| /* SPDX-License-Identifier: MIT */ |
| /* origin: core-math/src/binary64/cbrt/cbrt.c |
| * Copyright (c) 2021-2022 Alexei Sibidanov. |
| * Ported to Rust in 2025 by Trevor Gross. |
| */ |
| |
| use super::Float; |
| use super::support::{FpResult, Round, cold_path}; |
| |
| /// Compute the cube root of the argument. |
| #[cfg_attr(assert_no_panic, no_panic::no_panic)] |
| pub fn cbrt(x: f64) -> f64 { |
| cbrt_round(x, Round::Nearest).val |
| } |
| |
| pub fn cbrt_round(x: f64, round: Round) -> FpResult<f64> { |
| const ESCALE: [f64; 3] = [ |
| 1.0, |
| hf64!("0x1.428a2f98d728bp+0"), /* 2^(1/3) */ |
| hf64!("0x1.965fea53d6e3dp+0"), /* 2^(2/3) */ |
| ]; |
| |
| /* the polynomial c0+c1*x+c2*x^2+c3*x^3 approximates x^(1/3) on [1,2] |
| with maximal error < 9.2e-5 (attained at x=2) */ |
| const C: [f64; 4] = [ |
| hf64!("0x1.1b0babccfef9cp-1"), |
| hf64!("0x1.2c9a3e94d1da5p-1"), |
| hf64!("-0x1.4dc30b1a1ddbap-3"), |
| hf64!("0x1.7a8d3e4ec9b07p-6"), |
| ]; |
| |
| let u0: f64 = hf64!("0x1.5555555555555p-2"); |
| let u1: f64 = hf64!("0x1.c71c71c71c71cp-3"); |
| |
| let rsc = [1.0, -1.0, 0.5, -0.5, 0.25, -0.25]; |
| |
| let off = [hf64!("0x1p-53"), 0.0, 0.0, 0.0]; |
| |
| /* rm=0 for rounding to nearest, and other values for directed roundings */ |
| let hx: u64 = x.to_bits(); |
| let mut mant: u64 = hx & f64::SIG_MASK; |
| let sign: u64 = hx >> 63; |
| |
| let mut e: u32 = (hx >> f64::SIG_BITS) as u32 & f64::EXP_SAT; |
| |
| if ((e + 1) & f64::EXP_SAT) < 2 { |
| cold_path(); |
| |
| let ix: u64 = hx & !f64::SIGN_MASK; |
| |
| /* 0, inf, nan: we return x + x instead of simply x, |
| to that for x a signaling NaN, it correctly triggers |
| the invalid exception. */ |
| if e == f64::EXP_SAT || ix == 0 { |
| return FpResult::ok(x + x); |
| } |
| |
| let nz = ix.leading_zeros() - 11; /* subnormal */ |
| mant <<= nz; |
| mant &= f64::SIG_MASK; |
| e = e.wrapping_sub(nz - 1); |
| } |
| |
| e = e.wrapping_add(3072); |
| let cvt1: u64 = mant | (0x3ffu64 << 52); |
| let mut cvt5: u64 = cvt1; |
| |
| let et: u32 = e / 3; |
| let it: u32 = e % 3; |
| |
| /* 2^(3k+it) <= x < 2^(3k+it+1), with 0 <= it <= 3 */ |
| cvt5 += u64::from(it) << f64::SIG_BITS; |
| cvt5 |= sign << 63; |
| let zz: f64 = f64::from_bits(cvt5); |
| |
| /* cbrt(x) = cbrt(zz)*2^(et-1365) where 1 <= zz < 8 */ |
| let mut isc: u64 = ESCALE[it as usize].to_bits(); // todo: index |
| isc |= sign << 63; |
| let cvt2: u64 = isc; |
| let z: f64 = f64::from_bits(cvt1); |
| |
| /* cbrt(zz) = cbrt(z)*isc, where isc encodes 1, 2^(1/3) or 2^(2/3), |
| and 1 <= z < 2 */ |
| let r: f64 = 1.0 / z; |
| let rr: f64 = r * rsc[((it as usize) << 1) | sign as usize]; |
| let z2: f64 = z * z; |
| let c0: f64 = C[0] + z * C[1]; |
| let c2: f64 = C[2] + z * C[3]; |
| let mut y: f64 = c0 + z2 * c2; |
| let mut y2: f64 = y * y; |
| |
| /* y is an approximation of z^(1/3) */ |
| let mut h: f64 = y2 * (y * r) - 1.0; |
| |
| /* h determines the error between y and z^(1/3) */ |
| y -= (h * y) * (u0 - u1 * h); |
| |
| /* The correction y -= (h*y)*(u0 - u1*h) corresponds to a cubic variant |
| of Newton's method, with the function f(y) = 1-z/y^3. */ |
| y *= f64::from_bits(cvt2); |
| |
| /* Now y is an approximation of zz^(1/3), |
| * and rr an approximation of 1/zz. We now perform another iteration of |
| * Newton-Raphson, this time with a linear approximation only. */ |
| y2 = y * y; |
| let mut y2l: f64 = y.fma(y, -y2); |
| |
| /* y2 + y2l = y^2 exactly */ |
| let mut y3: f64 = y2 * y; |
| let mut y3l: f64 = y.fma(y2, -y3) + y * y2l; |
| |
| /* y3 + y3l approximates y^3 with about 106 bits of accuracy */ |
| h = ((y3 - zz) + y3l) * rr; |
| let mut dy: f64 = h * (y * u0); |
| |
| /* the approximation of zz^(1/3) is y - dy */ |
| let mut y1: f64 = y - dy; |
| dy = (y - y1) - dy; |
| |
| /* the approximation of zz^(1/3) is now y1 + dy, where |dy| < 1/2 ulp(y) |
| * (for rounding to nearest) */ |
| let mut ady: f64 = dy.abs(); |
| |
| /* For directed roundings, ady0 is tiny when dy is tiny, or ady0 is near |
| * from ulp(1); |
| * for rounding to nearest, ady0 is tiny when dy is near from 1/2 ulp(1), |
| * or from 3/2 ulp(1). */ |
| let mut ady0: f64 = (ady - off[round as usize]).abs(); |
| let mut ady1: f64 = (ady - (hf64!("0x1p-52") + off[round as usize])).abs(); |
| |
| if ady0 < hf64!("0x1p-75") || ady1 < hf64!("0x1p-75") { |
| cold_path(); |
| |
| y2 = y1 * y1; |
| y2l = y1.fma(y1, -y2); |
| y3 = y2 * y1; |
| y3l = y1.fma(y2, -y3) + y1 * y2l; |
| h = ((y3 - zz) + y3l) * rr; |
| dy = h * (y1 * u0); |
| y = y1 - dy; |
| dy = (y1 - y) - dy; |
| y1 = y; |
| ady = dy.abs(); |
| ady0 = (ady - off[round as usize]).abs(); |
| ady1 = (ady - (hf64!("0x1p-52") + off[round as usize])).abs(); |
| |
| if ady0 < hf64!("0x1p-98") || ady1 < hf64!("0x1p-98") { |
| cold_path(); |
| let azz: f64 = zz.abs(); |
| |
| // ~ 0x1.79d15d0e8d59b80000000000000ffc3dp+0 |
| if azz == hf64!("0x1.9b78223aa307cp+1") { |
| y1 = hf64!("0x1.79d15d0e8d59cp+0").copysign(zz); |
| } |
| |
| // ~ 0x1.de87aa837820e80000000000001c0f08p+0 |
| if azz == hf64!("0x1.a202bfc89ddffp+2") { |
| y1 = hf64!("0x1.de87aa837820fp+0").copysign(zz); |
| } |
| |
| if round != Round::Nearest { |
| let wlist = [ |
| (hf64!("0x1.3a9ccd7f022dbp+0"), hf64!("0x1.1236160ba9b93p+0")), // ~ 0x1.1236160ba9b930000000000001e7e8fap+0 |
| (hf64!("0x1.7845d2faac6fep+0"), hf64!("0x1.23115e657e49cp+0")), // ~ 0x1.23115e657e49c0000000000001d7a799p+0 |
| (hf64!("0x1.d1ef81cbbbe71p+0"), hf64!("0x1.388fb44cdcf5ap+0")), // ~ 0x1.388fb44cdcf5a0000000000002202c55p+0 |
| (hf64!("0x1.0a2014f62987cp+1"), hf64!("0x1.46bcbf47dc1e8p+0")), // ~ 0x1.46bcbf47dc1e8000000000000303aa2dp+0 |
| (hf64!("0x1.fe18a044a5501p+1"), hf64!("0x1.95decfec9c904p+0")), // ~ 0x1.95decfec9c9040000000000000159e8ep+0 |
| (hf64!("0x1.a6bb8c803147bp+2"), hf64!("0x1.e05335a6401dep+0")), // ~ 0x1.e05335a6401de00000000000027ca017p+0 |
| (hf64!("0x1.ac8538a031cbdp+2"), hf64!("0x1.e281d87098de8p+0")), // ~ 0x1.e281d87098de80000000000000ee9314p+0 |
| ]; |
| |
| for (a, b) in wlist { |
| if azz == a { |
| let tmp = if round as u64 + sign == 2 { |
| hf64!("0x1p-52") |
| } else { |
| 0.0 |
| }; |
| y1 = (b + tmp).copysign(zz); |
| } |
| } |
| } |
| } |
| } |
| |
| let mut cvt3: u64 = y1.to_bits(); |
| cvt3 = cvt3.wrapping_add(((et.wrapping_sub(342).wrapping_sub(1023)) as u64) << 52); |
| let m0: u64 = cvt3 << 30; |
| let m1 = m0 >> 63; |
| |
| if (m0 ^ m1) <= (1u64 << 30) { |
| cold_path(); |
| |
| let mut cvt4: u64 = y1.to_bits(); |
| cvt4 = (cvt4 + (164 << 15)) & 0xffffffffffff0000u64; |
| |
| if ((f64::from_bits(cvt4) - y1) - dy).abs() < hf64!("0x1p-60") || (zz).abs() == 1.0 { |
| cvt3 = (cvt3 + (1u64 << 15)) & 0xffffffffffff0000u64; |
| } |
| } |
| |
| FpResult::ok(f64::from_bits(cvt3)) |
| } |
| |
| #[cfg(test)] |
| mod tests { |
| use super::*; |
| |
| #[test] |
| fn spot_checks() { |
| if !cfg!(x86_no_sse) { |
| // Exposes a rounding mode problem. Ignored on i586 because of inaccurate FMA. |
| assert_biteq!( |
| cbrt(f64::from_bits(0xf7f792b28f600000)), |
| f64::from_bits(0xd29ce68655d962f3) |
| ); |
| } |
| } |
| } |
