pxfm/cube_roots/
rcbrt.rs

1/*
2 * // Copyright (c) Radzivon Bartoshyk 8/2025. All rights reserved.
3 * //
4 * // Redistribution and use in source and binary forms, with or without modification,
5 * // are permitted provided that the following conditions are met:
6 * //
7 * // 1.  Redistributions of source code must retain the above copyright notice, this
8 * // list of conditions and the following disclaimer.
9 * //
10 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
11 * // this list of conditions and the following disclaimer in the documentation
12 * // and/or other materials provided with the distribution.
13 * //
14 * // 3.  Neither the name of the copyright holder nor the names of its
15 * // contributors may be used to endorse or promote products derived from
16 * // this software without specific prior written permission.
17 * //
18 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
19 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
21 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
22 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
24 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
25 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
26 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
27 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
28 */
29use crate::common::f_fmla;
30use crate::double_double::DoubleDouble;
31use crate::exponents::fast_ldexp;
32use crate::polyeval::f_polyeval6;
33
34//
35// // y1 = y0 + 1/3 * y0 * (1 - a * y0 * y0 * y0)
36// #[inline]
37// fn raphson_step(x: f64, a: f64) -> f64 {
38//     let h = f_fmla(-a * x, x * x, 1.0);
39//     f_fmla(1. / 3. * h, x, x)
40// }
41
42// y1 = y0(k1 − c(k2 − k3c), c = x*y0*y0*y0
43// k1 = 14/9 , k2 = 7/9 , k3 = 2/9
44#[inline(always)]
45fn halleys_div_free(x: f64, a: f64) -> f64 {
46    const K3: f64 = 2. / 9.;
47    const K2: f64 = 7. / 9.;
48    const K1: f64 = 14. / 9.;
49    let c = a * x * x * x;
50    let mut y = f_fmla(-K3, c, K2);
51    y = f_fmla(-c, y, K1);
52    y * x
53}
54
55#[inline(always)]
56#[allow(unused)]
57fn halleys_div_free_fma(x: f64, a: f64) -> f64 {
58    const K3: f64 = 2. / 9.;
59    const K2: f64 = 7. / 9.;
60    const K1: f64 = 14. / 9.;
61    let c = a * x * x * x;
62    let mut y = f64::mul_add(-K3, c, K2);
63    y = f64::mul_add(-c, y, K1);
64    y * x
65}
66
67#[inline(always)]
68fn rcbrt_gen_impl(a: f64) -> f64 {
69    // Decompose a = m * 2^e, with m in [0.5, 1)
70    let xu = a.to_bits();
71    let exp = ((xu >> 52) & 0x7ff) as i32;
72    let mut e = ((xu >> 52) & 0x7ff) as i32;
73    let mut mant = xu & ((1u64 << 52) - 1);
74
75    if exp == 0x7ff {
76        if a.is_infinite() {
77            return if a.is_sign_negative() { -0.0 } else { 0.0 };
78        }
79        return a + a;
80    }
81    if exp == 0 && a == 0. {
82        return if a.is_sign_negative() {
83            f64::NEG_INFINITY
84        } else {
85            f64::INFINITY
86        };
87    }
88
89    // Normalize subnormal
90    if exp == 0 {
91        let norm = a * f64::from_bits(0x4350000000000000); // * 2^54
92        let norm_bits = norm.to_bits();
93        mant = norm_bits & ((1u64 << 52) - 1);
94        e = ((norm_bits >> 52) & 0x7ff) as i32 - 54;
95    }
96
97    e -= 1023;
98
99    mant |= 0x3ff << 52;
100    let m = f64::from_bits(mant);
101
102    // Polynomial for x^(-1/3) on [1.0; 2.0]
103    // Generated by Sollya:
104    // d = [1.0, 2.0];
105    // f_inv_cbrt = x^(-1/3);
106    // Q = fpminimax(f_inv_cbrt, 5, [|D...|], d, relative, floating);
107    // See ./notes/inv_cbrt.sollya
108
109    let p = f_polyeval6(
110        m,
111        f64::from_bits(0x3ffc7f365bceaf71),
112        f64::from_bits(0xbff90e741fb9c896),
113        f64::from_bits(0x3ff3e68b9b2cd237),
114        f64::from_bits(0xbfe321c5eb24a185),
115        f64::from_bits(0x3fc3fa269b897f69),
116        f64::from_bits(0xbf916d6f13849fd1),
117    );
118
119    // split exponent e = 3*q + r with r in {0,1,2}
120    // use div_euclid/rem_euclid to get r >= 0
121    let q = e.div_euclid(3);
122    let rem_scale = e.rem_euclid(3);
123
124    // 1; 2^{-1/3}; 2^{-2/3}
125    static ESCALE: [u64; 3] = [1.0f64.to_bits(), 0x3fe965fea53d6e3d, 0x3fe428a2f98d728b];
126
127    let z = p * f64::from_bits(ESCALE[rem_scale as usize]);
128
129    let mm = fast_ldexp(m, rem_scale); // bring domain into [1;8]
130
131    // One Halley's method step
132    // then refine in partial double-double precision with Newton-Raphson iteration
133    let y0 = halleys_div_free(z, mm);
134
135    let d2y = DoubleDouble::from_exact_mult(y0, y0);
136    let d3y = DoubleDouble::quick_mult_f64(d2y, y0);
137    let hb = DoubleDouble::quick_mult_f64(d3y, mm);
138
139    let y: f64;
140
141    #[cfg(any(
142        all(
143            any(target_arch = "x86", target_arch = "x86_64"),
144            target_feature = "fma"
145        ),
146        target_arch = "aarch64"
147    ))]
148    {
149        // decompose double-double in linear FMA sums
150        // r = (1.0 - hb.hi - hb.lo) * y0 = y0 - hb.hi * y0 - hb.lo * y0 = fma(-hb.lo, y0, fma(-hb.hi, y0, y0))
151        let r = f_fmla(-hb.lo, y0, f_fmla(hb.hi, -y0, y0));
152        // // y1 = y0 + 1/3 * y0 * (1 - a * y0 * y0 * y0) = y0 + 1/3 * r
153        y = f_fmla(1. / 3., r, y0);
154    }
155    #[cfg(not(any(
156        all(
157            any(target_arch = "x86", target_arch = "x86_64"),
158            target_feature = "fma"
159        ),
160        target_arch = "aarch64"
161    )))]
162    {
163        let m_hb = DoubleDouble::full_add_f64(-hb, 1.0);
164        let r = DoubleDouble::quick_mult_f64(m_hb, y0);
165        y = f_fmla(1. / 3., r.to_f64(), y0);
166    }
167    f64::copysign(fast_ldexp(y, -q), a)
168}
169
170#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
171#[target_feature(enable = "avx", enable = "fma")]
172unsafe fn rcbrt_fma_impl(a: f64) -> f64 {
173    // Decompose a = m * 2^e, with m in [0.5, 1)
174    let xu = a.to_bits();
175    let exp = ((xu >> 52) & 0x7ff) as i32;
176    let mut e = ((xu >> 52) & 0x7ff) as i32;
177    let mut mant = xu & ((1u64 << 52) - 1);
178
179    if exp == 0x7ff {
180        if a.is_infinite() {
181            return if a.is_sign_negative() { -0.0 } else { 0.0 };
182        }
183        return a + a;
184    }
185    if exp == 0 && a == 0. {
186        return if a.is_sign_negative() {
187            f64::NEG_INFINITY
188        } else {
189            f64::INFINITY
190        };
191    }
192
193    // Normalize subnormal
194    if exp == 0 {
195        let norm = a * f64::from_bits(0x4350000000000000); // * 2^54
196        let norm_bits = norm.to_bits();
197        mant = norm_bits & ((1u64 << 52) - 1);
198        e = ((norm_bits >> 52) & 0x7ff) as i32 - 54;
199    }
200
201    e -= 1023;
202
203    mant |= 0x3ff << 52;
204    let m = f64::from_bits(mant);
205
206    // Polynomial for x^(-1/3) on [1.0; 2.0]
207    // Generated by Sollya:
208    // d = [1.0, 2.0];
209    // f_inv_cbrt = x^(-1/3);
210    // Q = fpminimax(f_inv_cbrt, 5, [|D...|], d, relative, floating);
211    // See ./notes/inv_cbrt.sollya
212
213    use crate::polyeval::d_polyeval6;
214    let p = d_polyeval6(
215        m,
216        f64::from_bits(0x3ffc7f365bceaf71),
217        f64::from_bits(0xbff90e741fb9c896),
218        f64::from_bits(0x3ff3e68b9b2cd237),
219        f64::from_bits(0xbfe321c5eb24a185),
220        f64::from_bits(0x3fc3fa269b897f69),
221        f64::from_bits(0xbf916d6f13849fd1),
222    );
223
224    // split exponent e = 3*q + r with r in {0,1,2}
225    // use div_euclid/rem_euclid to get r >= 0
226    let q = e.div_euclid(3);
227    let rem_scale = e.rem_euclid(3);
228
229    // 1; 2^{-1/3}; 2^{-2/3}
230    static ESCALE: [u64; 3] = [1.0f64.to_bits(), 0x3fe965fea53d6e3d, 0x3fe428a2f98d728b];
231
232    let z = p * f64::from_bits(ESCALE[rem_scale as usize]);
233
234    let mm = fast_ldexp(m, rem_scale); // bring domain into [1;8]
235
236    // One Halley's method step
237    // then refine in partial double-double precision with Newton-Raphson iteration
238    let y0 = halleys_div_free_fma(z, mm);
239
240    let d2y = DoubleDouble::from_exact_mult_fma(y0, y0);
241    let d3y = DoubleDouble::quick_mult_f64_fma(d2y, y0);
242    let hb = DoubleDouble::quick_mult_f64_fma(d3y, mm);
243
244    // decompose double-double in linear FMA sums
245    // r = (1.0 - hb.hi - hb.lo) * y0 = y0 - hb.hi * y0 - hb.lo * y0 = fma(-hb.lo, y0, fma(-hb.hi, y0, y0))
246    let r = f64::mul_add(-hb.lo, y0, f64::mul_add(hb.hi, -y0, y0));
247    // // y1 = y0 + 1/3 * y0 * (1 - a * y0 * y0 * y0) = y0 + 1/3 * r
248    let y = f64::mul_add(1. / 3., r, y0);
249    f64::copysign(fast_ldexp(y, -q), a)
250}
251
252/// Computes 1/cbrt(x)
253///
254/// ULP 0.5
255pub fn f_rcbrt(a: f64) -> f64 {
256    #[cfg(not(any(target_arch = "x86", target_arch = "x86_64")))]
257    {
258        rcbrt_gen_impl(a)
259    }
260    #[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
261    {
262        use std::sync::OnceLock;
263        static EXECUTOR: OnceLock<unsafe fn(f64) -> f64> = OnceLock::new();
264        let q = EXECUTOR.get_or_init(|| {
265            if std::arch::is_x86_feature_detected!("avx")
266                && std::arch::is_x86_feature_detected!("fma")
267            {
268                rcbrt_fma_impl
269            } else {
270                fn def_rcbrt(x: f64) -> f64 {
271                    rcbrt_gen_impl(x)
272                }
273                def_rcbrt
274            }
275        });
276        unsafe { q(a) }
277    }
278}
279
280#[cfg(test)]
281mod tests {
282    use super::*;
283
284    #[test]
285    fn test_rcbrt() {
286        assert_eq!(f_rcbrt(0.9999999999999717), 1.0000000000000095);
287        assert_eq!(f_rcbrt(-68355745214719140000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.),
288                   -0.000000000000000000000000000000000000000002445728958868668);
289        assert_eq!(f_rcbrt(-96105972807656840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.),
290                   -0.0000000000000000000000000000000000000000000000000000000002183148143573148);
291        assert_eq!(f_rcbrt(0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000139491540182158),
292                   8949883389846071000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.);
293        assert_eq!(f_rcbrt(0.00008386280387617153), 22.846001824951983);
294        assert_eq!(f_rcbrt(-125.0), -0.2);
295        assert_eq!(f_rcbrt(125.0), 0.2);
296        assert_eq!(f_rcbrt(1.0), 1.0);
297        assert_eq!(f_rcbrt(-1.0), -1.0);
298        assert_eq!(f_rcbrt(0.0), f64::INFINITY);
299        assert_eq!(f_rcbrt(-27.0), -1. / 3.);
300        assert_eq!(
301            f_rcbrt(2417851639214765300000000.),
302            0.000000007450580596938716
303        );
304        assert_eq!(f_rcbrt(27.0), 1. / 3.);
305        assert_eq!(f_rcbrt(64.0), 0.25);
306        assert_eq!(f_rcbrt(-64.0), -0.25);
307        assert_eq!(f_rcbrt(f64::NEG_INFINITY), -0.0);
308        assert_eq!(f_rcbrt(f64::INFINITY), 0.0);
309        assert!(f_rcbrt(f64::NAN).is_nan());
310    }
311}
pxfm/cube_roots/rcbrt.rs

pxfm/cube_roots/
rcbrt.rs
