diff options
Diffstat (limited to 'libc/src/math/generic/cbrt.cpp')
| -rw-r--r-- | libc/src/math/generic/cbrt.cpp | 328 |
1 files changed, 2 insertions, 326 deletions
diff --git a/libc/src/math/generic/cbrt.cpp b/libc/src/math/generic/cbrt.cpp index ce227e6..e9b69bb 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'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 - -LLVM_LIBC_FUNCTION(double, cbrt, (double x)) { - 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 -} +LLVM_LIBC_FUNCTION(double, cbrt, (double x)) { return math::cbrt(x); } } // namespace LIBC_NAMESPACE_DECL |