
rhi += zhi + (rlo < zlo) as u64;
} else {
/* r -= z */
let (res, borrow) = rlo.overflowing_sub(zlo);
rlo = res;
rhi = rhi.wrapping_sub(zhi.wrapping_add(borrow as u64));
if (rhi >> 63) != 0 {
rlo = (rlo as i64).wrapping_neg() as u64;
rhi = (rhi as i64).wrapping_neg() as u64 - (rlo != 0) as u64;
sign = (sign == 0) as i32;
}
nonzero = (rhi != 0) as i32;
}
/* set rhi to top 63bit of the result (last bit is sticky) */
if nonzero != 0 {
e += 64;
d = rhi.leading_zeros() as i32 - 1;
/* note: d > 0 */
rhi = rhi << d | rlo >> (64 - d) | ((rlo << d) != 0) as u64;
} else if rlo != 0 {
d = rlo.leading_zeros() as i32 - 1;
if d < 0 {
rhi = rlo >> 1 | (rlo & 1);
} else {
rhi = rlo << d;
}
} else {
/* exact +-0 */
return x * y + z;
}
e -= d;
/* convert to double */
let mut i: i64 = rhi as i64; /* i is in [1<<62,(1<<63)-1] */
if sign != 0 {
i = -i;
}
let mut r: f64 = i as f64; /* |r| is in [0x1p62,0x1p63] */
if e < -1022 - 62 {
/* result is subnormal before rounding */
if e == -1022 - 63 {
let mut c: f64 = x1p63;
if sign != 0 {
c = -c;
}
if r == c {
/* min normal after rounding, underflow depends
on arch behaviour which can be imitated by
a double to float conversion */
let fltmin: f32 = (x0_ffffff8p_63 * f32::MIN_POSITIVE as f64 * r) as f32;
return f64::MIN_POSITIVE / f32::MIN_POSITIVE as f64 * fltmin as f64;
}
/* one bit is lost when scaled, add another top bit to
only round once at conversion if it is inexact */
if (rhi << 53) != 0 {
i = (rhi >> 1 | (rhi & 1) | 1 << 62) as i64;
if sign != 0 {
i = -i;
}
r = i as f64;
r = 2. * r - c; /* remove top bit */
/* raise underflow portably, such that it
cannot be optimized away */
{
let tiny: f64 = f64::MIN_POSITIVE / f32::MIN_POSITIVE as f64 * r;
r += (tiny * tiny) * (r - r);
}
}
} else {
/* only round once when scaled */
d = 10;
i = ((rhi >> d | ((rhi << (64 - d)) != 0) as u64) << d) as i64;
if sign != 0 {
i = -i;
}
r = i as f64;
}
}
scalbn(r, e)
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn fma_segfault() {
// These two inputs cause fma to segfault on release due to overflow:
assert_eq!(
fma(
-0.0000000000000002220446049250313,
-0.0000000000000002220446049250313,
-0.0000000000000002220446049250313
),
-0.00000000000000022204460492503126,
);
let result = fma(-0.992, -0.992, -0.992);
//force rounding to storage format on x87 to prevent superious errors.
#[cfg(all(target_arch = "x86", not(target_feature = "sse2")))]
let result = force_eval!(result);
assert_eq!(result, -0.007936000000000007,);
}
#[test]
fn fma_sbb() {
assert_eq!(
fma(-(1.0 - f64::EPSILON), f64::MIN, f64::MIN),
-3991680619069439e277
);
}
#[test]
fn fma_underflow() {
assert_eq!(
fma(1.1102230246251565e-16, -9.812526705433188e-305, 1.0894e-320),
0.0,
);
}
}
|
