llvmbot wrote:
<!--LLVM PR SUMMARY COMMENT--> @llvm/pr-subscribers-libc Author: Muhammad Bassiouni (bassiounix) <details> <summary>Changes</summary> --- Patch is 30.62 KiB, truncated to 20.00 KiB below, full version: https://github.com/llvm/llvm-project/pull/151837.diff 9 Files Affected: - (modified) libc/shared/math.h (+1) - (added) libc/shared/math/cbrt.h (+23) - (modified) libc/src/__support/math/CMakeLists.txt (+15) - (added) libc/src/__support/math/cbrt.h (+349) - (modified) libc/src/math/generic/CMakeLists.txt (+1-9) - (modified) libc/src/math/generic/cbrt.cpp (+2-326) - (modified) libc/test/shared/CMakeLists.txt (+1) - (modified) libc/test/shared/shared_math_test.cpp (+1) - (modified) utils/bazel/llvm-project-overlay/libc/BUILD.bazel (+13-3) ``````````diff diff --git a/libc/shared/math.h b/libc/shared/math.h index 7fb736b78efa5..3714f380a27dc 100644 --- a/libc/shared/math.h +++ b/libc/shared/math.h @@ -30,6 +30,7 @@ #include "math/atanf16.h" #include "math/atanhf.h" #include "math/atanhf16.h" +#include "math/cbrt.h" #include "math/erff.h" #include "math/exp.h" #include "math/exp10.h" diff --git a/libc/shared/math/cbrt.h b/libc/shared/math/cbrt.h new file mode 100644 index 0000000000000..2f49dbd364328 --- /dev/null +++ b/libc/shared/math/cbrt.h @@ -0,0 +1,23 @@ +//===-- Shared cbrt function ------------------------------------*- C++ -*-===// +// +// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. +// See https://llvm.org/LICENSE.txt for license information. +// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception +// +//===----------------------------------------------------------------------===// + +#ifndef LLVM_LIBC_SHARED_MATH_CBRT_H +#define LLVM_LIBC_SHARED_MATH_CBRT_H + +#include "shared/libc_common.h" +#include "src/__support/math/cbrt.h" + +namespace LIBC_NAMESPACE_DECL { +namespace shared { + +using math::cbrt; + +} // namespace shared +} // namespace LIBC_NAMESPACE_DECL + +#endif // LLVM_LIBC_SHARED_MATH_CBRT_H \ No newline at end of file diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt index 9631ab5be7d3b..e1076edf1e61c 100644 --- a/libc/src/__support/math/CMakeLists.txt +++ b/libc/src/__support/math/CMakeLists.txt @@ -331,6 +331,21 @@ add_header_library( libc.src.__support.macros.optimization ) +add_header_library( + cbrt + HDRS + cbrt.h + DEPENDS + libc.src.__support.FPUtil.double_double + libc.src.__support.FPUtil.dyadic_float + libc.src.__support.FPUtil.fenv_impl + libc.src.__support.FPUtil.fp_bits + libc.src.__support.FPUtil.multiply_add + libc.src.__support.FPUtil.polyeval + libc.src.__support.macros.optimization + libc.src.__support.integer_literals +) + add_header_library( erff HDRS diff --git a/libc/src/__support/math/cbrt.h b/libc/src/__support/math/cbrt.h new file mode 100644 index 0000000000000..2b9a73c823b14 --- /dev/null +++ b/libc/src/__support/math/cbrt.h @@ -0,0 +1,349 @@ +//===-- Implementation header for erff --------------------------*- C++ -*-===// +// +// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. +// See https://llvm.org/LICENSE.txt for license information. +// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception +// +//===----------------------------------------------------------------------===// + +#ifndef LIBC_SRC___SUPPORT_MATH_CBRT_H +#define LIBC_SRC___SUPPORT_MATH_CBRT_H + +#include "src/__support/FPUtil/FEnvImpl.h" +#include "src/__support/FPUtil/FPBits.h" +#include "src/__support/FPUtil/PolyEval.h" +#include "src/__support/FPUtil/double_double.h" +#include "src/__support/FPUtil/dyadic_float.h" +#include "src/__support/FPUtil/multiply_add.h" +#include "src/__support/integer_literals.h" +#include "src/__support/macros/config.h" +#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY + +namespace LIBC_NAMESPACE_DECL { + +namespace math { + +#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0) +#define LIBC_MATH_CBRT_SKIP_ACCURATE_PASS +#endif + +namespace cbrt_internal { +using namespace fputil; + +// Initial approximation of x^(-2/3) for 1 <= x < 2. +// Polynomial generated by Sollya with: +// > P = fpminimax(x^(-2/3), 7, [|D...|], [1, 2]); +// > dirtyinfnorm(P/x^(-2/3) - 1, [1, 2]); +// 0x1.28...p-21 +LIBC_INLINE static double intial_approximation(double x) { + constexpr double COEFFS[8] = { + 0x1.bc52aedead5c6p1, -0x1.b52bfebf110b3p2, 0x1.1d8d71d53d126p3, + -0x1.de2db9e81cf87p2, 0x1.0154ca06153bdp2, -0x1.5973c66ee6da7p0, + 0x1.07bf6ac832552p-2, -0x1.5e53d9ce41cb8p-6, + }; + + double x_sq = x * x; + + double c0 = fputil::multiply_add(x, COEFFS[1], COEFFS[0]); + double c1 = fputil::multiply_add(x, COEFFS[3], COEFFS[2]); + double c2 = fputil::multiply_add(x, COEFFS[5], COEFFS[4]); + double c3 = fputil::multiply_add(x, COEFFS[7], COEFFS[6]); + + double x_4 = x_sq * x_sq; + double d0 = fputil::multiply_add(x_sq, c1, c0); + double d1 = fputil::multiply_add(x_sq, c3, c2); + + return fputil::multiply_add(x_4, d1, d0); +} + +// Get the error term for Newton iteration: +// h(x) = x^3 * a^2 - 1, +#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE +LIBC_INLINE static double get_error(const DoubleDouble &x_3, + const DoubleDouble &a_sq) { + return fputil::multiply_add(x_3.hi, a_sq.hi, -1.0) + + fputil::multiply_add(x_3.lo, a_sq.hi, x_3.hi * a_sq.lo); +} +#else +LIBC_INLINE static constexpr double get_error(const DoubleDouble &x_3, + const DoubleDouble &a_sq) { + DoubleDouble x_3_a_sq = fputil::quick_mult(a_sq, x_3); + return (x_3_a_sq.hi - 1.0) + x_3_a_sq.lo; +} +#endif + +} // namespace cbrt_internal + +// Correctly rounded cbrt algorithm: +// +// === Step 1 - Range reduction === +// For x = (-1)^s * 2^e * (1.m), we get 2 reduced arguments x_r and a as: +// x_r = 1.m +// a = (-1)^s * 2^(e % 3) * (1.m) +// Then cbrt(x) = x^(1/3) can be computed as: +// x^(1/3) = 2^(e / 3) * a^(1/3). +// +// In order to avoid division, we compute a^(-2/3) using Newton method and then +// multiply the results by a: +// a^(1/3) = a * a^(-2/3). +// +// === Step 2 - First approximation to a^(-2/3) === +// First, we use a degree-7 minimax polynomial generated by Sollya to +// approximate x_r^(-2/3) for 1 <= x_r < 2. +// p = P(x_r) ~ x_r^(-2/3), +// with relative errors bounded by: +// | p / x_r^(-2/3) - 1 | < 1.16 * 2^-21. +// +// Then we multiply with 2^(e % 3) from a small lookup table to get: +// x_0 = 2^(-2*(e % 3)/3) * p +// ~ 2^(-2*(e % 3)/3) * x_r^(-2/3) +// = a^(-2/3) +// With relative errors: +// | x_0 / a^(-2/3) - 1 | < 1.16 * 2^-21. +// This step is done in double precision. +// +// === Step 3 - First Newton iteration === +// We follow the method described in: +// Sibidanov, A. and Zimmermann, P., "Correctly rounded cubic root evaluation +// in double precision", https://core-math.gitlabpages.inria.fr/cbrt64.pdf +// to derive multiplicative Newton iterations as below: +// Let x_n be the nth approximation to a^(-2/3). Define the n^th error as: +// h_n = x_n^3 * a^2 - 1 +// Then: +// a^(-2/3) = x_n / (1 + h_n)^(1/3) +// = x_n * (1 - (1/3) * h_n + (2/9) * h_n^2 - (14/81) * h_n^3 + ...) +// using the Taylor series expansion of (1 + h_n)^(-1/3). +// +// Apply to x_0 above: +// h_0 = x_0^3 * a^2 - 1 +// = a^2 * (x_0 - a^(-2/3)) * (x_0^2 + x_0 * a^(-2/3) + a^(-4/3)), +// it's bounded by: +// |h_0| < 4 * 3 * 1.16 * 2^-21 * 4 < 2^-17. +// So in the first iteration step, we use: +// x_1 = x_0 * (1 - (1/3) * h_n + (2/9) * h_n^2 - (14/81) * h_n^3) +// Its relative error is bounded by: +// | x_1 / a^(-2/3) - 1 | < 35/242 * |h_0|^4 < 2^-70. +// Then we perform Ziv's rounding test and check if the answer is exact. +// This step is done in double-double precision. +// +// === Step 4 - Second Newton iteration === +// If the Ziv's rounding test from the previous step fails, we define the error +// term: +// h_1 = x_1^3 * a^2 - 1, +// And perform another iteration: +// x_2 = x_1 * (1 - h_1 / 3) +// with the relative errors exceed the precision of double-double. +// We then check the Ziv's accuracy test with relative errors < 2^-102 to +// compensate for rounding errors. +// +// === Step 5 - Final iteration === +// If the Ziv's accuracy test from the previous step fails, we perform another +// iteration in 128-bit precision and check for exact outputs. +// +// TODO: It is possible to replace this costly computation step with special +// exceptional handling, similar to what was done in the CORE-MATH project: +// https://gitlab.inria.fr/core-math/core-math/-/blob/master/src/binary64/cbrt/cbrt.c + +LIBC_INLINE static constexpr double cbrt(double x) { + using DoubleDouble = fputil::DoubleDouble; + using Float128 = fputil::DyadicFloat<128>; + using namespace cbrt_internal; + using FPBits = fputil::FPBits<double>; + + uint64_t x_abs = FPBits(x).abs().uintval(); + + unsigned exp_bias_correction = 682; // 1023 * 2/3 + + if (LIBC_UNLIKELY(x_abs < FPBits::min_normal().uintval() || + x_abs >= FPBits::inf().uintval())) { + if (x == 0.0 || x_abs >= FPBits::inf().uintval()) + // x is 0, Inf, or NaN. + // Make sure it works for FTZ/DAZ modes. + return static_cast<double>(x + x); + + // x is non-zero denormal number. + // Normalize x. + x *= 0x1.0p60; + exp_bias_correction -= 20; + } + + FPBits x_bits(x); + + // When using biased exponent of x in double precision, + // x_e = real_exponent_of_x + 1023 + // Then: + // x_e / 3 = real_exponent_of_x / 3 + 1023/3 + // = real_exponent_of_x / 3 + 341 + // So to make it the correct biased exponent of x^(1/3), we add + // 1023 - 341 = 682 + // to the quotient x_e / 3. + unsigned x_e = static_cast<unsigned>(x_bits.get_biased_exponent()); + unsigned out_e = (x_e / 3 + exp_bias_correction); + unsigned shift_e = x_e % 3; + + // Set x_r = 1.mantissa + double x_r = + FPBits(x_bits.get_mantissa() | + (static_cast<uint64_t>(FPBits::EXP_BIAS) << FPBits::FRACTION_LEN)) + .get_val(); + + // Set a = (-1)^x_sign * 2^(x_e % 3) * (1.mantissa) + uint64_t a_bits = x_bits.uintval() & 0x800F'FFFF'FFFF'FFFF; + a_bits |= + (static_cast<uint64_t>(shift_e + static_cast<unsigned>(FPBits::EXP_BIAS)) + << FPBits::FRACTION_LEN); + double a = FPBits(a_bits).get_val(); + + // Initial approximation of x_r^(-2/3). + double p = intial_approximation(x_r); + + // Look up for 2^(-2*n/3) used for first approximation step. + constexpr double EXP2_M2_OVER_3[3] = {1.0, 0x1.428a2f98d728bp-1, + 0x1.965fea53d6e3dp-2}; + + // x0 is an initial approximation of a^(-2/3) for 1 <= |a| < 8. + // Relative error: < 1.16 * 2^(-21). + double x0 = static_cast<double>(EXP2_M2_OVER_3[shift_e] * p); + + // First iteration in double precision. + DoubleDouble a_sq = fputil::exact_mult(a, a); + + // h0 = x0^3 * a^2 - 1 + DoubleDouble x0_sq = fputil::exact_mult(x0, x0); + DoubleDouble x0_3 = fputil::quick_mult(x0, x0_sq); + + double h0 = get_error(x0_3, a_sq); + +#ifdef LIBC_MATH_CBRT_SKIP_ACCURATE_PASS + constexpr double REL_ERROR = 0; +#else + constexpr double REL_ERROR = 0x1.0p-51; +#endif // LIBC_MATH_CBRT_SKIP_ACCURATE_PASS + + // Taylor polynomial of (1 + h)^(-1/3): + // (1 + h)^(-1/3) = 1 - h/3 + 2 h^2 / 9 - 14 h^3 / 81 + ... + constexpr double ERR_COEFFS[3] = { + -0x1.5555555555555p-2 - REL_ERROR, // -1/3 - relative_error + 0x1.c71c71c71c71cp-3, // 2/9 + -0x1.61f9add3c0ca4p-3, // -14/81 + }; + // e0 = -14 * h^2 / 81 + 2 * h / 9 - 1/3 - relative_error. + double e0 = fputil::polyeval(h0, ERR_COEFFS[0], ERR_COEFFS[1], ERR_COEFFS[2]); + double x0_h0 = x0 * h0; + + // x1 = x0 (1 - h0/3 + 2 h0^2 / 9 - 14 h0^3 / 81) + // x1 approximate a^(-2/3) with relative errors bounded by: + // | x1 / a^(-2/3) - 1 | < (34/243) h0^4 < h0 * REL_ERROR + DoubleDouble x1_dd{x0_h0 * e0, x0}; + + // r1 = x1 * a ~ a^(-2/3) * a = a^(1/3). + DoubleDouble r1 = fputil::quick_mult(a, x1_dd); + + // Lambda function to update the exponent of the result. + auto update_exponent = [=](double r) -> double { + uint64_t r_m = FPBits(r).uintval() - 0x3FF0'0000'0000'0000; + // Adjust exponent and sign. + uint64_t r_bits = + r_m + (static_cast<uint64_t>(out_e) << FPBits::FRACTION_LEN); + return FPBits(r_bits).get_val(); + }; + +#ifdef LIBC_MATH_CBRT_SKIP_ACCURATE_PASS + // TODO: We probably don't need to use double-double if accurate tests and + // passes are skipped. + return update_exponent(r1.hi + r1.lo); +#else + // Accurate checks and passes. + double r1_lower = r1.hi + r1.lo; + double r1_upper = + r1.hi + fputil::multiply_add(x0_h0, 2.0 * REL_ERROR * a, r1.lo); + + // Ziv's accuracy test. + if (LIBC_LIKELY(r1_upper == r1_lower)) { + // Test for exact outputs. + // Check if lower (52 - 17 = 35) bits are 0's. + if (LIBC_UNLIKELY((FPBits(r1_lower).uintval() & 0x0000'0007'FFFF'FFFF) == + 0)) { + double r1_err = (r1_lower - r1.hi) - r1.lo; + if (FPBits(r1_err).abs().get_val() < 0x1.0p69) + fputil::clear_except_if_required(FE_INEXACT); + } + + return update_exponent(r1_lower); + } + + // Accuracy test failed, perform another Newton iteration. + double x1 = x1_dd.hi + (e0 + REL_ERROR) * x0_h0; + + // Second iteration in double-double precision. + // h1 = x1^3 * a^2 - 1. + DoubleDouble x1_sq = fputil::exact_mult(x1, x1); + DoubleDouble x1_3 = fputil::quick_mult(x1, x1_sq); + double h1 = get_error(x1_3, a_sq); + + // e1 = -x1*h1/3. + double e1 = h1 * (x1 * -0x1.5555555555555p-2); + // x2 = x1*(1 - h1/3) = x1 + e1 ~ a^(-2/3) with relative errors < 2^-101. + DoubleDouble x2 = fputil::exact_add(x1, e1); + // r2 = a * x2 ~ a * a^(-2/3) = a^(1/3) with relative errors < 2^-100. + DoubleDouble r2 = fputil::quick_mult(a, x2); + + double r2_upper = r2.hi + fputil::multiply_add(a, 0x1.0p-102, r2.lo); + double r2_lower = r2.hi + fputil::multiply_add(a, -0x1.0p-102, r2.lo); + + // Ziv's accuracy test. + if (LIBC_LIKELY(r2_upper == r2_lower)) + return update_exponent(r2_upper); + + // TODO: Investigate removing float128 and just list exceptional cases. + // Apply another Newton iteration with ~126-bit accuracy. + Float128 x2_f128 = fputil::quick_add(Float128(x2.hi), Float128(x2.lo)); + // x2^3 + Float128 x2_3 = + fputil::quick_mul(fputil::quick_mul(x2_f128, x2_f128), x2_f128); + // a^2 + Float128 a_sq_f128 = fputil::quick_mul(Float128(a), Float128(a)); + // x2^3 * a^2 + Float128 x2_3_a_sq = fputil::quick_mul(x2_3, a_sq_f128); + // h2 = x2^3 * a^2 - 1 + Float128 h2_f128 = fputil::quick_add(x2_3_a_sq, Float128(-1.0)); + double h2 = static_cast<double>(h2_f128); + // t2 = 1 - h2 / 3 + Float128 t2 = + fputil::quick_add(Float128(1.0), Float128(h2 * (-0x1.5555555555555p-2))); + // x3 = x2 * (1 - h2 / 3) ~ a^(-2/3) + Float128 x3 = fputil::quick_mul(x2_f128, t2); + // r3 = a * x3 ~ a * a^(-2/3) = a^(1/3) + Float128 r3 = fputil::quick_mul(Float128(a), x3); + + // Check for exact cases: + Float128::MantissaType rounding_bits = + r3.mantissa & 0x0000'0000'0000'03FF'FFFF'FFFF'FFFF'FFFF_u128; + + double result = static_cast<double>(r3); + if ((rounding_bits < 0x0000'0000'0000'0000'0000'0000'0000'000F_u128) || + (rounding_bits >= 0x0000'0000'0000'03FF'FFFF'FFFF'FFFF'FFF0_u128)) { + // Output is exact. + r3.mantissa &= 0xFFFF'FFFF'FFFF'FFFF'FFFF'FFFF'FFFF'FFF0_u128; + + if (rounding_bits >= 0x0000'0000'0000'03FF'FFFF'FFFF'FFFF'FFF0_u128) { + Float128 tmp{r3.sign, r3.exponent - 123, + 0x8000'0000'0000'0000'0000'0000'0000'0000_u128}; + Float128 r4 = fputil::quick_add(r3, tmp); + result = static_cast<double>(r4); + } else { + result = static_cast<double>(r3); + } + + fputil::clear_except_if_required(FE_INEXACT); + } + + return update_exponent(result); +#endif // LIBC_MATH_CBRT_SKIP_ACCURATE_PASS +} + +} // namespace math + +} // namespace LIBC_NAMESPACE_DECL + +#endif // LIBC_SRC___SUPPORT_MATH_CBRT_H diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt index 9df9973810c77..a86619576cfc6 100644 --- a/libc/src/math/generic/CMakeLists.txt +++ b/libc/src/math/generic/CMakeLists.txt @@ -4753,15 +4753,7 @@ add_entrypoint_object( HDRS ../cbrt.h DEPENDS - libc.hdr.fenv_macros - libc.src.__support.FPUtil.double_double - libc.src.__support.FPUtil.dyadic_float - libc.src.__support.FPUtil.fenv_impl - libc.src.__support.FPUtil.fp_bits - libc.src.__support.FPUtil.multiply_add - libc.src.__support.FPUtil.polyeval - libc.src.__support.macros.optimization - libc.src.__support.integer_literals + libc.src.__support.math.cbrt ) add_entrypoint_object( diff --git a/libc/src/math/generic/cbrt.cpp b/libc/src/math/generic/cbrt.cpp index ce227e6650c84..e9b69bbf35cf6 100644 --- a/libc/src/math/generic/cbrt.cpp +++ b/libc/src/math/generic/cbrt.cpp @@ -7,334 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/cbrt.h" -#include "hdr/fenv_macros.h" -#include "src/__support/FPUtil/FEnvImpl.h" -#include "src/__support/FPUtil/FPBits.h" -#include "src/__support/FPUtil/PolyEval.h" -#include "src/__support/FPUtil/double_double.h" -#include "src/__support/FPUtil/dyadic_float.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/common.h" -#include "src/__support/integer_literals.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY - -#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0) -#define LIBC_MATH_CBRT_SKIP_ACCURATE_PASS -#endif +#include "src/__support/math/cbrt.h" namespace LIBC_NAMESPACE_DECL { -using DoubleDouble = fputil::DoubleDouble; -using Float128 = fputil::DyadicFloat<128>; - -namespace { - -// Initial approximation of x^(-2/3) for 1 <= x < 2. -// Polynomial generated by Sollya with: -// > P = fpminimax(x^(-2/3), 7, [|D...|], [1, 2]); -// > dirtyinfnorm(P/x^(-2/3) - 1, [1, 2]); -// 0x1.28...p-21 -double intial_approximation(double x) { - constexpr double COEFFS[8] = { - 0x1.bc52aedead5c6p1, -0x1.b52bfebf110b3p2, 0x1.1d8d71d53d126p3, - -0x1.de2db9e81cf87p2, 0x1.0154ca06153bdp2, -0x1.5973c66ee6da7p0, - 0x1.07bf6ac832552p-2, -0x1.5e53d9ce41cb8p-6, - }; - - double x_sq = x * x; - - double c0 = fputil::multiply_add(x, COEFFS[1], COEFFS[0]); - double c1 = fputil::multiply_add(x, COEFFS[3], COEFFS[2]); - double c2 = fputil::multiply_add(x, COEFFS[5], COEFFS[4]); - double c3 = fputil::multiply_add(x, COEFFS[7], COEFFS[6]); - - double x_4 = x_sq * x_sq; - double d0 = fputil::multiply_add(x_sq, c1, c0); - double d1 = fputil::multiply_add(x_sq, c3, c2); - - return fputil::multiply_add(x_4, d1, d0); -} - -// Get the error term for Newton iteration: -// h(x) = x^3 * a^2 - 1, -#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE -double get_error(const DoubleDouble &x_3, const DoubleDouble &a_sq) { - return fputil::multiply_add(x_3.hi, a_sq.hi, -1.0) + - fputil::multiply_add(x_3.lo, a_sq.hi, x_3.hi * a_sq.lo); -} -#else -double get_error(const DoubleDouble &x_3, const DoubleDouble &a_sq) { - DoubleDouble x_3_a_sq = fputil::quick_mult(a_sq, x_3); - return (x_3_a_sq.hi - 1.0) + x_3_a_sq.lo; -} -#endif - -} // anonymous namespace - -// Correctly rounded cbrt algorithm: -// -// === Step 1 - Range reduction === -// For x = (-1)^s * 2^e * (1.m), we get 2 reduced arguments x_r and a as: -// x_r = 1.m -// a = (-1)^s * 2^(e % 3) * (1.m) -// Then cbrt(x) = x^(1/3) can be computed as: -// x^(1/3) = 2^(e / 3) * a^(1/3). -// -// In order to avoid division, we compute a^(-2/3) using Newton method and then -// multiply the results by a: -// a^(1/3) = a * a^(-2/3). -// -// === Step 2 - First approximation to a^(-2/3) === -// First, we use a degree-7 minimax polynomial generated by Sollya to -// approximate x_r^(-2/3) for 1 <= x_r < 2. -// p = P(x_r) ~ x_r^(-2/3), -// with relative errors bounded by: -// | p / x_r^(-2/3) - 1 | < 1.16 * 2^-21. -// -// Then we multiply with 2^(e % 3) from a small lookup table to get: -// x_0 = 2^(-2*(e % 3)/3) * p -// ~ 2^(-2*(e % 3)/3) * x_r^(-2/3) -// = a^(-2/3) -// With relative errors: -// | x_0 / a^(-2/3) - 1 | < 1.16 * 2^-21. -// This step is done in double precision. -// -// === Step 3 - First Newton iteration === -// We follow the method described in: -// Sibidanov, A. and Zimmermann, P., "Correctly rounded cubic root evaluation -// in double precision", https://core-math.gitlabpages.inria.fr/cbrt64.pdf -// to derive multiplicative Newton iterations as below: -// Let x_n be the nth approximation to a^(-2/3). Define the n^th error as: -// h_n = x_n^3 * a^2 - 1 -// Then: -// a^(-2/3) = x_n / (1 + h_n)^(1/3) -// = x_n * (1 - (1/3) * h_n + (2/9) * h_n^2 - (14/81) * h_n^3 + ...) -// using the Taylor series expansion of (1 + h_n)^(-1/3). -// -// Apply to x_0 above: -// h_0 = x_0^3 * a^2 - 1 -// = a^2 * (x_0 - a^(-2/3)) * (x_0^2 + x_0 * a^(-2/3) + a^(-4/3)), -// it... [truncated] `````````` </details> https://github.com/llvm/llvm-project/pull/151837 _______________________________________________ llvm-branch-commits mailing list llvm-branch-commits@lists.llvm.org https://lists.llvm.org/cgi-bin/mailman/listinfo/llvm-branch-commits
