libm/math/
fma.rs

1use core::{f32, f64};
2
3use super::scalbn;
4
5const ZEROINFNAN: i32 = 0x7ff - 0x3ff - 52 - 1;
6
7struct Num {
8    m: u64,
9    e: i32,
10    sign: i32,
11}
12
13fn normalize(x: f64) -> Num {
14    let x1p63: f64 = f64::from_bits(0x43e0000000000000); // 0x1p63 === 2 ^ 63
15
16    let mut ix: u64 = x.to_bits();
17    let mut e: i32 = (ix >> 52) as i32;
18    let sign: i32 = e & 0x800;
19    e &= 0x7ff;
20    if e == 0 {
21        ix = (x * x1p63).to_bits();
22        e = (ix >> 52) as i32 & 0x7ff;
23        e = if e != 0 { e - 63 } else { 0x800 };
24    }
25    ix &= (1 << 52) - 1;
26    ix |= 1 << 52;
27    ix <<= 1;
28    e -= 0x3ff + 52 + 1;
29    Num { m: ix, e, sign }
30}
31
32#[inline]
33fn mul(x: u64, y: u64) -> (u64, u64) {
34    let t = (x as u128).wrapping_mul(y as u128);
35    ((t >> 64) as u64, t as u64)
36}
37
38/// Floating multiply add (f64)
39///
40/// Computes `(x*y)+z`, rounded as one ternary operation:
41/// Computes the value (as if) to infinite precision and rounds once to the result format,
42/// according to the rounding mode characterized by the value of FLT_ROUNDS.
43#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
44pub fn fma(x: f64, y: f64, z: f64) -> f64 {
45    let x1p63: f64 = f64::from_bits(0x43e0000000000000); // 0x1p63 === 2 ^ 63
46    let x0_ffffff8p_63 = f64::from_bits(0x3bfffffff0000000); // 0x0.ffffff8p-63
47
48    /* normalize so top 10bits and last bit are 0 */
49    let nx = normalize(x);
50    let ny = normalize(y);
51    let nz = normalize(z);
52
53    if nx.e >= ZEROINFNAN || ny.e >= ZEROINFNAN {
54        return x * y + z;
55    }
56    if nz.e >= ZEROINFNAN {
57        if nz.e > ZEROINFNAN {
58            /* z==0 */
59            return x * y + z;
60        }
61        return z;
62    }
63
64    /* mul: r = x*y */
65    let zhi: u64;
66    let zlo: u64;
67    let (mut rhi, mut rlo) = mul(nx.m, ny.m);
68    /* either top 20 or 21 bits of rhi and last 2 bits of rlo are 0 */
69
70    /* align exponents */
71    let mut e: i32 = nx.e + ny.e;
72    let mut d: i32 = nz.e - e;
73    /* shift bits z<<=kz, r>>=kr, so kz+kr == d, set e = e+kr (== ez-kz) */
74    if d > 0 {
75        if d < 64 {
76            zlo = nz.m << d;
77            zhi = nz.m >> (64 - d);
78        } else {
79            zlo = 0;
80            zhi = nz.m;
81            e = nz.e - 64;
82            d -= 64;
83            if d == 0 {
84            } else if d < 64 {
85                rlo = rhi << (64 - d) | rlo >> d | ((rlo << (64 - d)) != 0) as u64;
86                rhi = rhi >> d;
87            } else {
88                rlo = 1;
89                rhi = 0;
90            }
91        }
92    } else {
93        zhi = 0;
94        d = -d;
95        if d == 0 {
96            zlo = nz.m;
97        } else if d < 64 {
98            zlo = nz.m >> d | ((nz.m << (64 - d)) != 0) as u64;
99        } else {
100            zlo = 1;
101        }
102    }
103
104    /* add */
105    let mut sign: i32 = nx.sign ^ ny.sign;
106    let samesign: bool = (sign ^ nz.sign) == 0;
107    let mut nonzero: i32 = 1;
108    if samesign {
109        /* r += z */
110        rlo = rlo.wrapping_add(zlo);
111        rhi += zhi + (rlo < zlo) as u64;
112    } else {
113        /* r -= z */
114        let (res, borrow) = rlo.overflowing_sub(zlo);
115        rlo = res;
116        rhi = rhi.wrapping_sub(zhi.wrapping_add(borrow as u64));
117        if (rhi >> 63) != 0 {
118            rlo = (rlo as i64).wrapping_neg() as u64;
119            rhi = (rhi as i64).wrapping_neg() as u64 - (rlo != 0) as u64;
120            sign = (sign == 0) as i32;
121        }
122        nonzero = (rhi != 0) as i32;
123    }
124
125    /* set rhi to top 63bit of the result (last bit is sticky) */
126    if nonzero != 0 {
127        e += 64;
128        d = rhi.leading_zeros() as i32 - 1;
129        /* note: d > 0 */
130        rhi = rhi << d | rlo >> (64 - d) | ((rlo << d) != 0) as u64;
131    } else if rlo != 0 {
132        d = rlo.leading_zeros() as i32 - 1;
133        if d < 0 {
134            rhi = rlo >> 1 | (rlo & 1);
135        } else {
136            rhi = rlo << d;
137        }
138    } else {
139        /* exact +-0 */
140        return x * y + z;
141    }
142    e -= d;
143
144    /* convert to double */
145    let mut i: i64 = rhi as i64; /* i is in [1<<62,(1<<63)-1] */
146    if sign != 0 {
147        i = -i;
148    }
149    let mut r: f64 = i as f64; /* |r| is in [0x1p62,0x1p63] */
150
151    if e < -1022 - 62 {
152        /* result is subnormal before rounding */
153        if e == -1022 - 63 {
154            let mut c: f64 = x1p63;
155            if sign != 0 {
156                c = -c;
157            }
158            if r == c {
159                /* min normal after rounding, underflow depends
160                on arch behaviour which can be imitated by
161                a double to float conversion */
162                let fltmin: f32 = (x0_ffffff8p_63 * f32::MIN_POSITIVE as f64 * r) as f32;
163                return f64::MIN_POSITIVE / f32::MIN_POSITIVE as f64 * fltmin as f64;
164            }
165            /* one bit is lost when scaled, add another top bit to
166            only round once at conversion if it is inexact */
167            if (rhi << 53) != 0 {
168                i = (rhi >> 1 | (rhi & 1) | 1 << 62) as i64;
169                if sign != 0 {
170                    i = -i;
171                }
172                r = i as f64;
173                r = 2. * r - c; /* remove top bit */
174
175                /* raise underflow portably, such that it
176                cannot be optimized away */
177                {
178                    let tiny: f64 = f64::MIN_POSITIVE / f32::MIN_POSITIVE as f64 * r;
179                    r += (tiny * tiny) * (r - r);
180                }
181            }
182        } else {
183            /* only round once when scaled */
184            d = 10;
185            i = ((rhi >> d | ((rhi << (64 - d)) != 0) as u64) << d) as i64;
186            if sign != 0 {
187                i = -i;
188            }
189            r = i as f64;
190        }
191    }
192    scalbn(r, e)
193}
194
195#[cfg(test)]
196mod tests {
197    use super::*;
198    #[test]
199    fn fma_segfault() {
200        // These two inputs cause fma to segfault on release due to overflow:
201        assert_eq!(
202            fma(
203                -0.0000000000000002220446049250313,
204                -0.0000000000000002220446049250313,
205                -0.0000000000000002220446049250313
206            ),
207            -0.00000000000000022204460492503126,
208        );
209
210        let result = fma(-0.992, -0.992, -0.992);
211        //force rounding to storage format on x87 to prevent superious errors.
212        #[cfg(all(target_arch = "x86", not(target_feature = "sse2")))]
213        let result = force_eval!(result);
214        assert_eq!(result, -0.007936000000000007,);
215    }
216
217    #[test]
218    fn fma_sbb() {
219        assert_eq!(fma(-(1.0 - f64::EPSILON), f64::MIN, f64::MIN), -3991680619069439e277);
220    }
221
222    #[test]
223    fn fma_underflow() {
224        assert_eq!(fma(1.1102230246251565e-16, -9.812526705433188e-305, 1.0894e-320), 0.0,);
225    }
226}
libm/math/fma.rs

libm/math/
fma.rs
