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Sixth root
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| /* | |
| Copyright 2024 Radzivon Bartoshyk | |
| Licensed under the Apache License, Version 2.0 (the "License"); | |
| you may not use this file except in compliance with the License. | |
| You may obtain a copy of the License at | |
| http://www.apache.org/licenses/LICENSE-2.0 | |
| Unless required by applicable law or agreed to in writing, software | |
| distributed under the License is distributed on an "AS IS" BASIS, | |
| WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. | |
| See the License for the specific language governing permissions and | |
| limitations under the License. | |
| */ | |
| /// Optional FMA, if it is available hardware FMA will use, if not then just scalar `c + a * b` | |
| #[inline(always)] | |
| pub(crate) fn fmla(a: f64, b: f64, c: f64) -> f64 { | |
| #[cfg(any( | |
| all( | |
| any(target_arch = "x86", target_arch = "x86_64"), | |
| target_feature = "fma" | |
| ), | |
| all(target_arch = "aarch64", target_feature = "neon") | |
| ))] | |
| { | |
| f64::mul_add(a, b, c) | |
| } | |
| #[cfg(not(any( | |
| all( | |
| any(target_arch = "x86", target_arch = "x86_64"), | |
| target_feature = "fma" | |
| ), | |
| all(target_arch = "aarch64", target_feature = "neon") | |
| )))] | |
| { | |
| a * b + c | |
| } | |
| } | |
| /// Optional FMA, if it is available hardware FMA will use, if not then just scalar `c + a * b` | |
| #[inline(always)] | |
| pub(crate) fn fmlaf(a: f32, b: f32, c: f32) -> f32 { | |
| #[cfg(any( | |
| all( | |
| any(target_arch = "x86", target_arch = "x86_64"), | |
| target_feature = "fma" | |
| ), | |
| all(target_arch = "aarch64", target_feature = "neon") | |
| ))] | |
| { | |
| f32::mul_add(a, b, c) | |
| } | |
| #[cfg(not(any( | |
| all( | |
| any(target_arch = "x86", target_arch = "x86_64"), | |
| target_feature = "fma" | |
| ), | |
| all(target_arch = "aarch64", target_feature = "neon") | |
| )))] | |
| { | |
| a * b + c | |
| } | |
| } | |
| #[inline] | |
| fn householders_method(y: f64, a: f64) -> f64 { | |
| let y2 = y * y; | |
| let y3 = y2 * y; | |
| let y6 = y3 * y3; | |
| let num = fmla(5., y6, 7. * a); | |
| let den = fmla(7., y6, 5. * a); | |
| // y (7 * a + 5 * y^6) / (5a + 7*y^6) | |
| y * (num / den) | |
| } | |
| #[inline] | |
| fn householders_methodf(y: f32, a: f32) -> f32 { | |
| let y2 = y * y; | |
| let y3 = y2 * y; | |
| let y6 = y3 * y3; | |
| let num = fmlaf(5., y6, 7. * a); | |
| let den = fmlaf(7., y6, 5. * a); | |
| // y (7 * a + 5 * y^6) / (5a + 7*y^6) | |
| y * (num / den) | |
| } | |
| #[inline] | |
| fn newtons_method(y: f64, a: f64) -> f64 { | |
| let y2 = y * y; | |
| let y3 = y2 * y; | |
| let y6 = y3 * y3; | |
| let r = a / y6; | |
| let num = r + 5.; | |
| // y (a/y^6 + 5) / 6 | |
| (num * (1. / 6.)) * y | |
| } | |
| #[inline] | |
| fn newtons_methodf(y: f32, a: f32) -> f32 { | |
| let y2 = y * y; | |
| let y3 = y2 * y; | |
| let y6 = y3 * y3; | |
| let r = a / y6; | |
| let num = r + 5.; | |
| // y (a/y^6 + 5) / 6 | |
| (num * (1. / 6.)) * y | |
| } | |
| #[inline] | |
| fn fast_ldexp(d: f64, i: i32) -> f64 { | |
| let mut u = d.to_bits(); | |
| u = u.wrapping_add((i as u64).wrapping_shl(52)); | |
| f64::from_bits(u) | |
| } | |
| #[inline] | |
| fn fast_ldexpf(d: f32, i: i32) -> f32 { | |
| let mut u = d.to_bits(); | |
| u = u.wrapping_add((i as u32).wrapping_shl(23)); | |
| f32::from_bits(u) | |
| } | |
| pub(crate) fn sixth_root(x: f32) -> f32 { | |
| let ix = x.to_bits(); | |
| // filter out exceptional cases | |
| if ix >= 0xffu32 << 23 || ix == 0 { | |
| if ix.wrapping_shl(1) == 0 { | |
| return 0.; // +/-0 | |
| } | |
| if (ix >> 31) != 0 { | |
| return f32::NAN; | |
| } | |
| if ix.wrapping_shl(9) == 0 { | |
| return f32::INFINITY; | |
| } | |
| return x + x; // nan | |
| } | |
| let exp = ((ix >> 23) & 0xff) as i32; | |
| let mut e = exp; | |
| let mut mant = ix & ((1u32 << 23) - 1); | |
| // Normalize subnormal | |
| if exp == 0 { | |
| let norm = x * f32::from_bits(0x4b800000); // * 2^24 | |
| let norm_bits = norm.to_bits(); | |
| mant = norm_bits & ((1u32 << 23) - 1); | |
| e = ((norm_bits >> 23) & 0x7ff) as i32 - 24; | |
| } | |
| // Unbias exponent | |
| e -= 127; | |
| // Restore implicit leading 1 for normal numbers | |
| mant |= 0x7f << 23; // 0x7f = 127 -> exponent of 0 in biased form | |
| let m = f32::from_bits(mant) as f64; | |
| // split exponent e = 6*q + r with r in {0,1,2,4,5} | |
| // use div_euclid/rem_euclid to get r >= 0 | |
| let q = e.div_euclid(6); | |
| let rem_scale = e.rem_euclid(6); | |
| // initial guess, estrin scheme | |
| // Polynomial generated by Sollya: | |
| // d = [1.0, 2.0]; | |
| // f_root6 = x^(1/6); | |
| // Q = fpminimax(f_root6, 4, [|D...|], d, relative, floating); | |
| const C: [u64; 5] = [ | |
| 0x3fe76acf3da4521c, | |
| 0x3fda7b37caf775d8, | |
| 0xbfc923c79d43baba, | |
| 0x3fadc0e4139bd21b, | |
| 0xbf7dbc63861d5ca5, | |
| ]; | |
| let m2 = m * m; | |
| let p01 = fmla(m, f64::from_bits(C[1]), f64::from_bits(C[0])); | |
| let p23 = fmla(m, f64::from_bits(C[3]), f64::from_bits(C[2])); | |
| let t = fmla(m2, f64::from_bits(C[4]), p23); | |
| let guess = fmla(m2, t, p01); | |
| // 1; 2^{1/6}; 2^{2/6}; 2^{3/6}; 2^{4/6}; 2^{5/6} | |
| static ESCALE: [u64; 6] = [ | |
| 1.0f64.to_bits(), | |
| 0x3ff1f59ac3c7d6c0, | |
| 0x3ff428a2f98d728b, | |
| 0x3ff6a09e667f3bcd, | |
| 0x3ff965fea53d6e3c, | |
| 0x3ffc823e074ec129, | |
| ]; | |
| let z = guess * f64::from_bits(ESCALE[rem_scale as usize]); | |
| let mm = fast_ldexp(m, rem_scale); // bring domain into [1;8] | |
| let y0 = householders_method(z, mm); | |
| fast_ldexp(y0, q) as f32 | |
| } | |
| pub(crate) fn sixth_rootf(x: f32) -> f32 { | |
| let ix = x.to_bits(); | |
| // filter out exceptional cases | |
| if ix >= 0xffu32 << 23 || ix == 0 { | |
| if ix.wrapping_shl(1) == 0 { | |
| return 0.; // +/-0 | |
| } | |
| if (ix >> 31) != 0 { | |
| return f32::NAN; | |
| } | |
| if ix.wrapping_shl(9) == 0 { | |
| return f32::INFINITY; | |
| } | |
| return x + x; // nan | |
| } | |
| let exp = ((ix >> 23) & 0xff) as i32; | |
| let mut e = exp; | |
| let mut mant = ix & ((1u32 << 23) - 1); | |
| // Normalize subnormal | |
| if exp == 0 { | |
| let norm = x * f32::from_bits(0x4b800000); // * 2^24 | |
| let norm_bits = norm.to_bits(); | |
| mant = norm_bits & ((1u32 << 23) - 1); | |
| e = ((norm_bits >> 23) & 0x7ff) as i32 - 24; | |
| } | |
| // Unbias exponent | |
| e -= 127; | |
| // Restore implicit leading 1 for normal numbers | |
| mant |= 0x7f << 23; // 0x7f = 127 -> exponent of 0 in biased form | |
| let m = f32::from_bits(mant); | |
| // split exponent e = 6*q + r with r in {0,1,2,4,5} | |
| // use div_euclid/rem_euclid to get r >= 0 | |
| let q = e.div_euclid(6); | |
| let rem_scale = e.rem_euclid(6); | |
| // initial guess, estrin scheme | |
| // Polynomial generated by Sollya: | |
| // d = [1.0, 2.0]; | |
| // f_root6 = x^(1/6); | |
| // Q = fpminimax(f_root6, 4, [|SG...|], d, relative, floating); | |
| const C: [u32; 5] = [0x3f3b567d, 0x3ed3d9ad, 0xbe491e19, 0x3d6e06e0, 0xbbede2c7]; | |
| let m2 = m * m; | |
| let p01 = fmlaf(m, f32::from_bits(C[1]), f32::from_bits(C[0])); | |
| let p23 = fmlaf(m, f32::from_bits(C[3]), f32::from_bits(C[2])); | |
| let t = fmlaf(m2, f32::from_bits(C[4]), p23); | |
| let guess = fmlaf(m2, t, p01); | |
| // 1; 2^{1/6}; 2^{2/6}; 2^{3/6}; 2^{4/6}; 2^{5/6} | |
| static ESCALE: [u32; 6] = [ | |
| 0x3f800000, 0x3f8facd6, 0x3fa14518, 0x3fb504f3, 0x3fcb2ff5, 0x3fe411f0, | |
| ]; | |
| let z = guess * f32::from_bits(ESCALE[rem_scale as usize]); | |
| let mm = fast_ldexpf(m, rem_scale); // bring domain into [1;8] | |
| let y0 = newtons_methodf(z, mm); | |
| fast_ldexpf(y0, q) | |
| } |
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