pxfm 0.1.29

Fast and accurate math
Documentation 
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

/*
 * // Copyright (c) Radzivon Bartoshyk 6/2025. All rights reserved.
 * //
 * // Redistribution and use in source and binary forms, with or without modification,
 * // are permitted provided that the following conditions are met:
 * //
 * // 1.  Redistributions of source code must retain the above copyright notice, this
 * // list of conditions and the following disclaimer.
 * //
 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
 * // this list of conditions and the following disclaimer in the documentation
 * // and/or other materials provided with the distribution.
 * //
 * // 3.  Neither the name of the copyright holder nor the names of its
 * // contributors may be used to endorse or promote products derived from
 * // this software without specific prior written permission.
 * //
 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */
use crate::common::f_fmla;
use crate::cube_roots::cbrtf::halley_refine_d;
use crate::double_double::DoubleDouble;
use crate::exponents::fast_ldexp;

pub(crate) trait CbrtBackend {
    fn fma(&self, x: f64, y: f64, z: f64) -> f64;
    fn polyeval4(&self, x: f64, a0: f64, a1: f64, a2: f64, a3: f64) -> f64;
    fn halley(&self, x: f64, a: f64) -> f64;
    fn exact_mul(&self, x: f64, y: f64) -> DoubleDouble;
    fn quick_mult_f64(&self, x: DoubleDouble, y: f64) -> DoubleDouble;
}

pub(crate) struct GenericCbrtBackend {}

impl CbrtBackend for GenericCbrtBackend {
    #[inline(always)]
    fn fma(&self, x: f64, y: f64, z: f64) -> f64 {
        f_fmla(x, y, z)
    }
    #[inline(always)]
    fn polyeval4(&self, x: f64, a0: f64, a1: f64, a2: f64, a3: f64) -> f64 {
        use crate::polyeval::f_polyeval4;
        f_polyeval4(x, a0, a1, a2, a3)
    }
    #[inline(always)]
    fn halley(&self, x: f64, a: f64) -> f64 {
        halley_refine_d(x, a)
    }
    #[inline(always)]
    fn exact_mul(&self, x: f64, y: f64) -> DoubleDouble {
        DoubleDouble::from_exact_mult(x, y)
    }
    #[inline(always)]
    fn quick_mult_f64(&self, x: DoubleDouble, y: f64) -> DoubleDouble {
        DoubleDouble::quick_mult_f64(x, y)
    }
}

#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
pub(crate) struct FmaCbrtBackend {}

#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
impl CbrtBackend for FmaCbrtBackend {
    #[inline(always)]
    fn fma(&self, x: f64, y: f64, z: f64) -> f64 {
        f64::mul_add(x, y, z)
    }
    #[inline(always)]
    fn polyeval4(&self, x: f64, a0: f64, a1: f64, a2: f64, a3: f64) -> f64 {
        use crate::polyeval::d_polyeval4;
        d_polyeval4(x, a0, a1, a2, a3)
    }
    #[inline(always)]
    fn halley(&self, x: f64, a: f64) -> f64 {
        use crate::cube_roots::cbrtf::halley_refine_d_fma;
        halley_refine_d_fma(x, a)
    }
    #[inline(always)]
    fn exact_mul(&self, x: f64, y: f64) -> DoubleDouble {
        DoubleDouble::from_exact_mult_fma(x, y)
    }
    #[inline(always)]
    fn quick_mult_f64(&self, x: DoubleDouble, y: f64) -> DoubleDouble {
        DoubleDouble::quick_mult_f64_fma(x, y)
    }
}

#[inline(always)]
fn cbrt_gen_impl<B: CbrtBackend>(x: f64, backend: B) -> f64 {
    // 1; 2^{1/3}; 2^{2/3}
    static ESCALE: [f64; 3] = [
        1.0,
        f64::from_bits(0x3ff428a2f98d728b),
        f64::from_bits(0x3ff965fea53d6e3d),
    ];

    let bits = x.to_bits();
    let mut exp = ((bits >> 52) & 0x7ff) as i32;
    let mut mant = bits & ((1u64 << 52) - 1);

    if exp == 0x7ff || x == 0.0 {
        return x + x;
    }

    // Normalize subnormal
    if exp == 0 && x != 0.0 {
        let norm = x * f64::from_bits(0x4350000000000000); // * 2^54
        let norm_bits = norm.to_bits();
        mant = norm_bits & ((1u64 << 52) - 1);
        exp = ((norm_bits >> 52) & 0x7ff) as i32 - 54;
    }

    exp -= 1023;

    mant |= 0x3ff << 52;
    let m = f64::from_bits(mant);

    // Polynomial for x^(1/3) on [1.0; 2.0]
    // Generated by Sollya:
    // d = [1.0, 2.0];
    // f_cbrt = x^(1/3);
    // Q = fpminimax(f_cbrt, 4, [|D...|], d, relative, floating);
    // See ./notes/cbrt.sollya

    let p = backend.polyeval4(
        m,
        f64::from_bits(0x3fe1b0babceeaafa),
        f64::from_bits(0x3fe2c9a3e8e06a3c),
        f64::from_bits(0xbfc4dc30afb71885),
        f64::from_bits(0x3f97a8d3e05458e4),
    );

    // split exponent e = 3*q + r with r in {0,1,2}
    // use div_euclid/rem_euclid to get r >= 0
    let q = exp.div_euclid(3);
    let rem_scale = exp.rem_euclid(3);

    let z = p * ESCALE[rem_scale as usize];

    let mm = fast_ldexp(m, rem_scale); // bring mantissa into [1;8]

    let r = 1.0 / mm;

    // One Halley's method step
    // then refine in partial double-double precision with Newton-Raphson iteration
    let y0 = backend.halley(z, mm);
    let d2y = backend.exact_mul(y0, y0);
    let d3y = backend.quick_mult_f64(d2y, y0);
    // Newton-Raphson step
    // h = (x^3 - a) * r
    // y1 = y0 - 1/3 * h * y0
    let h = ((d3y.hi - mm) + d3y.lo) * r;
    // y1 = y0 - 1/3*y0*(h.lo + h.hi) = y0 - 1/3 *y0*h.lo - 1/3 * y0 * h.hi
    let y = backend.fma(-f64::from_bits(0x3fd5555555555555), y0 * h, y0);

    f64::copysign(fast_ldexp(y, q), x)
}

#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
#[target_feature(enable = "avx", enable = "fma")]
unsafe fn cbrt_fma_impl(x: f64) -> f64 {
    cbrt_gen_impl(x, FmaCbrtBackend {})
}

/// Computes cube root
///
/// Max found ULP 0.5
pub fn f_cbrt(x: f64) -> f64 {
    #[cfg(not(any(target_arch = "x86", target_arch = "x86_64")))]
    {
        cbrt_gen_impl(x, GenericCbrtBackend {})
    }
    #[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
    {
        use std::sync::OnceLock;
        static EXECUTOR: OnceLock<unsafe fn(f64) -> f64> = OnceLock::new();
        let q = EXECUTOR.get_or_init(|| {
            if std::arch::is_x86_feature_detected!("avx")
                && std::arch::is_x86_feature_detected!("fma")
            {
                cbrt_fma_impl
            } else {
                fn def_cbrt(x: f64) -> f64 {
                    cbrt_gen_impl(x, GenericCbrtBackend {})
                }
                def_cbrt
            }
        });
        unsafe { q(x) }
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_cbrt() {
        assert_eq!(f_cbrt(0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005432309223745),
                   0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000017579026781511548);
        assert_eq!(f_cbrt(1.225158611559834), 1.0700336588124544);
        assert_eq!(f_cbrt(0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000139491540182158), 1.1173329935611586e-103);
        assert_eq!(f_cbrt(27.0), 3.0);
        assert_eq!(f_cbrt(64.0), 4.0);
        assert_eq!(f_cbrt(125.0), 5.0);
        assert_eq!(f_cbrt(216.0), 6.0);
        assert_eq!(f_cbrt(343.0), 7.0);
        assert_eq!(f_cbrt(512.0), 8.0);
        assert_eq!(f_cbrt(729.0), 9.0);
        assert_eq!(f_cbrt(-729.0), -9.0);
        assert_eq!(f_cbrt(-512.0), -8.0);
        assert_eq!(f_cbrt(-343.0), -7.0);
        assert_eq!(f_cbrt(-216.0), -6.0);
        assert_eq!(f_cbrt(-125.0), -5.0);
        assert_eq!(f_cbrt(-64.0), -4.0);
        assert_eq!(f_cbrt(-27.0), -3.0);
        assert_eq!(f_cbrt(0.0), 0.0);
        assert_eq!(f_cbrt(f64::INFINITY), f64::INFINITY);
        assert_eq!(f_cbrt(f64::NEG_INFINITY), f64::NEG_INFINITY);
        assert!(f_cbrt(f64::NAN).is_nan());
    }
}