diff options
Diffstat (limited to 'libc/src/math/generic')
24 files changed, 53 insertions, 2037 deletions
diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt index 99db743..a001d99 100644 --- a/libc/src/math/generic/CMakeLists.txt +++ b/libc/src/math/generic/CMakeLists.txt @@ -1295,12 +1295,8 @@ add_entrypoint_object( HDRS ../erff.h DEPENDS - .common_constants - libc.src.__support.FPUtil.fp_bits - libc.src.__support.FPUtil.except_value_utils - libc.src.__support.FPUtil.multiply_add - libc.src.__support.FPUtil.polyeval - libc.src.__support.macros.optimization + libc.src.__support.math.erff + libc.src.errno.errno ) add_entrypoint_object( @@ -1477,20 +1473,8 @@ add_entrypoint_object( HDRS ../exp10f16.h DEPENDS - .expxf16 - libc.hdr.errno_macros - libc.hdr.fenv_macros - libc.src.__support.CPP.array - libc.src.__support.FPUtil.cast - libc.src.__support.FPUtil.except_value_utils - libc.src.__support.FPUtil.fenv_impl - libc.src.__support.FPUtil.fp_bits - libc.src.__support.FPUtil.multiply_add - libc.src.__support.FPUtil.nearest_integer - libc.src.__support.FPUtil.polyeval - libc.src.__support.FPUtil.rounding_mode - libc.src.__support.macros.optimization - libc.src.__support.macros.properties.cpu_features + libc.src.__support.math.exp10f16 + libc.src.errno.errno ) add_entrypoint_object( @@ -1519,7 +1503,6 @@ add_entrypoint_object( HDRS ../exp10m1f16.h DEPENDS - .expxf16 libc.hdr.errno_macros libc.hdr.fenv_macros libc.src.__support.FPUtil.cast @@ -1531,6 +1514,7 @@ add_entrypoint_object( libc.src.__support.FPUtil.rounding_mode libc.src.__support.macros.optimization libc.src.__support.macros.properties.cpu_features + libc.src.__support.math.exp10f16_utils ) add_entrypoint_object( @@ -1910,6 +1894,7 @@ add_object_library( common_constants.cpp DEPENDS libc.src.__support.math.exp_constants + libc.src.__support.math.acosh_float_constants libc.src.__support.number_pair ) @@ -3773,7 +3758,7 @@ add_header_library( DEPENDS .common_constants libc.src.__support.math.exp_utils - libc.src.__support.math.exp10f_utils + libc.src.__support.math.acoshf_utils libc.src.__support.macros.properties.cpu_features libc.src.errno.errno ) @@ -3883,12 +3868,7 @@ add_entrypoint_object( ../acoshf.h DEPENDS .explogxf - 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.FPUtil.sqrt - libc.src.__support.macros.optimization + libc.src.__support.math.acoshf ) add_entrypoint_object( @@ -3898,18 +3878,8 @@ add_entrypoint_object( HDRS ../acoshf16.h DEPENDS - .explogxf - libc.hdr.errno_macros - libc.hdr.fenv_macros - libc.src.__support.FPUtil.cast - libc.src.__support.FPUtil.except_value_utils - 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.FPUtil.sqrt - libc.src.__support.macros.optimization - libc.src.__support.macros.properties.types + libc.src.__support.math.acoshf16 + libc.src.errno.errno ) add_entrypoint_object( @@ -3981,18 +3951,6 @@ add_entrypoint_object( libc.src.__support.macros.properties.types ) -add_object_library( - inv_trigf_utils - HDRS - inv_trigf_utils.h - SRCS - inv_trigf_utils.cpp - DEPENDS - libc.src.__support.FPUtil.multiply_add - libc.src.__support.FPUtil.polyeval - libc.src.__support.common -) - add_entrypoint_object( asinf SRCS @@ -4006,7 +3964,7 @@ add_entrypoint_object( libc.src.__support.FPUtil.polyeval libc.src.__support.FPUtil.sqrt libc.src.__support.macros.optimization - .inv_trigf_utils + libc.src.__support.math.inv_trigf_utils ) add_entrypoint_object( @@ -4028,20 +3986,6 @@ add_entrypoint_object( libc.src.__support.macros.properties.types ) -add_header_library( - asin_utils - HDRS - atan_utils.h - DEPENDS - libc.src.__support.integer_literals - libc.src.__support.FPUtil.double_double - libc.src.__support.FPUtil.dyadic_float - libc.src.__support.FPUtil.multiply_add - libc.src.__support.FPUtil.nearest_integer - libc.src.__support.FPUtil.polyeval - libc.src.__support.macros.optimization -) - add_entrypoint_object( asin SRCS @@ -4049,7 +3993,7 @@ add_entrypoint_object( HDRS ../asin.h DEPENDS - .asin_utils + libc.src.__support.math.asin_utils libc.src.__support.FPUtil.double_double libc.src.__support.FPUtil.dyadic_float libc.src.__support.FPUtil.fenv_impl @@ -4068,13 +4012,7 @@ add_entrypoint_object( HDRS ../acosf.h DEPENDS - libc.src.__support.FPUtil.except_value_utils - libc.src.__support.FPUtil.fp_bits - libc.src.__support.FPUtil.multiply_add - libc.src.__support.FPUtil.polyeval - libc.src.__support.FPUtil.sqrt - libc.src.__support.macros.optimization - .inv_trigf_utils + libc.src.__support.math.acosf ) add_entrypoint_object( @@ -4084,17 +4022,8 @@ add_entrypoint_object( HDRS ../acosf16.h DEPENDS - libc.hdr.errno_macros - libc.hdr.fenv_macros - libc.src.__support.FPUtil.cast - libc.src.__support.FPUtil.except_value_utils - 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.FPUtil.sqrt - libc.src.__support.macros.optimization - libc.src.__support.macros.properties.types + libc.src.__support.math.acosf16 + libc.src.errno.errno ) add_entrypoint_object( @@ -4104,17 +4033,7 @@ add_entrypoint_object( HDRS ../acos.h DEPENDS - .asin_utils - 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.FPUtil.sqrt - libc.src.__support.macros.optimization - libc.src.__support.macros.properties.types - libc.src.__support.macros.properties.cpu_features + libc.src.__support.math.acos ) add_entrypoint_object( @@ -4156,7 +4075,6 @@ add_entrypoint_object( HDRS ../atanf.h DEPENDS - .inv_trigf_utils libc.src.__support.FPUtil.except_value_utils libc.src.__support.FPUtil.fp_bits libc.src.__support.FPUtil.multiply_add @@ -4164,6 +4082,7 @@ add_entrypoint_object( libc.src.__support.FPUtil.polyeval libc.src.__support.FPUtil.rounding_mode libc.src.__support.macros.optimization + libc.src.__support.math.inv_trigf_utils ) add_entrypoint_object( @@ -4212,7 +4131,6 @@ add_entrypoint_object( ../atan2f.h atan2f_float.h DEPENDS - .inv_trigf_utils libc.hdr.fenv_macros libc.src.__support.FPUtil.double_double libc.src.__support.FPUtil.fenv_impl @@ -4222,6 +4140,7 @@ add_entrypoint_object( libc.src.__support.FPUtil.polyeval libc.src.__support.FPUtil.rounding_mode libc.src.__support.macros.optimization + libc.src.__support.math.inv_trigf_utils ) add_entrypoint_object( @@ -5023,10 +4942,12 @@ add_header_library( HDRS expxf16.h DEPENDS - libc.src.__support.FPUtil.cast + libc.hdr.stdint_proxy libc.src.__support.FPUtil.fp_bits + libc.src.__support.FPUtil.cast libc.src.__support.FPUtil.multiply_add libc.src.__support.FPUtil.nearest_integer libc.src.__support.macros.attributes libc.src.__support.math.expf16_utils + libc.src.__support.math.exp10_float16_constants ) diff --git a/libc/src/math/generic/acos.cpp b/libc/src/math/generic/acos.cpp index c14721f..3a59642 100644 --- a/libc/src/math/generic/acos.cpp +++ b/libc/src/math/generic/acos.cpp @@ -7,272 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/acos.h" -#include "asin_utils.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/FPUtil/sqrt.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY -#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA +#include "src/__support/math/acos.h" namespace LIBC_NAMESPACE_DECL { -using DoubleDouble = fputil::DoubleDouble; -using Float128 = fputil::DyadicFloat<128>; - -LLVM_LIBC_FUNCTION(double, acos, (double x)) { - using FPBits = fputil::FPBits<double>; - - FPBits xbits(x); - int x_exp = xbits.get_biased_exponent(); - - // |x| < 0.5. - if (x_exp < FPBits::EXP_BIAS - 1) { - // |x| < 2^-55. - if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 55)) { - // When |x| < 2^-55, acos(x) = pi/2 -#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) - return PI_OVER_TWO.hi; -#else - // Force the evaluation and prevent constant propagation so that it - // is rounded correctly for FE_UPWARD rounding mode. - return (xbits.abs().get_val() + 0x1.0p-160) + PI_OVER_TWO.hi; -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - } - -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - // acos(x) = pi/2 - asin(x) - // = pi/2 - x * P(x^2) - double p = asin_eval(x * x); - return PI_OVER_TWO.hi + fputil::multiply_add(-x, p, PI_OVER_TWO.lo); -#else - unsigned idx; - DoubleDouble x_sq = fputil::exact_mult(x, x); - double err = xbits.abs().get_val() * 0x1.0p-51; - // Polynomial approximation: - // p ~ asin(x)/x - DoubleDouble p = asin_eval(x_sq, idx, err); - // asin(x) ~ x * p - DoubleDouble r0 = fputil::exact_mult(x, p.hi); - // acos(x) = pi/2 - asin(x) - // ~ pi/2 - x * p - // = pi/2 - x * (p.hi + p.lo) - double r_hi = fputil::multiply_add(-x, p.hi, PI_OVER_TWO.hi); - // Use Dekker's 2SUM algorithm to compute the lower part. - double r_lo = ((PI_OVER_TWO.hi - r_hi) - r0.hi) - r0.lo; - r_lo = fputil::multiply_add(-x, p.lo, r_lo + PI_OVER_TWO.lo); - - // Ziv's accuracy test. - - double r_upper = r_hi + (r_lo + err); - double r_lower = r_hi + (r_lo - err); - - if (LIBC_LIKELY(r_upper == r_lower)) - return r_upper; - - // Ziv's accuracy test failed, perform 128-bit calculation. - - // Recalculate mod 1/64. - idx = static_cast<unsigned>(fputil::nearest_integer(x_sq.hi * 0x1.0p6)); - - // Get x^2 - idx/64 exactly. When FMA is available, double-double - // multiplication will be correct for all rounding modes. Otherwise we use - // Float128 directly. - Float128 x_f128(x); - -#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE - // u = x^2 - idx/64 - Float128 u_hi( - fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, x_sq.hi)); - Float128 u = fputil::quick_add(u_hi, Float128(x_sq.lo)); -#else - Float128 x_sq_f128 = fputil::quick_mul(x_f128, x_f128); - Float128 u = fputil::quick_add( - x_sq_f128, Float128(static_cast<double>(idx) * (-0x1.0p-6))); -#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE - - Float128 p_f128 = asin_eval(u, idx); - // Flip the sign of x_f128 to perform subtraction. - x_f128.sign = x_f128.sign.negate(); - Float128 r = - fputil::quick_add(PI_OVER_TWO_F128, fputil::quick_mul(x_f128, p_f128)); - - return static_cast<double>(r); -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - } - // |x| >= 0.5 - - double x_abs = xbits.abs().get_val(); - - // Maintaining the sign: - constexpr double SIGN[2] = {1.0, -1.0}; - double x_sign = SIGN[xbits.is_neg()]; - // |x| >= 1 - if (LIBC_UNLIKELY(x_exp >= FPBits::EXP_BIAS)) { - // x = +-1, asin(x) = +- pi/2 - if (x_abs == 1.0) { - // x = 1, acos(x) = 0, - // x = -1, acos(x) = pi - return x == 1.0 ? 0.0 : fputil::multiply_add(-x_sign, PI.hi, PI.lo); - } - // |x| > 1, return NaN. - if (xbits.is_quiet_nan()) - return x; - - // Set domain error for non-NaN input. - if (!xbits.is_nan()) - fputil::set_errno_if_required(EDOM); - - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - - // When |x| >= 0.5, we perform range reduction as follow: - // - // When 0.5 <= x < 1, let: - // y = acos(x) - // We will use the double angle formula: - // cos(2y) = 1 - 2 sin^2(y) - // and the complement angle identity: - // x = cos(y) = 1 - 2 sin^2 (y/2) - // So: - // sin(y/2) = sqrt( (1 - x)/2 ) - // And hence: - // y/2 = asin( sqrt( (1 - x)/2 ) ) - // Equivalently: - // acos(x) = y = 2 * asin( sqrt( (1 - x)/2 ) ) - // Let u = (1 - x)/2, then: - // acos(x) = 2 * asin( sqrt(u) ) - // Moreover, since 0.5 <= x < 1: - // 0 < u <= 1/4, and 0 < sqrt(u) <= 0.5, - // And hence we can reuse the same polynomial approximation of asin(x) when - // |x| <= 0.5: - // acos(x) ~ 2 * sqrt(u) * P(u). - // - // When -1 < x <= -0.5, we reduce to the previous case using the formula: - // acos(x) = pi - acos(-x) - // = pi - 2 * asin ( sqrt( (1 + x)/2 ) ) - // ~ pi - 2 * sqrt(u) * P(u), - // where u = (1 - |x|)/2. - - // u = (1 - |x|)/2 - double u = fputil::multiply_add(x_abs, -0.5, 0.5); - // v_hi + v_lo ~ sqrt(u). - // Let: - // h = u - v_hi^2 = (sqrt(u) - v_hi) * (sqrt(u) + v_hi) - // Then: - // sqrt(u) = v_hi + h / (sqrt(u) + v_hi) - // ~ v_hi + h / (2 * v_hi) - // So we can use: - // v_lo = h / (2 * v_hi). - double v_hi = fputil::sqrt<double>(u); - -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - constexpr DoubleDouble CONST_TERM[2] = {{0.0, 0.0}, PI}; - DoubleDouble const_term = CONST_TERM[xbits.is_neg()]; - - double p = asin_eval(u); - double scale = x_sign * 2.0 * v_hi; - double r = const_term.hi + fputil::multiply_add(scale, p, const_term.lo); - return r; -#else - -#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE - double h = fputil::multiply_add(v_hi, -v_hi, u); -#else - DoubleDouble v_hi_sq = fputil::exact_mult(v_hi, v_hi); - double h = (u - v_hi_sq.hi) - v_hi_sq.lo; -#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE - - // Scale v_lo and v_hi by 2 from the formula: - // vh = v_hi * 2 - // vl = 2*v_lo = h / v_hi. - double vh = v_hi * 2.0; - double vl = h / v_hi; - - // Polynomial approximation: - // p ~ asin(sqrt(u))/sqrt(u) - unsigned idx; - double err = vh * 0x1.0p-51; - - DoubleDouble p = asin_eval(DoubleDouble{0.0, u}, idx, err); - - // Perform computations in double-double arithmetic: - // asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p) - DoubleDouble r0 = fputil::quick_mult(DoubleDouble{vl, vh}, p); - - double r_hi, r_lo; - if (xbits.is_pos()) { - r_hi = r0.hi; - r_lo = r0.lo; - } else { - DoubleDouble r = fputil::exact_add(PI.hi, -r0.hi); - r_hi = r.hi; - r_lo = (PI.lo - r0.lo) + r.lo; - } - - // Ziv's accuracy test. - - double r_upper = r_hi + (r_lo + err); - double r_lower = r_hi + (r_lo - err); - - if (LIBC_LIKELY(r_upper == r_lower)) - return r_upper; - - // Ziv's accuracy test failed, we redo the computations in Float128. - // Recalculate mod 1/64. - idx = static_cast<unsigned>(fputil::nearest_integer(u * 0x1.0p6)); - - // After the first step of Newton-Raphson approximating v = sqrt(u), we have - // that: - // sqrt(u) = v_hi + h / (sqrt(u) + v_hi) - // v_lo = h / (2 * v_hi) - // With error: - // sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) ) - // = -h^2 / (2*v * (sqrt(u) + v)^2). - // Since: - // (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u, - // we can add another correction term to (v_hi + v_lo) that is: - // v_ll = -h^2 / (2*v_hi * 4u) - // = -v_lo * (h / 4u) - // = -vl * (h / 8u), - // making the errors: - // sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3) - // well beyond 128-bit precision needed. - - // Get the rounding error of vl = 2 * v_lo ~ h / vh - // Get full product of vh * vl -#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE - double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi; -#else - DoubleDouble vh_vl = fputil::exact_mult(v_hi, vl); - double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi; -#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE - // vll = 2*v_ll = -vl * (h / (4u)). - double t = h * (-0.25) / u; - double vll = fputil::multiply_add(vl, t, vl_lo); - // m_v = -(v_hi + v_lo + v_ll). - Float128 m_v = fputil::quick_add( - Float128(vh), fputil::quick_add(Float128(vl), Float128(vll))); - m_v.sign = xbits.sign(); - - // Perform computations in Float128: - // acos(x) = (v_hi + v_lo + vll) * P(u) , when 0.5 <= x < 1, - // = pi - (v_hi + v_lo + vll) * P(u) , when -1 < x <= -0.5. - Float128 y_f128(fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, u)); - - Float128 p_f128 = asin_eval(y_f128, idx); - Float128 r_f128 = fputil::quick_mul(m_v, p_f128); - - if (xbits.is_neg()) - r_f128 = fputil::quick_add(PI_F128, r_f128); - - return static_cast<double>(r_f128); -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS -} +LLVM_LIBC_FUNCTION(double, acos, (double x)) { return math::acos(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/acosf.cpp b/libc/src/math/generic/acosf.cpp index 8dd6de2..7afc7d6 100644 --- a/libc/src/math/generic/acosf.cpp +++ b/libc/src/math/generic/acosf.cpp @@ -7,127 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/acosf.h" -#include "src/__support/FPUtil/FEnvImpl.h" -#include "src/__support/FPUtil/FPBits.h" -#include "src/__support/FPUtil/PolyEval.h" -#include "src/__support/FPUtil/except_value_utils.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/FPUtil/sqrt.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY - -#include "inv_trigf_utils.h" +#include "src/__support/math/acosf.h" namespace LIBC_NAMESPACE_DECL { -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS -static constexpr size_t N_EXCEPTS = 4; - -// Exceptional values when |x| <= 0.5 -static constexpr fputil::ExceptValues<float, N_EXCEPTS> ACOSF_EXCEPTS = {{ - // (inputs, RZ output, RU offset, RD offset, RN offset) - // x = 0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ) - {0x328885a3, 0x3fc90fda, 1, 0, 1}, - // x = -0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ) - {0xb28885a3, 0x3fc90fda, 1, 0, 1}, - // x = 0x1.04c444p-12, acosf(x) = 0x1.920f68p0 (RZ) - {0x39826222, 0x3fc907b4, 1, 0, 1}, - // x = -0x1.04c444p-12, acosf(x) = 0x1.923p0 (RZ) - {0xb9826222, 0x3fc91800, 1, 0, 1}, -}}; -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -LLVM_LIBC_FUNCTION(float, acosf, (float x)) { - using FPBits = typename fputil::FPBits<float>; - - FPBits xbits(x); - uint32_t x_uint = xbits.uintval(); - uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU; - uint32_t x_sign = x_uint >> 31; - - // |x| <= 0.5 - if (LIBC_UNLIKELY(x_abs <= 0x3f00'0000U)) { - // |x| < 0x1p-10 - if (LIBC_UNLIKELY(x_abs < 0x3a80'0000U)) { - // When |x| < 2^-10, we use the following approximation: - // acos(x) = pi/2 - asin(x) - // ~ pi/2 - x - x^3 / 6 - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - // Check for exceptional values - if (auto r = ACOSF_EXCEPTS.lookup(x_uint); LIBC_UNLIKELY(r.has_value())) - return r.value(); -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - - double xd = static_cast<double>(x); - return static_cast<float>(fputil::multiply_add( - -0x1.5555555555555p-3 * xd, xd * xd, M_MATH_PI_2 - xd)); - } - - // For |x| <= 0.5, we approximate acosf(x) by: - // acos(x) = pi/2 - asin(x) = pi/2 - x * P(x^2) - // Where P(X^2) = Q(X) is a degree-20 minimax even polynomial approximating - // asin(x)/x on [0, 0.5] generated by Sollya with: - // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|], - // [|1, D...|], [0, 0.5]); - double xd = static_cast<double>(x); - double xsq = xd * xd; - double x3 = xd * xsq; - double r = asin_eval(xsq); - return static_cast<float>(fputil::multiply_add(-x3, r, M_MATH_PI_2 - xd)); - } - - // |x| >= 1, return 0, 2pi, or NaNs. - if (LIBC_UNLIKELY(x_abs >= 0x3f80'0000U)) { - if (x_abs == 0x3f80'0000U) - return x_sign ? /* x == -1.0f */ fputil::round_result_slightly_down( - 0x1.921fb6p+1f) - : /* x == 1.0f */ 0.0f; - - if (xbits.is_signaling_nan()) { - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - - // |x| <= +/-inf - if (x_abs <= 0x7f80'0000U) { - fputil::set_errno_if_required(EDOM); - fputil::raise_except_if_required(FE_INVALID); - } - - return x + FPBits::quiet_nan().get_val(); - } - - // When 0.5 < |x| < 1, we perform range reduction as follow: - // - // Assume further that 0.5 < x <= 1, and let: - // y = acos(x) - // We use the double angle formula: - // x = cos(y) = 1 - 2 sin^2(y/2) - // So: - // sin(y/2) = sqrt( (1 - x)/2 ) - // And hence: - // y = 2 * asin( sqrt( (1 - x)/2 ) ) - // Let u = (1 - x)/2, then - // acos(x) = 2 * asin( sqrt(u) ) - // Moreover, since 0.5 < x <= 1, - // 0 <= u < 1/4, and 0 <= sqrt(u) < 0.5, - // And hence we can reuse the same polynomial approximation of asin(x) when - // |x| <= 0.5: - // acos(x) ~ 2 * sqrt(u) * P(u). - // - // When -1 < x <= -0.5, we use the identity: - // acos(x) = pi - acos(-x) - // which is reduced to the postive case. - - xbits.set_sign(Sign::POS); - double xd = static_cast<double>(xbits.get_val()); - double u = fputil::multiply_add(-0.5, xd, 0.5); - double cv = 2 * fputil::sqrt<double>(u); - - double r3 = asin_eval(u); - double r = fputil::multiply_add(cv * u, r3, cv); - return static_cast<float>(x_sign ? M_MATH_PI - r : r); -} +LLVM_LIBC_FUNCTION(float, acosf, (float x)) { return math::acosf(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/acosf16.cpp b/libc/src/math/generic/acosf16.cpp index 202a950..0bf85f8 100644 --- a/libc/src/math/generic/acosf16.cpp +++ b/libc/src/math/generic/acosf16.cpp @@ -8,144 +8,10 @@ //===----------------------------------------------------------------------===// #include "src/math/acosf16.h" -#include "hdr/errno_macros.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/cast.h" -#include "src/__support/FPUtil/except_value_utils.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/FPUtil/sqrt.h" -#include "src/__support/macros/optimization.h" +#include "src/__support/math/acosf16.h" namespace LIBC_NAMESPACE_DECL { -// Generated by Sollya using the following command: -// > round(pi/2, SG, RN); -// > round(pi, SG, RN); -static constexpr float PI_OVER_2 = 0x1.921fb6p0f; -static constexpr float PI = 0x1.921fb6p1f; +LLVM_LIBC_FUNCTION(float16, acosf16, (float16 x)) { return math::acosf16(x); } -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS -static constexpr size_t N_EXCEPTS = 2; - -static constexpr fputil::ExceptValues<float16, N_EXCEPTS> ACOSF16_EXCEPTS{{ - // (input, RZ output, RU offset, RD offset, RN offset) - {0xacaf, 0x3e93, 1, 0, 0}, - {0xb874, 0x4052, 1, 0, 1}, -}}; -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -LLVM_LIBC_FUNCTION(float16, acosf16, (float16 x)) { - using FPBits = fputil::FPBits<float16>; - FPBits xbits(x); - - uint16_t x_u = xbits.uintval(); - uint16_t x_abs = x_u & 0x7fff; - uint16_t x_sign = x_u >> 15; - - // |x| > 0x1p0, |x| > 1, or x is NaN. - if (LIBC_UNLIKELY(x_abs > 0x3c00)) { - // acosf16(NaN) = NaN - if (xbits.is_nan()) { - if (xbits.is_signaling_nan()) { - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - - return x; - } - - // 1 < |x| <= +/-inf - fputil::raise_except_if_required(FE_INVALID); - fputil::set_errno_if_required(EDOM); - - return FPBits::quiet_nan().get_val(); - } - - float xf = x; - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - // Handle exceptional values - if (auto r = ACOSF16_EXCEPTS.lookup(x_u); LIBC_UNLIKELY(r.has_value())) - return r.value(); -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - - // |x| == 0x1p0, x is 1 or -1 - // if x is (-)1, return pi, else - // if x is (+)1, return 0 - if (LIBC_UNLIKELY(x_abs == 0x3c00)) - return fputil::cast<float16>(x_sign ? PI : 0.0f); - - float xsq = xf * xf; - - // |x| <= 0x1p-1, |x| <= 0.5 - if (x_abs <= 0x3800) { - // if x is 0, return pi/2 - if (LIBC_UNLIKELY(x_abs == 0)) - return fputil::cast<float16>(PI_OVER_2); - - // Note that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x) - // Degree-6 minimax polynomial of asin(x) generated by Sollya with: - // > P = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8|], [|SG...|], [0, 0.5]); - float interm = - fputil::polyeval(xsq, 0x1.000002p0f, 0x1.554c2ap-3f, 0x1.3541ccp-4f, - 0x1.43b2d6p-5f, 0x1.a0d73ep-5f); - return fputil::cast<float16>(fputil::multiply_add(-xf, interm, PI_OVER_2)); - } - - // When |x| > 0.5, assume that 0.5 < |x| <= 1 - // - // Step-by-step range-reduction proof: - // 1: Let y = asin(x), such that, x = sin(y) - // 2: From complimentary angle identity: - // x = sin(y) = cos(pi/2 - y) - // 3: Let z = pi/2 - y, such that x = cos(z) - // 4: From double angle formula; cos(2A) = 1 - 2 * sin^2(A): - // z = 2A, z/2 = A - // cos(z) = 1 - 2 * sin^2(z/2) - // 5: Make sin(z/2) subject of the formula: - // sin(z/2) = sqrt((1 - cos(z))/2) - // 6: Recall [3]; x = cos(z). Therefore: - // sin(z/2) = sqrt((1 - x)/2) - // 7: Let u = (1 - x)/2 - // 8: Therefore: - // asin(sqrt(u)) = z/2 - // 2 * asin(sqrt(u)) = z - // 9: Recall [3]; z = pi/2 - y. Therefore: - // y = pi/2 - z - // y = pi/2 - 2 * asin(sqrt(u)) - // 10: Recall [1], y = asin(x). Therefore: - // asin(x) = pi/2 - 2 * asin(sqrt(u)) - // 11: Recall that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x) - // Therefore: - // acos(x) = pi/2 - (pi/2 - 2 * asin(sqrt(u))) - // acos(x) = 2 * asin(sqrt(u)) - // - // THE RANGE REDUCTION, HOW? - // 12: Recall [7], u = (1 - x)/2 - // 13: Since 0.5 < x <= 1, therefore: - // 0 <= u <= 0.25 and 0 <= sqrt(u) <= 0.5 - // - // Hence, we can reuse the same [0, 0.5] domain polynomial approximation for - // Step [11] as `sqrt(u)` is in range. - // When -1 < x <= -0.5, the identity: - // acos(x) = pi - acos(-x) - // allows us to compute for the negative x value (lhs) - // with a positive x value instead (rhs). - - float xf_abs = (xf < 0 ? -xf : xf); - float u = fputil::multiply_add(-0.5f, xf_abs, 0.5f); - float sqrt_u = fputil::sqrt<float>(u); - - // Degree-6 minimax polynomial of asin(x) generated by Sollya with: - // > P = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8|], [|SG...|], [0, 0.5]); - float asin_sqrt_u = - sqrt_u * fputil::polyeval(u, 0x1.000002p0f, 0x1.554c2ap-3f, - 0x1.3541ccp-4f, 0x1.43b2d6p-5f, 0x1.a0d73ep-5f); - - return fputil::cast<float16>( - x_sign ? fputil::multiply_add(-2.0f, asin_sqrt_u, PI) : 2 * asin_sqrt_u); -} } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/acoshf.cpp b/libc/src/math/generic/acoshf.cpp index c4927fa..5c04583 100644 --- a/libc/src/math/generic/acoshf.cpp +++ b/libc/src/math/generic/acoshf.cpp @@ -7,73 +7,11 @@ //===----------------------------------------------------------------------===// #include "src/math/acoshf.h" -#include "src/__support/FPUtil/FEnvImpl.h" -#include "src/__support/FPUtil/FPBits.h" -#include "src/__support/FPUtil/PolyEval.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/FPUtil/sqrt.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY -#include "src/math/generic/common_constants.h" -#include "src/math/generic/explogxf.h" -namespace LIBC_NAMESPACE_DECL { - -LLVM_LIBC_FUNCTION(float, acoshf, (float x)) { - using FPBits_t = typename fputil::FPBits<float>; - FPBits_t xbits(x); - - if (LIBC_UNLIKELY(x <= 1.0f)) { - if (x == 1.0f) - return 0.0f; - // x < 1. - fputil::set_errno_if_required(EDOM); - fputil::raise_except_if_required(FE_INVALID); - return FPBits_t::quiet_nan().get_val(); - } +#include "src/__support/math/acoshf.h" -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - uint32_t x_u = xbits.uintval(); - if (LIBC_UNLIKELY(x_u >= 0x4f8ffb03)) { - if (LIBC_UNLIKELY(xbits.is_inf_or_nan())) - return x; - - // Helper functions to set results for exceptional cases. - auto round_result_slightly_down = [](float r) -> float { - volatile float tmp = r; - tmp = tmp - 0x1.0p-25f; - return tmp; - }; - auto round_result_slightly_up = [](float r) -> float { - volatile float tmp = r; - tmp = tmp + 0x1.0p-25f; - return tmp; - }; - - switch (x_u) { - case 0x4f8ffb03: // x = 0x1.1ff606p32f - return round_result_slightly_up(0x1.6fdd34p4f); - case 0x5c569e88: // x = 0x1.ad3d1p57f - return round_result_slightly_up(0x1.45c146p5f); - case 0x5e68984e: // x = 0x1.d1309cp61f - return round_result_slightly_up(0x1.5c9442p5f); - case 0x655890d3: // x = 0x1.b121a6p75f - return round_result_slightly_down(0x1.a9a3f2p5f); - case 0x6eb1a8ec: // x = 0x1.6351d8p94f - return round_result_slightly_down(0x1.08b512p6f); - case 0x7997f30a: // x = 0x1.2fe614p116f - return round_result_slightly_up(0x1.451436p6f); - } - } -#else - if (LIBC_UNLIKELY(xbits.is_inf_or_nan())) - return x; -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS +namespace LIBC_NAMESPACE_DECL { - double x_d = static_cast<double>(x); - // acosh(x) = log(x + sqrt(x^2 - 1)) - return static_cast<float>(log_eval( - x_d + fputil::sqrt<double>(fputil::multiply_add(x_d, x_d, -1.0)))); -} +LLVM_LIBC_FUNCTION(float, acoshf, (float x)) { return math::acoshf(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/acoshf16.cpp b/libc/src/math/generic/acoshf16.cpp index 44783a8..bb3a91f 100644 --- a/libc/src/math/generic/acoshf16.cpp +++ b/libc/src/math/generic/acoshf16.cpp @@ -7,104 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/acoshf16.h" -#include "explogxf.h" -#include "hdr/errno_macros.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/cast.h" -#include "src/__support/FPUtil/except_value_utils.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/FPUtil/sqrt.h" -#include "src/__support/common.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" +#include "src/__support/math/acoshf16.h" namespace LIBC_NAMESPACE_DECL { -static constexpr size_t N_EXCEPTS = 2; -static constexpr fputil::ExceptValues<float16, N_EXCEPTS> ACOSHF16_EXCEPTS{{ - // (input, RZ output, RU offset, RD offset, RN offset) - // x = 0x1.6dcp+1, acoshf16(x) = 0x1.b6p+0 (RZ) - {0x41B7, 0x3ED8, 1, 0, 0}, - // x = 0x1.39p+0, acoshf16(x) = 0x1.4f8p-1 (RZ) - {0x3CE4, 0x393E, 1, 0, 1}, -}}; - -LLVM_LIBC_FUNCTION(float16, acoshf16, (float16 x)) { - using FPBits = fputil::FPBits<float16>; - FPBits xbits(x); - uint16_t x_u = xbits.uintval(); - - // Check for NaN input first. - if (LIBC_UNLIKELY(xbits.is_inf_or_nan())) { - if (xbits.is_signaling_nan()) { - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - if (xbits.is_neg()) { - fputil::set_errno_if_required(EDOM); - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - return x; - } - - // Domain error for inputs less than 1.0. - if (LIBC_UNLIKELY(x <= 1.0f)) { - if (x == 1.0f) - return FPBits::zero().get_val(); - fputil::set_errno_if_required(EDOM); - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - - if (auto r = ACOSHF16_EXCEPTS.lookup(xbits.uintval()); - LIBC_UNLIKELY(r.has_value())) - return r.value(); - - float xf = x; - // High-precision polynomial approximation for inputs close to 1.0 - // ([1, 1.25)). - // - // Brief derivation: - // 1. Expand acosh(1 + delta) using Taylor series around delta=0: - // acosh(1 + delta) ≈ sqrt(2 * delta) * [1 - delta/12 + 3*delta^2/160 - // - 5*delta^3/896 + 35*delta^4/18432 + ...] - // 2. Truncate the series to fit accurately for delta in [0, 0.25]. - // 3. Polynomial coefficients (from sollya) used here are: - // P(delta) ≈ 1 - 0x1.555556p-4 * delta + 0x1.333334p-6 * delta^2 - // - 0x1.6db6dcp-8 * delta^3 + 0x1.f1c71cp-10 * delta^4 - // 4. The Sollya commands used to generate these coefficients were: - // > display = hexadecimal; - // > round(1/12, SG, RN); - // > round(3/160, SG, RN); - // > round(5/896, SG, RN); - // > round(35/18432, SG, RN); - // With hexadecimal display mode enabled, the outputs were: - // 0x1.555556p-4 - // 0x1.333334p-6 - // 0x1.6db6dcp-8 - // 0x1.f1c71cp-10 - // 5. The maximum absolute error, estimated using: - // dirtyinfnorm(acosh(1 + x) - sqrt(2*x) * P(x), [0, 0.25]) - // is: - // 0x1.d84281p-22 - if (LIBC_UNLIKELY(x_u < 0x3D00U)) { - float delta = xf - 1.0f; - float sqrt_2_delta = fputil::sqrt<float>(2.0 * delta); - float pe = fputil::polyeval(delta, 0x1p+0f, -0x1.555556p-4f, 0x1.333334p-6f, - -0x1.6db6dcp-8f, 0x1.f1c71cp-10f); - float approx = sqrt_2_delta * pe; - return fputil::cast<float16>(approx); - } - - // acosh(x) = log(x + sqrt(x^2 - 1)) - float sqrt_term = fputil::sqrt<float>(fputil::multiply_add(xf, xf, -1.0f)); - float result = static_cast<float>(log_eval(xf + sqrt_term)); - - return fputil::cast<float16>(result); -} +LLVM_LIBC_FUNCTION(float16, acoshf16, (float16 x)) { return math::acoshf16(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/asin.cpp b/libc/src/math/generic/asin.cpp index ad77683..d286fce 100644 --- a/libc/src/math/generic/asin.cpp +++ b/libc/src/math/generic/asin.cpp @@ -7,7 +7,6 @@ //===----------------------------------------------------------------------===// #include "src/math/asin.h" -#include "asin_utils.h" #include "src/__support/FPUtil/FEnvImpl.h" #include "src/__support/FPUtil/FPBits.h" #include "src/__support/FPUtil/PolyEval.h" @@ -18,6 +17,7 @@ #include "src/__support/macros/config.h" #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA +#include "src/__support/math/asin_utils.h" namespace LIBC_NAMESPACE_DECL { @@ -25,6 +25,7 @@ using DoubleDouble = fputil::DoubleDouble; using Float128 = fputil::DyadicFloat<128>; LLVM_LIBC_FUNCTION(double, asin, (double x)) { + using namespace asin_internal; using FPBits = fputil::FPBits<double>; FPBits xbits(x); diff --git a/libc/src/math/generic/asin_utils.h b/libc/src/math/generic/asin_utils.h deleted file mode 100644 index 44913d5..0000000 --- a/libc/src/math/generic/asin_utils.h +++ /dev/null @@ -1,574 +0,0 @@ -//===-- Collection of utils for asin/acos -----------------------*- 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_SRC_MATH_GENERIC_ASIN_UTILS_H -#define LLVM_LIBC_SRC_MATH_GENERIC_ASIN_UTILS_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/FPUtil/nearest_integer.h" -#include "src/__support/integer_literals.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" - -namespace LIBC_NAMESPACE_DECL { - -namespace { - -using DoubleDouble = fputil::DoubleDouble; -using Float128 = fputil::DyadicFloat<128>; - -constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p1}; - -constexpr DoubleDouble PI_OVER_TWO = {0x1.1a62633145c07p-54, - 0x1.921fb54442d18p0}; - -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -// When correct rounding is not needed, we use a degree-22 minimax polynomial to -// approximate asin(x)/x on [0, 0.5] using Sollya with: -// > P = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22|], -// [|1, D...|], [0, 0.5]); -// > dirtyinfnorm(asin(x)/x - P, [0, 0.5]); -// 0x1.1a71ef0a0f26a9fb7ed7e41dee788b13d1770db3dp-52 - -constexpr double ASIN_COEFFS[12] = { - 0x1.0000000000000p0, 0x1.5555555556dcfp-3, 0x1.3333333082e11p-4, - 0x1.6db6dd14099edp-5, 0x1.f1c69b35bf81fp-6, 0x1.6e97194225a67p-6, - 0x1.1babddb82ce12p-6, 0x1.d55bd078600d6p-7, 0x1.33328959e63d6p-7, - 0x1.2b5993bda1d9bp-6, -0x1.806aff270bf25p-7, 0x1.02614e5ed3936p-5, -}; - -LIBC_INLINE double asin_eval(double u) { - double u2 = u * u; - double c0 = fputil::multiply_add(u, ASIN_COEFFS[1], ASIN_COEFFS[0]); - double c1 = fputil::multiply_add(u, ASIN_COEFFS[3], ASIN_COEFFS[2]); - double c2 = fputil::multiply_add(u, ASIN_COEFFS[5], ASIN_COEFFS[4]); - double c3 = fputil::multiply_add(u, ASIN_COEFFS[7], ASIN_COEFFS[6]); - double c4 = fputil::multiply_add(u, ASIN_COEFFS[9], ASIN_COEFFS[8]); - double c5 = fputil::multiply_add(u, ASIN_COEFFS[11], ASIN_COEFFS[10]); - - double u4 = u2 * u2; - double d0 = fputil::multiply_add(u2, c1, c0); - double d1 = fputil::multiply_add(u2, c3, c2); - double d2 = fputil::multiply_add(u2, c5, c4); - - return fputil::polyeval(u4, d0, d1, d2); -} - -#else - -// The Taylor expansion of asin(x) around 0 is: -// asin(x) = x + x^3/6 + 3x^5/40 + ... -// ~ x * P(x^2). -// Let u = x^2, then P(x^2) = P(u), and |x| = sqrt(u). Note that when -// |x| <= 0.5, we have |u| <= 0.25. -// We approximate P(u) by breaking it down by performing range reduction mod -// 2^-5 = 1/32. -// So for: -// k = round(u * 32), -// y = u - k/32, -// we have that: -// x = sqrt(u) = sqrt(k/32 + y), -// |y| <= 2^-5 = 1/32, -// and: -// P(u) = P(k/32 + y) = Q_k(y). -// Hence : -// asin(x) = sqrt(k/32 + y) * Q_k(y), -// Or equivalently: -// Q_k(y) = asin(sqrt(k/32 + y)) / sqrt(k/32 + y). -// We generate the coefficients of Q_k by Sollya as following: -// > procedure ASIN_APPROX(N, Deg) { -// abs_error = 0; -// rel_error = 0; -// deg = [||]; -// for i from 2 to Deg do deg = deg :. i; -// for i from 1 to N/4 do { -// F = asin(sqrt(i/N + x))/sqrt(i/N + x); -// T = taylor(F, 1, 0); -// T_DD = roundcoefficients(T, [|DD...|]); -// I = [-1/(2*N), 1/(2*N)]; -// Q = fpminimax(F, deg, [|D...|], I, T_DD); -// abs_err = dirtyinfnorm(F - Q, I); -// rel_err = dirtyinfnorm((F - Q)/x^2, I); -// if (abs_err > abs_error) then abs_error = abs_err; -// if (rel_err > rel_error) then rel_error = rel_err; -// d0 = D(coeff(Q, 0)); -// d1 = coeff(Q, 0) - d0; -// write("{", d0, ", ", d1); -// d0 = D(coeff(Q, 1)); d1 = coeff(Q, 1) - d0; write(", ", d0, ", ", d1); -// for j from 2 to Deg do { -// write(", ", coeff(Q, j)); -// }; -// print("},"); -// }; -// print("Absolute Errors:", D(abs_error)); -// print("Relative Errors:", D(rel_error)); -// }; -// > ASIN_APPROX(32, 9); -// Absolute Errors: 0x1.69837b5183654p-72 -// Relative Errors: 0x1.4d7f82835bf64p-55 - -// For k = 0, we use the degree-18 Taylor polynomial of asin(x)/x: -// -// > P = 1 + x^2 * DD(1/6) + x^4 * D(3/40) + x^6 * D(5/112) + x^8 * D(35/1152) + -// x^10 * D(63/2816) + x^12 * D(231/13312) + x^14 * D(143/10240) + -// x^16 * D(6435/557056) + x^18 * D(12155/1245184); -// > dirtyinfnorm(asin(x)/x - P, [-1/64, 1/64]); -// 0x1.999075402cafp-83 - -constexpr double ASIN_COEFFS[9][12] = { - {1.0, 0.0, 0x1.5555555555555p-3, 0x1.5555555555555p-57, - 0x1.3333333333333p-4, 0x1.6db6db6db6db7p-5, 0x1.f1c71c71c71c7p-6, - 0x1.6e8ba2e8ba2e9p-6, 0x1.1c4ec4ec4ec4fp-6, 0x1.c99999999999ap-7, - 0x1.7a87878787878p-7, 0x1.3fde50d79435ep-7}, - {0x1.015a397cf0f1cp0, -0x1.eebd6ccfe3ee3p-55, 0x1.5f3581be7b08bp-3, - -0x1.5df80d0e7237dp-57, 0x1.4519ddf1ae53p-4, 0x1.8eb4b6eeb1696p-5, - 0x1.17bc85420fec8p-5, 0x1.a8e39b5dcad81p-6, 0x1.53f8df127539bp-6, - 0x1.1a485a0b0130ap-6, 0x1.e20e6e493002p-7, 0x1.a466a7030f4c9p-7}, - {0x1.02be9ce0b87cdp0, 0x1.e5d09da2e0f04p-56, 0x1.69ab5325bc359p-3, - -0x1.92f480cfede2dp-57, 0x1.58a4c3097aab1p-4, 0x1.b3db36068dd8p-5, - 0x1.3b9482184625p-5, 0x1.eedc823765d21p-6, 0x1.98e35d756be6bp-6, - 0x1.5ea4f1b32731ap-6, 0x1.355115764148ep-6, 0x1.16a5853847c91p-6}, - {0x1.042dc6a65ffbfp0, -0x1.c7ea28dce95d1p-55, 0x1.74c4bd7412f9dp-3, - 0x1.447024c0a3c87p-58, 0x1.6e09c6d2b72b9p-4, 0x1.ddd9dcdae5315p-5, - 0x1.656f1f64058b8p-5, 0x1.21a42e4437101p-5, 0x1.eed0350b7edb2p-6, - 0x1.b6bc877e58c52p-6, 0x1.903a0872eb2a4p-6, 0x1.74da839ddd6d8p-6}, - {0x1.05a8621feb16bp0, -0x1.e5b33b1407c5fp-56, 0x1.809186c2e57ddp-3, - -0x1.3dcb4d6069407p-60, 0x1.8587d99442dc5p-4, 0x1.06c23d1e75be3p-4, - 0x1.969024051c67dp-5, 0x1.54e4f934aacfdp-5, 0x1.2d60a732dbc9cp-5, - 0x1.149f0c046eac7p-5, 0x1.053a56dba1fbap-5, 0x1.f7face3343992p-6}, - {0x1.072f2b6f1e601p0, -0x1.2dcbb0541997p-54, 0x1.8d2397127aebap-3, - 0x1.ead0c497955fbp-57, 0x1.9f68df88da518p-4, 0x1.21ee26a5900d7p-4, - 0x1.d08e7081b53a9p-5, 0x1.938dd661713f7p-5, 0x1.71b9f299b72e6p-5, - 0x1.5fbc7d2450527p-5, 0x1.58573247ec325p-5, 0x1.585a174a6a4cep-5}, - {0x1.08c2f1d638e4cp0, 0x1.b47c159534a3dp-56, 0x1.9a8f592078624p-3, - -0x1.ea339145b65cdp-57, 0x1.bc04165b57aabp-4, 0x1.410df5f58441dp-4, - 0x1.0ab6bdf5f8f7p-4, 0x1.e0b92eea1fce1p-5, 0x1.c9094e443a971p-5, - 0x1.c34651d64bc74p-5, 0x1.caa008d1af08p-5, 0x1.dc165bc0c4fc5p-5}, - {0x1.0a649a73e61f2p0, 0x1.74ac0d817e9c7p-55, 0x1.a8ec30dc9389p-3, - -0x1.8ab1c0eef300cp-59, 0x1.dbc11ea95061bp-4, 0x1.64e371d661328p-4, - 0x1.33e0023b3d895p-4, 0x1.2042269c243cep-4, 0x1.1cce74bda223p-4, - 0x1.244d425572ce9p-4, 0x1.34d475c7f1e3ep-4, 0x1.4d4e653082ad3p-4}, - {0x1.0c152382d7366p0, -0x1.ee6913347c2a6p-54, 0x1.b8550d62bfb6dp-3, - -0x1.d10aec3f116d5p-57, 0x1.ff1bde0fa3cap-4, 0x1.8e5f3ab69f6a4p-4, - 0x1.656be8b6527cep-4, 0x1.5c39755dc041ap-4, 0x1.661e6ebd40599p-4, - 0x1.7ea3dddee2a4fp-4, 0x1.a4f439abb4869p-4, 0x1.d9181c0fda658p-4}, -}; - -// We calculate the lower part of the approximation P(u). -LIBC_INLINE DoubleDouble asin_eval(const DoubleDouble &u, unsigned &idx, - double &err) { - using fputil::multiply_add; - // k = round(u * 32). - double k = fputil::nearest_integer(u.hi * 0x1.0p5); - idx = static_cast<unsigned>(k); - // y = u - k/32. - double y_hi = multiply_add(k, -0x1.0p-5, u.hi); // Exact - DoubleDouble y = fputil::exact_add(y_hi, u.lo); - double y2 = y.hi * y.hi; - // Add double-double errors in addition to the relative errors from y2. - err = fputil::multiply_add(err, y2, 0x1.0p-102); - DoubleDouble c0 = fputil::quick_mult( - y, DoubleDouble{ASIN_COEFFS[idx][3], ASIN_COEFFS[idx][2]}); - double c1 = multiply_add(y.hi, ASIN_COEFFS[idx][5], ASIN_COEFFS[idx][4]); - double c2 = multiply_add(y.hi, ASIN_COEFFS[idx][7], ASIN_COEFFS[idx][6]); - double c3 = multiply_add(y.hi, ASIN_COEFFS[idx][9], ASIN_COEFFS[idx][8]); - double c4 = multiply_add(y.hi, ASIN_COEFFS[idx][11], ASIN_COEFFS[idx][10]); - - double y4 = y2 * y2; - double d0 = multiply_add(y2, c2, c1); - double d1 = multiply_add(y2, c4, c3); - - DoubleDouble r = fputil::exact_add(ASIN_COEFFS[idx][0], c0.hi); - - double e1 = multiply_add(y4, d1, d0); - - r.lo = multiply_add(y2, e1, ASIN_COEFFS[idx][1] + c0.lo + r.lo); - - return r; -} - -// Follow the discussion above, we generate the coefficients of Q_k by Sollya as -// following: -// > procedure PRINTF128(a) { -// write("{"); -// if (a < 0) -// then write("Sign::NEG, ") else write("Sign::POS, "); -// a_exp = floor(log2(a)) + 1; -// write((a + 2 ^ a_exp) * 2 ^ -128); -// print("},"); -// }; -// > verbosity = 0; -// > procedure ASIN_APPROX(N, Deg) { -// abs_error = 0; -// rel_error = 0; -// for i from 1 to N / 4 do { -// Q = fpminimax(asin(sqrt(i / N + x)) / sqrt(i / N + x), Deg, -// [| 128... | ], [ -1 / (2 * N), 1 / (2 * N) ]); -// abs_err = dirtyinfnorm(asin(sqrt(i / N + x)) - sqrt(i / N + x) * Q, -// [ -1 / (2 * N), 1 / (2 * N) ]); -// rel_err = dirtyinfnorm(asin(sqrt(i / N + x)) / sqrt(i / N + x) - Q, -// [ -1 / (2 * N), 1 / (2 * N) ]); -// if (abs_err > abs_error) then abs_error = abs_err; -// if (rel_err > rel_error) then rel_error = rel_err; -// write("{"); -// for j from 0 to Deg do PRINTF128(coeff(Q, j)); -// print("},"); -// }; -// print("Absolute Errors:", abs_error); -// print("Relative Errors:", rel_error); -// }; -// > ASIN_APPROX(64, 15); -// ... -// Absolute Errors: 0x1.0b3...p-129 -// Relative Errors: 0x1.1db...p-128 -// -// For k = 0, we use Taylor polynomial of asin(x)/x around x = 0. -// asin(x)/x ~ 1 + x^2/6 + (3 x^4)/40 + (5 x^6)/112 + (35 x^8)/1152 + -// + (63 x^10)/2816 + (231 x^12)/13312 + (143 x^14)/10240 + -// + (6435 x^16)/557056 + (12155 x^18)/1245184 + -// + (46189 x^20)/5505024 + (88179 x^22)/12058624 + -// + (676039 x^24)/104857600 + (1300075 x^26)/226492416 + -// + (5014575 x^28)/973078528 + (9694845 x^30)/2080374784. - -constexpr Float128 ASIN_COEFFS_F128[17][16] = { - { - {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, - {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, - {Sign::POS, -131, 0x99999999'99999999'99999999'9999999a_u128}, - {Sign::POS, -132, 0xb6db6db6'db6db6db'6db6db6d'b6db6db7_u128}, - {Sign::POS, -133, 0xf8e38e38'e38e38e3'8e38e38e'38e38e39_u128}, - {Sign::POS, -133, 0xb745d174'5d1745d1'745d1745'd1745d17_u128}, - {Sign::POS, -133, 0x8e276276'27627627'62762762'76276276_u128}, - {Sign::POS, -134, 0xe4cccccc'cccccccc'cccccccc'cccccccd_u128}, - {Sign::POS, -134, 0xbd43c3c3'c3c3c3c3'c3c3c3c3'c3c3c3c4_u128}, - {Sign::POS, -134, 0x9fef286b'ca1af286'bca1af28'6bca1af3_u128}, - {Sign::POS, -134, 0x89779e79'e79e79e7'9e79e79e'79e79e7a_u128}, - {Sign::POS, -135, 0xef9de9bd'37a6f4de'9bd37a6f'4de9bd38_u128}, - {Sign::POS, -135, 0xd3431eb8'51eb851e'b851eb85'1eb851ec_u128}, - {Sign::POS, -135, 0xbc16ed09'7b425ed0'97b425ed'097b425f_u128}, - {Sign::POS, -135, 0xa8dd1846'9ee58469'ee58469e'e58469ee_u128}, - {Sign::POS, -135, 0x98b41def'7bdef7bd'ef7bdef7'bdef7bdf_u128}, - }, - { - {Sign::POS, -127, 0x8055f060'94f0f05f'3ac3b927'50a701d9_u128}, - {Sign::POS, -130, 0xad19c2ea'e3dd2429'8d04f71d'b965ee1b_u128}, - {Sign::POS, -131, 0x9dfa882b'7b31af17'f9f19d33'0c45d24b_u128}, - {Sign::POS, -132, 0xbedd3b58'c9e605ef'1404e1f0'4ba57940_u128}, - {Sign::POS, -132, 0x83df2581'cb4fea82'b406f201'2fde6d5c_u128}, - {Sign::POS, -133, 0xc534fe61'9b82dd16'ed5d8a43'f7710526_u128}, - {Sign::POS, -133, 0x9b56fa62'88295ddf'ce8425fe'a04d733e_u128}, - {Sign::POS, -134, 0xfdeddb19'4a030da7'27158080'd24caf46_u128}, - {Sign::POS, -134, 0xd55827db'ff416ea8'042c4d8c'07cddeeb_u128}, - {Sign::POS, -134, 0xb71d73a9'f2ba0688'5eaeeae9'413a0f5f_u128}, - {Sign::POS, -134, 0x9fde87e2'ace91274'38f82666'd619c1ba_u128}, - {Sign::POS, -134, 0x8d876557'5e4626a1'1b621336'93587847_u128}, - {Sign::POS, -135, 0xfd801840'c8710595'6880fe13'a9657f8f_u128}, - {Sign::POS, -135, 0xe54245a9'4c8c2ebb'30488494'64b0e34d_u128}, - {Sign::POS, -135, 0xd11eb46f'4095a661'8890d123'15c96482_u128}, - {Sign::POS, -135, 0xc01a4201'467fbc0b'960618d5'ec2adaa8_u128}, - }, - { - {Sign::POS, -127, 0x80ad1cbe'7878de11'4293301c'11ce9d49_u128}, - {Sign::POS, -130, 0xaf9ac0df'3d845544'0fe5e31b'9051d03e_u128}, - {Sign::POS, -131, 0xa28ceef8'd7297e05'f94773ad'f4a695c6_u128}, - {Sign::POS, -132, 0xc75a5b77'58b4b11d'396c68ad'6733022b_u128}, - {Sign::POS, -132, 0x8bde42a1'084a6674'50c5bceb'005d4b62_u128}, - {Sign::POS, -133, 0xd471cdae'e2f35a96'bd4bc513'e0ccdf2c_u128}, - {Sign::POS, -133, 0xa9fc6fd5'd204a4e3'e609940c'6b991b67_u128}, - {Sign::POS, -133, 0x8d242d97'ba12b492'e25c7e7c'0c3fcf60_u128}, - {Sign::POS, -134, 0xf0f1ba74'b149afc3'2f0bbab5'a20c6199_u128}, - {Sign::POS, -134, 0xd21b42fb'd8e9098d'19612692'9a043332_u128}, - {Sign::POS, -134, 0xba5e5492'7896a3e7'193a74d5'78631587_u128}, - {Sign::POS, -134, 0xa7a17ae7'fc707f45'910e7a5d'c95251f4_u128}, - {Sign::POS, -134, 0x98889a6a'b0370464'50c950d3'61d79ed7_u128}, - {Sign::POS, -134, 0x8c29330e'4318fd29'25c5b528'84e39e7c_u128}, - {Sign::POS, -134, 0x81e7bf48'b25bc7c0'b9204a4f'd4f5fa8b_u128}, - {Sign::POS, -135, 0xf2801b09'11bf0768'773996dd'5224d852_u128}, - }, - { - {Sign::POS, -127, 0x81058e3e'f82ba622'ab81cd63'e1a91d57_u128}, - {Sign::POS, -130, 0xb22e7055'c80dd354'8a2f2e8e'860d3f33_u128}, - 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{Sign::POS, -131, 0xde020b2d'abd5628c'b88634e5'73f312fc_u128}, - {Sign::POS, -131, 0xa086fafa'c220fb73'9939cae3'2d69683f_u128}, - {Sign::POS, -131, 0x855b5efa'f6963d73'e4664cb1'd43f03a9_u128}, - {Sign::POS, -132, 0xf05c9774'fe0de25c'ccf1c1df'd2ed9941_u128}, - {Sign::POS, -132, 0xe484a941'19639229'f06ae955'f8edc7d1_u128}, - {Sign::POS, -132, 0xe1a32bb2'52ca122c'bf2f0904'cfc476cb_u128}, - {Sign::POS, -132, 0xe528e091'7bb8a01a'9218ce3e'1e85af60_u128}, - {Sign::POS, -132, 0xeddd556a'faa2d46f'e91c61fa'adf12aec_u128}, - {Sign::POS, -132, 0xfb390fa3'15e9d55f'5683c0c4'c7719f81_u128}, - {Sign::POS, -131, 0x868e5fa4'15597c8f'7c42a262'8f2d6332_u128}, - {Sign::POS, -131, 0x91d79767'a3d037f9'cd84ead5'c0714310_u128}, - {Sign::POS, -131, 0x9fa6a035'915bc052'377a8abb'faf4e3c6_u128}, - {Sign::POS, -131, 0xb04edefd'6ac2a93e'ec33e6f6'3d53e7c2_u128}, - {Sign::POS, -131, 0xc416980d'dc5c186b'7bdcded6'97ea5844_u128}, - }, - { - {Sign::POS, -127, 0x84c8fd4d'ffdf9fc6'bdd7ebca'88183d7b_u128}, - {Sign::POS, -130, 0xd0cf0544'11dbf845'cb6eeae5'bc980e2f_u128}, - {Sign::POS, -131, 0xe5bb9480'7ce0eaca'74300a46'8398e944_u128}, - {Sign::POS, -131, 0xa92a18f8'd611860b'5f2ef8c6'8e8ca002_u128}, - {Sign::POS, -131, 0x8f2e1684'17eb4e6c'1ec44b9b'e4b1c3e5_u128}, - {Sign::POS, -131, 0x837f1764'0ee8f416'8694b4a1'c647af0c_u128}, - {Sign::POS, -132, 0xfed7e2a9'05a5190e'b7d70a61'a24ad801_u128}, - {Sign::POS, -131, 0x803f29ff'dc6fd2bc'3c3c4b50'a9dc860c_u128}, - {Sign::POS, -131, 0x84c61e09'b8aa35e4'96239f9c'b1d00b3c_u128}, - {Sign::POS, -131, 0x8c7ed311'f77980d6'842ddf90'6a68a0bc_u128}, - {Sign::POS, -131, 0x9746077b'd397c2d1'038a4744'a76f5fb5_u128}, - {Sign::POS, -131, 0xa5341277'c4185ace'54f26328'322158e8_u128}, - {Sign::POS, -131, 0xb68d78f5'0972f6de'9189aa23'd3ecefc2_u128}, - {Sign::POS, -131, 0xcbbcefc2'15bade4e'f1d36947'c8b6e460_u128}, - {Sign::POS, -131, 0xe564a459'c851390d'd45a4748'f29f182b_u128}, - {Sign::POS, -130, 0x820ea28b'c89662c3'2a64ccdc'efb2b259_u128}, - }, - { - {Sign::POS, -127, 0x85324d39'f30f9174'ac0d817e'9c744b0b_u128}, - {Sign::POS, -130, 0xd476186e'49c47f3a'a71f8886'7f9f21c4_u128}, - {Sign::POS, -131, 0xede08f54'a830e87b'07881700'65e57b6c_u128}, - {Sign::POS, -131, 0xb271b8eb'309963ee'89187c73'0b92f7d5_u128}, - {Sign::POS, -131, 0x99f0011d'95d3a6dd'282bd00a'db808151_u128}, - {Sign::POS, -131, 0x9021134e'02b479e7'3aabf9bb'b7ab6cf3_u128}, - {Sign::POS, -131, 0x8e673bf2'f11db54a'909c4c72'6389499f_u128}, - {Sign::POS, -131, 0x9226a371'88dd55f7'bfe21777'4a42a7ae_u128}, - {Sign::POS, -131, 0x9a4d78fc'9df79d9a'44609c02'a625808a_u128}, - {Sign::POS, -131, 0xa68335fb'41d2d91c'e7bbd2a3'31a1d17b_u128}, - {Sign::POS, -131, 0xb6d89c39'28d0cb26'809d4df6'e55cba1a_u128}, - {Sign::POS, -131, 0xcba71468'9177fc2d'7f23df2f'37226488_u128}, - {Sign::POS, -131, 0xe5846de8'44833ae9'34416c87'0315eb9e_u128}, - {Sign::POS, -130, 0x82a07032'64e6226b'200d94a1'66fc7951_u128}, - {Sign::POS, -130, 0x9602695c'b6fa8886'68ca0cba'b59ea683_u128}, - {Sign::POS, -130, 0xad7d185a'ab3d14dd'd908a7b1'c57352bb_u128}, - }, - { - {Sign::POS, -127, 0x859d78fa'4405d8fa'287dbc69'95d0975e_u128}, - {Sign::POS, -130, 0xd83ea3bc'131d6baa'67c51d88'4c4dae01_u128}, - {Sign::POS, -131, 0xf6790edb'df07342b'aad85870'167af128_u128}, - {Sign::POS, -131, 0xbc6daa33'12be0f85'bc7fa753'52b10a83_u128}, - {Sign::POS, -131, 0xa5bd41bc'9c986b13'1af2542e'92aacb59_u128}, - {Sign::POS, -131, 0x9e4358bc'24e04364'b4539b76'e444b790_u128}, - {Sign::POS, -131, 0x9f7fc21b'dca1f2b5'f3f6d44b'c5a37626_u128}, - {Sign::POS, -131, 0xa6fd793c'0b9c44c1'30a518cc'66b5e511_u128}, - {Sign::POS, -131, 0xb3dccfac'cd1592b3'bcd6b7c0'9749993d_u128}, - {Sign::POS, -131, 0xc6056c3a'4a5f329a'48f1429d'27f930fc_u128}, - {Sign::POS, -131, 0xddd9e529'858a4502'6e7f3d1c'1e7dcb89_u128}, - {Sign::POS, -131, 0xfc1bccee'dc8d2567'1721c468'6f7f53ec_u128}, - {Sign::POS, -130, 0x90f2bb21'5cdbe7e2'f9ef8e12'059cc66a_u128}, - {Sign::POS, -130, 0xa857d5df'5b4da940'15ce4e95'7201fc79_u128}, - {Sign::POS, -130, 0xc54119c0'10c02bf4'd87ece17'1ef85c5f_u128}, - {Sign::POS, -130, 0xe8c50ebc'880356de'2c1f4c42'9ee9748f_u128}, - }, - { - {Sign::POS, -127, 0x860a91c1'6b9b2c23'2dd99707'ab3d688b_u128}, - {Sign::POS, -130, 0xdc2a86b1'5fdb645d'ea2781dd'25555f49_u128}, - {Sign::POS, -131, 0xff8def07'd1e514d7'b2e8ebb6'5c3afe5e_u128}, - {Sign::POS, -131, 0xc72f9d5b'4fb559e3'20db92e3'a5ae3f73_u128}, - {Sign::POS, -131, 0xb2b5f45b'1d26f4dd'0b210309'fb68914f_u128}, - {Sign::POS, -131, 0xae1cbaae'c7b55465'4da858f5'47e62a37_u128}, - {Sign::POS, -131, 0xb30f3998'10202a0d'a52ec085'a7d63289_u128}, - {Sign::POS, -131, 0xbf51f27f'b7aff89d'dc24e2aa'208d2054_u128}, - {Sign::POS, -131, 0xd250735e'87d0b527'6f99bcc9'bd6fc717_u128}, - {Sign::POS, -131, 0xec543ec2'bddb2efb'36d9ce81'a7c84336_u128}, - {Sign::POS, -130, 0x871f73e3'298ef45c'eed83998'2bc731b9_u128}, - {Sign::POS, -130, 0x9cbb5447'af8574f1'21fa4cda'93d82b7e_u128}, - {Sign::POS, -130, 0xb7f5a6c0'430a347f'11b22cde'91de0885_u128}, - {Sign::POS, -130, 0xda153cc4'14abdb96'840df7c2'3299fec0_u128}, - {Sign::POS, -129, 0x826c129b'3e4a2612'b2cd11f1'4d2ba60c_u128}, - {Sign::POS, -129, 0x9d19c289'fc0e8aa4'f351418b'b760ce90_u128}, - }, -}; - -constexpr Float128 PI_OVER_TWO_F128 = { - Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}; - -constexpr Float128 PI_F128 = {Sign::POS, -126, - 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}; - -LIBC_INLINE Float128 asin_eval(const Float128 &u, unsigned idx) { - return fputil::polyeval(u, ASIN_COEFFS_F128[idx][0], ASIN_COEFFS_F128[idx][1], - ASIN_COEFFS_F128[idx][2], ASIN_COEFFS_F128[idx][3], - ASIN_COEFFS_F128[idx][4], ASIN_COEFFS_F128[idx][5], - ASIN_COEFFS_F128[idx][6], ASIN_COEFFS_F128[idx][7], - ASIN_COEFFS_F128[idx][8], ASIN_COEFFS_F128[idx][9], - ASIN_COEFFS_F128[idx][10], ASIN_COEFFS_F128[idx][11], - ASIN_COEFFS_F128[idx][12], ASIN_COEFFS_F128[idx][13], - ASIN_COEFFS_F128[idx][14], ASIN_COEFFS_F128[idx][15]); -} - -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -} // anonymous namespace - -} // namespace LIBC_NAMESPACE_DECL - -#endif // LLVM_LIBC_SRC_MATH_GENERIC_ASIN_UTILS_H diff --git a/libc/src/math/generic/asinf.cpp b/libc/src/math/generic/asinf.cpp index 12383bf..77d6de9 100644 --- a/libc/src/math/generic/asinf.cpp +++ b/libc/src/math/generic/asinf.cpp @@ -17,7 +17,7 @@ #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA -#include "inv_trigf_utils.h" +#include "src/__support/math/inv_trigf_utils.h" namespace LIBC_NAMESPACE_DECL { @@ -44,6 +44,7 @@ static constexpr fputil::ExceptValues<float, N_EXCEPTS> ASINF_EXCEPTS_HI = {{ #endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS LLVM_LIBC_FUNCTION(float, asinf, (float x)) { + using namespace inv_trigf_utils_internal; using FPBits = typename fputil::FPBits<float>; FPBits xbits(x); diff --git a/libc/src/math/generic/asinhf.cpp b/libc/src/math/generic/asinhf.cpp index 0bb7065..3aed3bc 100644 --- a/libc/src/math/generic/asinhf.cpp +++ b/libc/src/math/generic/asinhf.cpp @@ -19,6 +19,7 @@ namespace LIBC_NAMESPACE_DECL { LLVM_LIBC_FUNCTION(float, asinhf, (float x)) { + using namespace acoshf_internal; using FPBits_t = typename fputil::FPBits<float>; FPBits_t xbits(x); uint32_t x_u = xbits.uintval(); diff --git a/libc/src/math/generic/asinhf16.cpp b/libc/src/math/generic/asinhf16.cpp index 7878632..0a0b471 100644 --- a/libc/src/math/generic/asinhf16.cpp +++ b/libc/src/math/generic/asinhf16.cpp @@ -49,6 +49,7 @@ static constexpr fputil::ExceptValues<float16, N_EXCEPTS> ASINHF16_EXCEPTS{{ #endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS LLVM_LIBC_FUNCTION(float16, asinhf16, (float16 x)) { + using namespace acoshf_internal; using FPBits = fputil::FPBits<float16>; FPBits xbits(x); diff --git a/libc/src/math/generic/atan2f.cpp b/libc/src/math/generic/atan2f.cpp index c04b0eb..32b977f 100644 --- a/libc/src/math/generic/atan2f.cpp +++ b/libc/src/math/generic/atan2f.cpp @@ -8,7 +8,6 @@ #include "src/math/atan2f.h" #include "hdr/fenv_macros.h" -#include "inv_trigf_utils.h" #include "src/__support/FPUtil/FEnvImpl.h" #include "src/__support/FPUtil/FPBits.h" #include "src/__support/FPUtil/PolyEval.h" @@ -18,6 +17,7 @@ #include "src/__support/FPUtil/rounding_mode.h" #include "src/__support/macros/config.h" #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY +#include "src/__support/math/inv_trigf_utils.h" #if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \ defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT) @@ -236,6 +236,7 @@ float atan2f_double_double(double num_d, double den_d, double q_d, int idx, // which is about rounding errors of double-double (2^-104). LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) { + using namespace inv_trigf_utils_internal; using FPBits = typename fputil::FPBits<float>; constexpr double IS_NEG[2] = {1.0, -1.0}; constexpr double PI = 0x1.921fb54442d18p1; diff --git a/libc/src/math/generic/atanf.cpp b/libc/src/math/generic/atanf.cpp index 46196dbe..22f962e 100644 --- a/libc/src/math/generic/atanf.cpp +++ b/libc/src/math/generic/atanf.cpp @@ -7,7 +7,6 @@ //===----------------------------------------------------------------------===// #include "src/math/atanf.h" -#include "inv_trigf_utils.h" #include "src/__support/FPUtil/FPBits.h" #include "src/__support/FPUtil/PolyEval.h" #include "src/__support/FPUtil/except_value_utils.h" @@ -16,10 +15,12 @@ #include "src/__support/FPUtil/rounding_mode.h" #include "src/__support/macros/config.h" #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY +#include "src/__support/math/inv_trigf_utils.h" namespace LIBC_NAMESPACE_DECL { LLVM_LIBC_FUNCTION(float, atanf, (float x)) { + using namespace inv_trigf_utils_internal; using FPBits = typename fputil::FPBits<float>; constexpr double FINAL_SIGN[2] = {1.0, -1.0}; diff --git a/libc/src/math/generic/atanhf.cpp b/libc/src/math/generic/atanhf.cpp index f6fde76..602a8f0 100644 --- a/libc/src/math/generic/atanhf.cpp +++ b/libc/src/math/generic/atanhf.cpp @@ -16,6 +16,7 @@ namespace LIBC_NAMESPACE_DECL { LLVM_LIBC_FUNCTION(float, atanhf, (float x)) { + using namespace acoshf_internal; using FPBits = typename fputil::FPBits<float>; FPBits xbits(x); diff --git a/libc/src/math/generic/common_constants.cpp b/libc/src/math/generic/common_constants.cpp index 4dcf84d..42e3ff0 100644 --- a/libc/src/math/generic/common_constants.cpp +++ b/libc/src/math/generic/common_constants.cpp @@ -51,52 +51,6 @@ const float ONE_OVER_F_FLOAT[128] = { 0x1.08421p-1f, 0x1.07326p-1f, 0x1.0624dep-1f, 0x1.05198p-1f, 0x1.041042p-1f, 0x1.03091cp-1f, 0x1.020408p-1f, 0x1.010102p-1f}; -// Lookup table for (1/f) where f = 1 + n*2^(-7), n = 0..127. -const double ONE_OVER_F[128] = { - 0x1.0000000000000p+0, 0x1.fc07f01fc07f0p-1, 0x1.f81f81f81f820p-1, - 0x1.f44659e4a4271p-1, 0x1.f07c1f07c1f08p-1, 0x1.ecc07b301ecc0p-1, - 0x1.e9131abf0b767p-1, 0x1.e573ac901e574p-1, 0x1.e1e1e1e1e1e1ep-1, - 0x1.de5d6e3f8868ap-1, 0x1.dae6076b981dbp-1, 0x1.d77b654b82c34p-1, - 0x1.d41d41d41d41dp-1, 0x1.d0cb58f6ec074p-1, 0x1.cd85689039b0bp-1, - 0x1.ca4b3055ee191p-1, 0x1.c71c71c71c71cp-1, 0x1.c3f8f01c3f8f0p-1, - 0x1.c0e070381c0e0p-1, 0x1.bdd2b899406f7p-1, 0x1.bacf914c1bad0p-1, - 0x1.b7d6c3dda338bp-1, 0x1.b4e81b4e81b4fp-1, 0x1.b2036406c80d9p-1, - 0x1.af286bca1af28p-1, 0x1.ac5701ac5701bp-1, 0x1.a98ef606a63bep-1, - 0x1.a6d01a6d01a6dp-1, 0x1.a41a41a41a41ap-1, 0x1.a16d3f97a4b02p-1, - 0x1.9ec8e951033d9p-1, 0x1.9c2d14ee4a102p-1, 0x1.999999999999ap-1, - 0x1.970e4f80cb872p-1, 0x1.948b0fcd6e9e0p-1, 0x1.920fb49d0e229p-1, - 0x1.8f9c18f9c18fap-1, 0x1.8d3018d3018d3p-1, 0x1.8acb90f6bf3aap-1, - 0x1.886e5f0abb04ap-1, 0x1.8618618618618p-1, 0x1.83c977ab2beddp-1, - 0x1.8181818181818p-1, 0x1.7f405fd017f40p-1, 0x1.7d05f417d05f4p-1, - 0x1.7ad2208e0ecc3p-1, 0x1.78a4c8178a4c8p-1, 0x1.767dce434a9b1p-1, - 0x1.745d1745d1746p-1, 0x1.724287f46debcp-1, 0x1.702e05c0b8170p-1, - 0x1.6e1f76b4337c7p-1, 0x1.6c16c16c16c17p-1, 0x1.6a13cd1537290p-1, - 0x1.6816816816817p-1, 0x1.661ec6a5122f9p-1, 0x1.642c8590b2164p-1, - 0x1.623fa77016240p-1, 0x1.6058160581606p-1, 0x1.5e75bb8d015e7p-1, - 0x1.5c9882b931057p-1, 0x1.5ac056b015ac0p-1, 0x1.58ed2308158edp-1, - 0x1.571ed3c506b3ap-1, 0x1.5555555555555p-1, 0x1.5390948f40febp-1, - 0x1.51d07eae2f815p-1, 0x1.5015015015015p-1, 0x1.4e5e0a72f0539p-1, - 0x1.4cab88725af6ep-1, 0x1.4afd6a052bf5bp-1, 0x1.49539e3b2d067p-1, - 0x1.47ae147ae147bp-1, 0x1.460cbc7f5cf9ap-1, 0x1.446f86562d9fbp-1, - 0x1.42d6625d51f87p-1, 0x1.4141414141414p-1, 0x1.3fb013fb013fbp-1, - 0x1.3e22cbce4a902p-1, 0x1.3c995a47babe7p-1, 0x1.3b13b13b13b14p-1, - 0x1.3991c2c187f63p-1, 0x1.3813813813814p-1, 0x1.3698df3de0748p-1, - 0x1.3521cfb2b78c1p-1, 0x1.33ae45b57bcb2p-1, 0x1.323e34a2b10bfp-1, - 0x1.30d190130d190p-1, 0x1.2f684bda12f68p-1, 0x1.2e025c04b8097p-1, - 0x1.2c9fb4d812ca0p-1, 0x1.2b404ad012b40p-1, 0x1.29e4129e4129ep-1, - 0x1.288b01288b013p-1, 0x1.27350b8812735p-1, 0x1.25e22708092f1p-1, - 0x1.2492492492492p-1, 0x1.23456789abcdfp-1, 0x1.21fb78121fb78p-1, - 0x1.20b470c67c0d9p-1, 0x1.1f7047dc11f70p-1, 0x1.1e2ef3b3fb874p-1, - 0x1.1cf06ada2811dp-1, 0x1.1bb4a4046ed29p-1, 0x1.1a7b9611a7b96p-1, - 0x1.19453808ca29cp-1, 0x1.1811811811812p-1, 0x1.16e0689427379p-1, - 0x1.15b1e5f75270dp-1, 0x1.1485f0e0acd3bp-1, 0x1.135c81135c811p-1, - 0x1.12358e75d3033p-1, 0x1.1111111111111p-1, 0x1.0fef010fef011p-1, - 0x1.0ecf56be69c90p-1, 0x1.0db20a88f4696p-1, 0x1.0c9714fbcda3bp-1, - 0x1.0b7e6ec259dc8p-1, 0x1.0a6810a6810a7p-1, 0x1.0953f39010954p-1, - 0x1.0842108421084p-1, 0x1.073260a47f7c6p-1, 0x1.0624dd2f1a9fcp-1, - 0x1.05197f7d73404p-1, 0x1.0410410410410p-1, 0x1.03091b51f5e1ap-1, - 0x1.0204081020408p-1, 0x1.0101010101010p-1}; - // Lookup table for log(f) = log(1 + n*2^(-7)) where n = 0..127, // computed and stored as float precision constants. // Generated by Sollya with the following commands: @@ -136,52 +90,6 @@ const float LOG_F_FLOAT[128] = { 0x1.52a2d2p-1f, 0x1.54b246p-1f, 0x1.56bf9ep-1f, 0x1.58cadcp-1f, 0x1.5ad404p-1f, 0x1.5cdb1ep-1f, 0x1.5ee02ap-1f, 0x1.60e33p-1f}; -// Lookup table for log(f) = log(1 + n*2^(-7)) where n = 0..127. -const double LOG_F[128] = { - 0x0.0000000000000p+0, 0x1.fe02a6b106788p-8, 0x1.fc0a8b0fc03e3p-7, - 0x1.7b91b07d5b11ap-6, 0x1.f829b0e783300p-6, 0x1.39e87b9febd5fp-5, - 0x1.77458f632dcfcp-5, 0x1.b42dd711971bep-5, 0x1.f0a30c01162a6p-5, - 0x1.16536eea37ae0p-4, 0x1.341d7961bd1d0p-4, 0x1.51b073f06183fp-4, - 0x1.6f0d28ae56b4bp-4, 0x1.8c345d6319b20p-4, 0x1.a926d3a4ad563p-4, - 0x1.c5e548f5bc743p-4, 0x1.e27076e2af2e5p-4, 0x1.fec9131dbeabap-4, - 0x1.0d77e7cd08e59p-3, 0x1.1b72ad52f67a0p-3, 0x1.29552f81ff523p-3, - 0x1.371fc201e8f74p-3, 0x1.44d2b6ccb7d1ep-3, 0x1.526e5e3a1b437p-3, - 0x1.5ff3070a793d3p-3, 0x1.6d60fe719d21cp-3, 0x1.7ab890210d909p-3, - 0x1.87fa06520c910p-3, 0x1.9525a9cf456b4p-3, 0x1.a23bc1fe2b563p-3, - 0x1.af3c94e80bff2p-3, 0x1.bc286742d8cd6p-3, 0x1.c8ff7c79a9a21p-3, - 0x1.d5c216b4fbb91p-3, 0x1.e27076e2af2e5p-3, 0x1.ef0adcbdc5936p-3, - 0x1.fb9186d5e3e2ap-3, 0x1.0402594b4d040p-2, 0x1.0a324e27390e3p-2, - 0x1.1058bf9ae4ad5p-2, 0x1.1675cababa60ep-2, 0x1.1c898c16999fap-2, - 0x1.22941fbcf7965p-2, 0x1.2895a13de86a3p-2, 0x1.2e8e2bae11d30p-2, - 0x1.347dd9a987d54p-2, 0x1.3a64c556945e9p-2, 0x1.404308686a7e3p-2, - 0x1.4618bc21c5ec2p-2, 0x1.4be5f957778a0p-2, 0x1.51aad872df82dp-2, - 0x1.5767717455a6cp-2, 0x1.5d1bdbf5809cap-2, 0x1.62c82f2b9c795p-2, - 0x1.686c81e9b14aep-2, 0x1.6e08eaa2ba1e3p-2, 0x1.739d7f6bbd006p-2, - 0x1.792a55fdd47a2p-2, 0x1.7eaf83b82afc3p-2, 0x1.842d1da1e8b17p-2, - 0x1.89a3386c1425ap-2, 0x1.8f11e873662c7p-2, 0x1.947941c2116fap-2, - 0x1.99d958117e08ap-2, 0x1.9f323ecbf984bp-2, 0x1.a484090e5bb0ap-2, - 0x1.a9cec9a9a0849p-2, 0x1.af1293247786bp-2, 0x1.b44f77bcc8f62p-2, - 0x1.b9858969310fbp-2, 0x1.beb4d9da71b7bp-2, 0x1.c3dd7a7cdad4dp-2, - 0x1.c8ff7c79a9a21p-2, 0x1.ce1af0b85f3ebp-2, 0x1.d32fe7e00ebd5p-2, - 0x1.d83e7258a2f3ep-2, 0x1.dd46a04c1c4a0p-2, 0x1.e24881a7c6c26p-2, - 0x1.e744261d68787p-2, 0x1.ec399d2468cc0p-2, 0x1.f128f5faf06ecp-2, - 0x1.f6123fa7028acp-2, 0x1.faf588f78f31ep-2, 0x1.ffd2e0857f498p-2, - 0x1.02552a5a5d0fep-1, 0x1.04bdf9da926d2p-1, 0x1.0723e5c1cdf40p-1, - 0x1.0986f4f573520p-1, 0x1.0be72e4252a82p-1, 0x1.0e44985d1cc8bp-1, - 0x1.109f39e2d4c96p-1, 0x1.12f719593efbcp-1, 0x1.154c3d2f4d5e9p-1, - 0x1.179eabbd899a0p-1, 0x1.19ee6b467c96ep-1, 0x1.1c3b81f713c24p-1, - 0x1.1e85f5e7040d0p-1, 0x1.20cdcd192ab6dp-1, 0x1.23130d7bebf42p-1, - 0x1.2555bce98f7cbp-1, 0x1.2795e1289b11ap-1, 0x1.29d37fec2b08ap-1, - 0x1.2c0e9ed448e8bp-1, 0x1.2e47436e40268p-1, 0x1.307d7334f10bep-1, - 0x1.32b1339121d71p-1, 0x1.34e289d9ce1d3p-1, 0x1.37117b54747b5p-1, - 0x1.393e0d3562a19p-1, 0x1.3b68449fffc22p-1, 0x1.3d9026a7156fap-1, - 0x1.3fb5b84d16f42p-1, 0x1.41d8fe84672aep-1, 0x1.43f9fe2f9ce67p-1, - 0x1.4618bc21c5ec2p-1, 0x1.48353d1ea88dfp-1, 0x1.4a4f85db03ebbp-1, - 0x1.4c679afccee39p-1, 0x1.4e7d811b75bb0p-1, 0x1.50913cc01686bp-1, - 0x1.52a2d265bc5aap-1, 0x1.54b2467999497p-1, 0x1.56bf9d5b3f399p-1, - 0x1.58cadb5cd7989p-1, 0x1.5ad404c359f2cp-1, 0x1.5cdb1dc6c1764p-1, - 0x1.5ee02a9241675p-1, 0x1.60e32f44788d8p-1}; - // Range reduction constants for logarithms. // r(0) = 1, r(127) = 0.5 // r(k) = 2^-8 * ceil(2^8 * (1 - 2^-8) / (1 + k*2^-7)) diff --git a/libc/src/math/generic/common_constants.h b/libc/src/math/generic/common_constants.h index 291816a..72b1d564 100644 --- a/libc/src/math/generic/common_constants.h +++ b/libc/src/math/generic/common_constants.h @@ -11,6 +11,7 @@ #include "src/__support/FPUtil/triple_double.h" #include "src/__support/macros/config.h" +#include "src/__support/math/acosh_float_constants.h" #include "src/__support/math/exp_constants.h" #include "src/__support/number_pair.h" @@ -20,16 +21,10 @@ namespace LIBC_NAMESPACE_DECL { // computed and stored as float precision constants. extern const float ONE_OVER_F_FLOAT[128]; -// Lookup table for (1/f) where f = 1 + n*2^(-7), n = 0..127. -extern const double ONE_OVER_F[128]; - // Lookup table for log(f) = log(1 + n*2^(-7)) where n = 0..127, // computed and stored as float precision constants. extern const float LOG_F_FLOAT[128]; -// Lookup table for log(f) = log(1 + n*2^(-7)) where n = 0..127. -extern const double LOG_F[128]; - // Lookup table for range reduction constants r for logarithms. extern const float R[128]; diff --git a/libc/src/math/generic/erff.cpp b/libc/src/math/generic/erff.cpp index 44607a5..003b346 100644 --- a/libc/src/math/generic/erff.cpp +++ b/libc/src/math/generic/erff.cpp @@ -7,180 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/erff.h" -#include "src/__support/FPUtil/FPBits.h" -#include "src/__support/FPUtil/PolyEval.h" -#include "src/__support/FPUtil/except_value_utils.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/common.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY +#include "src/__support/math/erff.h" namespace LIBC_NAMESPACE_DECL { -// Polynomials approximating erf(x)/x on ( k/8, (k + 1)/8 ) generated by Sollya -// with: -// > P = fpminimax(erf(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14|], [|D...|], -// [k/8, (k + 1)/8]); -// for k = 0..31. -constexpr double COEFFS[32][8] = { - {0x1.20dd750429b6dp0, -0x1.812746b037753p-2, 0x1.ce2f219e8596ap-4, - -0x1.b82cdacb78fdap-6, 0x1.56479297dfda5p-8, -0x1.8b3ac5455ef02p-11, - -0x1.126fcac367e3bp-8, 0x1.2d0bdb3ba4984p-4}, - {0x1.20dd750429b6dp0, -0x1.812746b0379a8p-2, 0x1.ce2f21a03cf2ap-4, - -0x1.b82ce30de083ep-6, 0x1.565bcad3eb60fp-8, -0x1.c02c66f659256p-11, - 0x1.f92f673385229p-14, -0x1.def402648ae9p-17}, - {0x1.20dd750429b34p0, -0x1.812746b032dcep-2, 0x1.ce2f219d84aaep-4, - -0x1.b82ce22dcf139p-6, 0x1.565b9efcd4af1p-8, -0x1.c021f1af414bcp-11, - 0x1.f7c6d177eff82p-14, -0x1.c9e4410dcf865p-17}, - {0x1.20dd750426eabp0, -0x1.812746ae592c7p-2, 0x1.ce2f211525f14p-4, - -0x1.b82ccc125e63fp-6, 0x1.56596f261cfd3p-8, -0x1.bfde1ff8eeecfp-11, - 0x1.f31a9d15dc5d8p-14, -0x1.a5a4362844b3cp-17}, - {0x1.20dd75039c705p0, -0x1.812746777e74dp-2, 0x1.ce2f17af98a1bp-4, - -0x1.b82be4b817cbep-6, 0x1.564bec2e2962ep-8, -0x1.bee86f9da3558p-11, - 0x1.e9443689dc0ccp-14, -0x1.79c0f230805d8p-17}, - {0x1.20dd74f811211p0, -0x1.81274371a3e8fp-2, 0x1.ce2ec038262e5p-4, - -0x1.b8265b82c5e1fp-6, 0x1.5615a2e239267p-8, -0x1.bc63ae023dcebp-11, - 0x1.d87c2102f7e06p-14, -0x1.49584bea41d62p-17}, - {0x1.20dd746d063e3p0, -0x1.812729a8a950fp-2, 0x1.ce2cb0a2df232p-4, - -0x1.b80eca1f51278p-6, 0x1.5572e26c46815p-8, -0x1.b715e5638b65ep-11, - 0x1.bfbb195484968p-14, -0x1.177a565c15c52p-17}, - {0x1.20dd701b44486p0, -0x1.812691145f237p-2, 0x1.ce23a06b8cfd9p-4, - -0x1.b7c1dc7245288p-6, 0x1.53e92f7f397ddp-8, -0x1.ad97cc4acf0b2p-11, - 0x1.9f028b2b09b71p-14, -0x1.cdc4da08da8c1p-18}, - {0x1.20dd5715ac332p0, -0x1.8123e680bd0ebp-2, 0x1.ce0457aded691p-4, - -0x1.b6f52d52bed4p-6, 0x1.50c291b84414cp-8, -0x1.9ea246b1ad4a9p-11, - 0x1.77654674e0cap-14, -0x1.737c11a1bcebbp-18}, - {0x1.20dce6593e114p0, -0x1.811a59c02eadcp-2, 0x1.cdab53c7cd7d5p-4, - -0x1.b526d2e321eedp-6, 0x1.4b1d32cd8b994p-8, -0x1.8963143ec0a1ep-11, - 0x1.4ad5700e4db91p-14, -0x1.231e100e43ef2p-18}, - {0x1.20db48bfd5a62p0, -0x1.80fdd84f9e308p-2, 0x1.ccd340d462983p-4, - -0x1.b196a2928768p-6, 0x1.4210c2c13a0f7p-8, -0x1.6dbdfb4ff71aep-11, - 0x1.1bca2d17fbd71p-14, -0x1.bca36f90c7cf5p-19}, - {0x1.20d64b2f8f508p0, -0x1.80b4d4f19fa8bp-2, 0x1.cb088197262e3p-4, - -0x1.ab51fd02e5b99p-6, 0x1.34e1e5e81a632p-8, -0x1.4c66377b502cep-11, - 0x1.d9ad25066213cp-15, -0x1.4b0df7dd0cfa1p-19}, - {0x1.20c8fc1243576p0, -0x1.8010cb2009e27p-2, 0x1.c7a47e9299315p-4, - -0x1.a155be5683654p-6, 0x1.233502694997bp-8, -0x1.26c94b7d813p-11, - 0x1.8094f1de25fb9p-15, -0x1.e0e3d776c6eefp-20}, - {0x1.20a9bd1611bc1p0, -0x1.7ec7fbce83f9p-2, 0x1.c1d757d7317b7p-4, - -0x1.92c160cd589fp-6, 0x1.0d307269cc5c2p-8, -0x1.fda5b0d2d1879p-12, - 0x1.2fdd7b3b14a7fp-15, -0x1.54eed4a26af5ap-20}, - {0x1.20682834f943dp0, -0x1.7c73f747bf5a9p-2, 0x1.b8c2db4a9ffd1p-4, - -0x1.7f0e4ffe989ecp-6, 0x1.e7061eae4166ep-9, -0x1.ad36e873fff2dp-12, - 0x1.d39222396128ep-16, -0x1.d83dacec5ea6bp-21}, - {0x1.1feb8d12676d7p0, -0x1.7898347284afep-2, 0x1.aba3466b34451p-4, - -0x1.663adc573e2f9p-6, 0x1.ae99fb17c3e08p-9, -0x1.602f950ad5535p-12, - 0x1.5e9717490609dp-16, -0x1.3fca107bbc8d5p-21}, - {0x1.1f12fe3c536fap0, -0x1.72b1d1f22e6d3p-2, 0x1.99fc0eed4a896p-4, - -0x1.48db0a87bd8c6p-6, 0x1.73e368895aa61p-9, -0x1.19b35d5301fc8p-12, - 0x1.007987e4bb033p-16, -0x1.a7edcd4c2dc7p-22}, - {0x1.1db7b0df84d5dp0, -0x1.6a4e4a41cde02p-2, 0x1.83bbded16455dp-4, - -0x1.2809b3b36977ep-6, 0x1.39c08bab44679p-9, -0x1.b7b45a70ed119p-13, - 0x1.6e99b36410e7bp-17, -0x1.13619bb7ebc0cp-22}, - {0x1.1bb1c85c4a527p0, -0x1.5f23b99a249a3p-2, 0x1.694c91fa0d12cp-4, - -0x1.053e1ce11c72dp-6, 0x1.02bf72c50ea78p-9, -0x1.4f478fb56cb02p-13, - 0x1.005f80ecbe213p-17, -0x1.5f2446bde7f5bp-23}, - {0x1.18dec3bd51f9dp0, -0x1.5123f58346186p-2, 0x1.4b8a1ca536ab4p-4, - -0x1.c4243015cc723p-7, 0x1.a1a8a01d351efp-10, -0x1.f466b34f1d86bp-14, - 0x1.5f835eea0bf6ap-18, -0x1.b83165b939234p-24}, - {0x1.152804c3369f4p0, -0x1.4084cd4afd4bcp-2, 0x1.2ba2e836e47aap-4, - -0x1.800f2dfc6904bp-7, 0x1.4a6daf0669c59p-10, -0x1.6e326ab872317p-14, - 0x1.d9761a6a755a5p-19, -0x1.0fca33f9dd4b5p-24}, - {0x1.1087ad68356aap0, -0x1.2dbb044707459p-2, 0x1.0aea8ceaa0384p-4, - -0x1.40b516d52b3d2p-7, 0x1.00c9e05f01d22p-10, -0x1.076afb0dc0ff7p-14, - 0x1.39fadec400657p-19, -0x1.4b5761352e7e3p-25}, - {0x1.0b0a7a8ba4a22p0, -0x1.196990d22d4a1p-2, 0x1.d5551e6ac0c4dp-5, - -0x1.07cce1770bd1ap-7, 0x1.890347b8848bfp-11, -0x1.757ec96750b6ap-15, - 0x1.9b258a1e06bcep-20, -0x1.8fc6d22da7572p-26}, - {0x1.04ce2be70fb47p0, -0x1.0449e4b0b9cacp-2, 0x1.97f7424f4b0e7p-5, - -0x1.ac825439c42f4p-8, 0x1.28f5f65426dfbp-11, -0x1.05b699a90f90fp-15, - 0x1.0a888eecf4593p-20, -0x1.deace2b32bb31p-27}, - {0x1.fbf9fb0e11cc8p-1, -0x1.de2640856545ap-3, 0x1.5f5b1f47f851p-5, - -0x1.588bc71eb41b9p-8, 0x1.bc6a0a772f56dp-12, -0x1.6b9fad1f1657ap-16, - 0x1.573204ba66504p-21, -0x1.1d38065c94e44p-27}, - {0x1.ed8f18c99e031p-1, -0x1.b4cb6acd903b4p-3, 0x1.2c7f3dddd6fc1p-5, - -0x1.13052067df4ep-8, 0x1.4a5027444082fp-12, -0x1.f672bab0e2554p-17, - 0x1.b83c756348cc9p-22, -0x1.534f1a1079499p-28}, - {0x1.debd33044166dp-1, -0x1.8d7cd9053f7d8p-3, 0x1.ff9957fb3d6e7p-6, - -0x1.b50be55de0f36p-9, 0x1.e92c8ec53a628p-13, -0x1.5a4b88d508007p-17, - 0x1.1a27737559e26p-22, -0x1.942ae62cb2c14p-29}, - {0x1.cfdbf0386f3bdp-1, -0x1.68e33d93b0dc4p-3, 0x1.b2683d58f53dep-6, - -0x1.5a9174e70d26fp-9, 0x1.69ddd326d49cdp-13, -0x1.dd8f397a8219cp-18, - 0x1.6a755016ad4ddp-23, -0x1.e366e0139187dp-30}, - {0x1.c132adb8d7464p-1, -0x1.475a899f61b46p-3, 0x1.70a431397a77cp-6, - -0x1.12e3d35beeee2p-9, 0x1.0c16b05738333p-13, -0x1.4a47f873e144ep-18, - 0x1.d3d494c698c02p-24, -0x1.2302c59547fe5p-30}, - {0x1.b2f5fd05555e7p-1, -0x1.28feefbe03ec7p-3, 0x1.3923acbb3a676p-6, - -0x1.b4ff793cd6358p-10, 0x1.8ea0eb8c913bcp-14, -0x1.cb31ec2baceb1p-19, - 0x1.30011e7e80c04p-24, -0x1.617710635cb1dp-31}, - {0x1.a54853cd9593ep-1, -0x1.0dbdbaea4dc8ep-3, 0x1.0a93e2c20a0fdp-6, - -0x1.5c969ff401ea8p-10, 0x1.29e0cc64fe627p-14, -0x1.4160d8e9d3c2ap-19, - 0x1.8e7b67594624ap-25, -0x1.b1cf2c975b09bp-32}, - {0x1.983ceece09ff8p-1, -0x1.eacc78f7a2dp-4, 0x1.c74418410655fp-7, - -0x1.1756a050e441ep-10, 0x1.bff3650f7f548p-15, -0x1.c56c0217d3adap-20, - 0x1.07b4918d0b489p-25, -0x1.0d4be8c1c50f8p-32}, -}; - -LLVM_LIBC_FUNCTION(float, erff, (float x)) { - using FPBits = typename fputil::FPBits<float>; - FPBits xbits(x); - - uint32_t x_u = xbits.uintval(); - uint32_t x_abs = x_u & 0x7fff'ffffU; - - if (LIBC_UNLIKELY(x_abs >= 0x4080'0000U)) { - const float ONE[2] = {1.0f, -1.0f}; - const float SMALL[2] = {-0x1.0p-25f, 0x1.0p-25f}; - - int sign = xbits.is_neg() ? 1 : 0; - - if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) { - if (xbits.is_signaling_nan()) { - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - return (x_abs > 0x7f80'0000) ? x : ONE[sign]; - } - - return ONE[sign] + SMALL[sign]; - } - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - // Exceptional mask = common 0 bits of 2 exceptional values. - constexpr uint32_t EXCEPT_MASK = 0x809a'6184U; - - if (LIBC_UNLIKELY((x_abs & EXCEPT_MASK) == 0)) { - // Exceptional values - if (LIBC_UNLIKELY(x_abs == 0x3f65'9229U)) // |x| = 0x1.cb2452p-1f - return x < 0.0f ? fputil::round_result_slightly_down(-0x1.972ea8p-1f) - : fputil::round_result_slightly_up(0x1.972ea8p-1f); - if (LIBC_UNLIKELY(x_abs == 0x4004'1e6aU)) // |x| = 0x1.083cd4p+1f - return x < 0.0f ? fputil::round_result_slightly_down(-0x1.fe3462p-1f) - : fputil::round_result_slightly_up(0x1.fe3462p-1f); - if (x_abs == 0U) - return x; - } -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - - // Polynomial approximation: - // erf(x) ~ x * (c0 + c1 * x^2 + c2 * x^4 + ... + c7 * x^14) - double xd = static_cast<double>(x); - double xsq = xd * xd; - - const uint32_t EIGHT = 3 << FPBits::FRACTION_LEN; - int idx = static_cast<int>(FPBits(x_abs + EIGHT).get_val()); - - double x4 = xsq * xsq; - double c0 = fputil::multiply_add(xsq, COEFFS[idx][1], COEFFS[idx][0]); - double c1 = fputil::multiply_add(xsq, COEFFS[idx][3], COEFFS[idx][2]); - double c2 = fputil::multiply_add(xsq, COEFFS[idx][5], COEFFS[idx][4]); - double c3 = fputil::multiply_add(xsq, COEFFS[idx][7], COEFFS[idx][6]); - - double x8 = x4 * x4; - double p0 = fputil::multiply_add(x4, c1, c0); - double p1 = fputil::multiply_add(x4, c3, c2); - - return static_cast<float>(xd * fputil::multiply_add(x8, p1, p0)); -} +LLVM_LIBC_FUNCTION(float, erff, (float x)) { return math::erff(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/exp10f16.cpp b/libc/src/math/generic/exp10f16.cpp index 31abf3b..cb3c859 100644 --- a/libc/src/math/generic/exp10f16.cpp +++ b/libc/src/math/generic/exp10f16.cpp @@ -7,128 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/exp10f16.h" -#include "expxf16.h" -#include "hdr/errno_macros.h" -#include "hdr/fenv_macros.h" -#include "src/__support/CPP/array.h" -#include "src/__support/FPUtil/FEnvImpl.h" -#include "src/__support/FPUtil/FPBits.h" -#include "src/__support/FPUtil/PolyEval.h" -#include "src/__support/FPUtil/cast.h" -#include "src/__support/FPUtil/except_value_utils.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/FPUtil/nearest_integer.h" -#include "src/__support/FPUtil/rounding_mode.h" -#include "src/__support/common.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" -#include "src/__support/macros/properties/cpu_features.h" +#include "src/__support/math/exp10f16.h" namespace LIBC_NAMESPACE_DECL { -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS -#ifdef LIBC_TARGET_CPU_HAS_FMA_FLOAT -static constexpr size_t N_EXP10F16_EXCEPTS = 5; -#else -static constexpr size_t N_EXP10F16_EXCEPTS = 8; -#endif - -static constexpr fputil::ExceptValues<float16, N_EXP10F16_EXCEPTS> - EXP10F16_EXCEPTS = {{ - // x = 0x1.8f4p-2, exp10f16(x) = 0x1.3ap+1 (RZ) - {0x363dU, 0x40e8U, 1U, 0U, 1U}, - // x = 0x1.95cp-2, exp10f16(x) = 0x1.3ecp+1 (RZ) - {0x3657U, 0x40fbU, 1U, 0U, 0U}, - // x = -0x1.018p-4, exp10f16(x) = 0x1.bbp-1 (RZ) - {0xac06U, 0x3aecU, 1U, 0U, 0U}, - // x = -0x1.c28p+0, exp10f16(x) = 0x1.1ccp-6 (RZ) - {0xbf0aU, 0x2473U, 1U, 0U, 0U}, - // x = -0x1.e1cp+1, exp10f16(x) = 0x1.694p-13 (RZ) - {0xc387U, 0x09a5U, 1U, 0U, 0U}, -#ifndef LIBC_TARGET_CPU_HAS_FMA_FLOAT - // x = 0x1.0cp+1, exp10f16(x) = 0x1.f04p+6 (RZ) - {0x4030U, 0x57c1U, 1U, 0U, 1U}, - // x = 0x1.1b8p+1, exp10f16(x) = 0x1.47cp+7 (RZ) - {0x406eU, 0x591fU, 1U, 0U, 1U}, - // x = 0x1.1b8p+2, exp10f16(x) = 0x1.a4p+14 (RZ) - {0x446eU, 0x7690U, 1U, 0U, 1U}, -#endif - }}; -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -LLVM_LIBC_FUNCTION(float16, exp10f16, (float16 x)) { - using FPBits = fputil::FPBits<float16>; - FPBits x_bits(x); - - uint16_t x_u = x_bits.uintval(); - uint16_t x_abs = x_u & 0x7fffU; - - // When |x| >= 5, or x is NaN. - if (LIBC_UNLIKELY(x_abs >= 0x4500U)) { - // exp10(NaN) = NaN - if (x_bits.is_nan()) { - if (x_bits.is_signaling_nan()) { - fputil::raise_except_if_required(FE_INVALID); - return FPBits::quiet_nan().get_val(); - } - - return x; - } - - // When x >= 5. - if (x_bits.is_pos()) { - // exp10(+inf) = +inf - if (x_bits.is_inf()) - return FPBits::inf().get_val(); - - switch (fputil::quick_get_round()) { - case FE_TONEAREST: - case FE_UPWARD: - fputil::set_errno_if_required(ERANGE); - fputil::raise_except_if_required(FE_OVERFLOW); - return FPBits::inf().get_val(); - default: - return FPBits::max_normal().get_val(); - } - } - - // When x <= -8. - if (x_u >= 0xc800U) { - // exp10(-inf) = +0 - if (x_bits.is_inf()) - return FPBits::zero().get_val(); - - fputil::set_errno_if_required(ERANGE); - fputil::raise_except_if_required(FE_UNDERFLOW | FE_INEXACT); - - if (fputil::fenv_is_round_up()) - return FPBits::min_subnormal().get_val(); - return FPBits::zero().get_val(); - } - } - - // When x is 1, 2, 3, or 4. These are hard-to-round cases with exact results. - if (LIBC_UNLIKELY((x_u & ~(0x3c00U | 0x4000U | 0x4200U | 0x4400U)) == 0)) { - switch (x_u) { - case 0x3c00U: // x = 1.0f16 - return fputil::cast<float16>(10.0); - case 0x4000U: // x = 2.0f16 - return fputil::cast<float16>(100.0); - case 0x4200U: // x = 3.0f16 - return fputil::cast<float16>(1'000.0); - case 0x4400U: // x = 4.0f16 - return fputil::cast<float16>(10'000.0); - } - } - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - if (auto r = EXP10F16_EXCEPTS.lookup(x_u); LIBC_UNLIKELY(r.has_value())) - return r.value(); -#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS - - // 10^x = 2^((hi + mid) * log2(10)) * 10^lo - auto [exp2_hi_mid, exp10_lo] = exp10_range_reduction(x); - return fputil::cast<float16>(exp2_hi_mid * exp10_lo); -} +LLVM_LIBC_FUNCTION(float16, exp10f16, (float16 x)) { return math::exp10f16(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/exp10m1f16.cpp b/libc/src/math/generic/exp10m1f16.cpp index 545c479..6c2fdbe 100644 --- a/libc/src/math/generic/exp10m1f16.cpp +++ b/libc/src/math/generic/exp10m1f16.cpp @@ -7,7 +7,6 @@ //===----------------------------------------------------------------------===// #include "src/math/exp10m1f16.h" -#include "expxf16.h" #include "hdr/errno_macros.h" #include "hdr/fenv_macros.h" #include "src/__support/FPUtil/FEnvImpl.h" @@ -21,6 +20,7 @@ #include "src/__support/macros/config.h" #include "src/__support/macros/optimization.h" #include "src/__support/macros/properties/cpu_features.h" +#include "src/__support/math/exp10f16_utils.h" namespace LIBC_NAMESPACE_DECL { diff --git a/libc/src/math/generic/explogxf.h b/libc/src/math/generic/explogxf.h index be4328a..a2a6d60 100644 --- a/libc/src/math/generic/explogxf.h +++ b/libc/src/math/generic/explogxf.h @@ -13,6 +13,7 @@ #include "src/__support/common.h" #include "src/__support/macros/properties/cpu_features.h" +#include "src/__support/math/acoshf_utils.h" #include "src/__support/math/exp10f_utils.h" #include "src/__support/math/exp_utils.h" @@ -163,41 +164,6 @@ LIBC_INLINE static float log_eval_f(float x) { return result; } -// x should be positive, normal finite value -LIBC_INLINE static double log_eval(double x) { - // For x = 2^ex * (1 + mx) - // log(x) = ex * log(2) + log(1 + mx) - using FPB = fputil::FPBits<double>; - FPB bs(x); - - double ex = static_cast<double>(bs.get_exponent()); - - // p1 is the leading 7 bits of mx, i.e. - // p1 * 2^(-7) <= m_x < (p1 + 1) * 2^(-7). - int p1 = static_cast<int>(bs.get_mantissa() >> (FPB::FRACTION_LEN - 7)); - - // Set bs to (1 + (mx - p1*2^(-7)) - bs.set_uintval(bs.uintval() & (FPB::FRACTION_MASK >> 7)); - bs.set_biased_exponent(FPB::EXP_BIAS); - // dx = (mx - p1*2^(-7)) / (1 + p1*2^(-7)). - double dx = (bs.get_val() - 1.0) * ONE_OVER_F[p1]; - - // Minimax polynomial of log(1 + dx) generated by Sollya with: - // > P = fpminimax(log(1 + x)/x, 6, [|D...|], [0, 2^-7]); - const double COEFFS[6] = {-0x1.ffffffffffffcp-2, 0x1.5555555552ddep-2, - -0x1.ffffffefe562dp-3, 0x1.9999817d3a50fp-3, - -0x1.554317b3f67a5p-3, 0x1.1dc5c45e09c18p-3}; - double dx2 = dx * dx; - double c1 = fputil::multiply_add(dx, COEFFS[1], COEFFS[0]); - double c2 = fputil::multiply_add(dx, COEFFS[3], COEFFS[2]); - double c3 = fputil::multiply_add(dx, COEFFS[5], COEFFS[4]); - - double p = fputil::polyeval(dx2, dx, c1, c2, c3); - double result = - fputil::multiply_add(ex, /*log(2)*/ 0x1.62e42fefa39efp-1, LOG_F[p1] + p); - return result; -} - } // namespace LIBC_NAMESPACE_DECL #endif // LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H diff --git a/libc/src/math/generic/expxf16.h b/libc/src/math/generic/expxf16.h index 05ac95d..562a427 100644 --- a/libc/src/math/generic/expxf16.h +++ b/libc/src/math/generic/expxf16.h @@ -9,26 +9,18 @@ #ifndef LLVM_LIBC_SRC_MATH_GENERIC_EXPXF16_H #define LLVM_LIBC_SRC_MATH_GENERIC_EXPXF16_H +#include "hdr/stdint_proxy.h" #include "src/__support/FPUtil/FPBits.h" #include "src/__support/FPUtil/cast.h" #include "src/__support/FPUtil/multiply_add.h" #include "src/__support/FPUtil/nearest_integer.h" #include "src/__support/macros/attributes.h" #include "src/__support/macros/config.h" -#include <stdint.h> - +#include "src/__support/math/exp10_float16_constants.h" #include "src/__support/math/expf16_utils.h" namespace LIBC_NAMESPACE_DECL { -// Generated by Sollya with the following commands: -// > display = hexadecimal; -// > for i from 0 to 7 do printsingle(round(2^(i * 2^-3), SG, RN)); -constexpr cpp::array<uint32_t, 8> EXP2_MID_BITS = { - 0x3f80'0000U, 0x3f8b'95c2U, 0x3f98'37f0U, 0x3fa5'fed7U, - 0x3fb5'04f3U, 0x3fc5'672aU, 0x3fd7'44fdU, 0x3fea'c0c7U, -}; - LIBC_INLINE ExpRangeReduction exp2_range_reduction(float16 x) { // For -25 < x < 16, to compute 2^x, we perform the following range reduction: // find hi, mid, lo, such that: @@ -68,53 +60,6 @@ LIBC_INLINE ExpRangeReduction exp2_range_reduction(float16 x) { // Generated by Sollya with the following commands: // > display = hexadecimal; -// > round(log2(10), SG, RN); -static constexpr float LOG2F_10 = 0x1.a934fp+1f; - -// Generated by Sollya with the following commands: -// > display = hexadecimal; -// > round(log10(2), SG, RN); -static constexpr float LOG10F_2 = 0x1.344136p-2f; - -LIBC_INLINE ExpRangeReduction exp10_range_reduction(float16 x) { - // For -8 < x < 5, to compute 10^x, we perform the following range reduction: - // find hi, mid, lo, such that: - // x = (hi + mid) * log2(10) + lo, in which - // hi is an integer, - // mid * 2^3 is an integer, - // -2^(-4) <= lo < 2^(-4). - // In particular, - // hi + mid = round(x * 2^3) * 2^(-3). - // Then, - // 10^x = 10^(hi + mid + lo) = 2^((hi + mid) * log2(10)) + 10^lo - // We store 2^mid in the lookup table EXP2_MID_BITS, and compute 2^hi * 2^mid - // by adding hi to the exponent field of 2^mid. 10^lo is computed using a - // degree-4 minimax polynomial generated by Sollya. - - float xf = x; - float kf = fputil::nearest_integer(xf * (LOG2F_10 * 0x1.0p+3f)); - int x_hi_mid = static_cast<int>(kf); - unsigned x_hi = static_cast<unsigned>(x_hi_mid) >> 3; - unsigned x_mid = static_cast<unsigned>(x_hi_mid) & 0x7; - // lo = x - (hi + mid) = round(x * 2^3 * log2(10)) * log10(2) * (-2^(-3)) + x - float lo = fputil::multiply_add(kf, LOG10F_2 * -0x1.0p-3f, xf); - - uint32_t exp2_hi_mid_bits = - EXP2_MID_BITS[x_mid] + - static_cast<uint32_t>(x_hi << fputil::FPBits<float>::FRACTION_LEN); - float exp2_hi_mid = fputil::FPBits<float>(exp2_hi_mid_bits).get_val(); - // Degree-4 minimax polynomial generated by Sollya with the following - // commands: - // > display = hexadecimal; - // > P = fpminimax((10^x - 1)/x, 3, [|SG...|], [-2^-4, 2^-4]); - // > 1 + x * P; - float exp10_lo = fputil::polyeval(lo, 0x1p+0f, 0x1.26bb14p+1f, 0x1.53526p+1f, - 0x1.04b434p+1f, 0x1.2bcf9ep+0f); - return {exp2_hi_mid, exp10_lo}; -} - -// Generated by Sollya with the following commands: -// > display = hexadecimal; // > round(log2(exp(1)), SG, RN); static constexpr float LOG2F_E = 0x1.715476p+0f; diff --git a/libc/src/math/generic/inv_trigf_utils.cpp b/libc/src/math/generic/inv_trigf_utils.cpp deleted file mode 100644 index f23028b..0000000 --- a/libc/src/math/generic/inv_trigf_utils.cpp +++ /dev/null @@ -1,86 +0,0 @@ -//===-- Single-precision general exp/log functions ------------------------===// -// -// 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 -// -//===----------------------------------------------------------------------===// - -#include "inv_trigf_utils.h" -#include "src/__support/macros/config.h" - -namespace LIBC_NAMESPACE_DECL { - -// Polynomial approximation for 0 <= x <= 1: -// atan(x) ~ atan((i/16) + (x - (i/16)) * Q(x - i/16) -// = P(x - i/16) -// Generated by Sollya with: -// > for i from 1 to 16 do { -// mid_point = i/16; -// P = fpminimax(atan(mid_point + x), 8, [|D...|], [-1/32, 1/32]); -// print("{", coeff(P, 0), ",", coeff(P, 1), ",", coeff(P, 2), ",", -// coeff(P, 3), ",", coeff(P, 4), ",", coeff(P, 5), ",", coeff(P, 6), -// ",", coeff(P, 7), ",", coeff(P, 8), "},"); -// }; -// For i = 0, the polynomial is generated by: -// > P = fpminimax(atan(x)/x, 7, [|1, D...|], [0, 1/32]); -// > dirtyinfnorm((atan(x) - x*P)/x, [0, 1/32]); -// 0x1.feb2fcdba66447ccbe28a1a0f935b51678a718fb1p-59 -// Notice that degree-7 is good enough for atanf, but degree-8 helps reduce the -// error bounds for atan2f's fast pass 16 times, and it does not affect the -// performance of atanf much. -double ATAN_COEFFS[17][9] = { - {0.0, 1.0, 0x1.3f8d76d26d61bp-47, -0x1.5555555574cd8p-2, - 0x1.0dde5d06878eap-29, 0x1.99997738acc77p-3, 0x1.2c43eac9797cap-16, - -0x1.25fb020007dbdp-3, 0x1.c1b6c31d7b0aep-7}, - {0x1.ff55bb72cfde9p-5, 0x1.fe01fe01fe007p-1, -0x1.fc05f809ed8dap-5, - -0x1.4d69303afe04ep-2, 0x1.f61bc3e8349cp-5, 0x1.820839278756bp-3, - -0x1.eda4de1c6bf3fp-5, -0x1.0514d42d64a63p-3, 0x1.db3746a442dcbp-5}, - {0x1.fd5ba9aac2f6ep-4, 0x1.f81f81f81f813p-1, -0x1.f05e09d0dc378p-4, - -0x1.368c3aa719215p-2, 0x1.d9b16b33ff9c9p-4, 0x1.40488f9c6262ap-3, - -0x1.ba55933e62ea5p-4, -0x1.64c6a15cd9116p-4, 0x1.9273d5939a75ap-4}, - {0x1.7b97b4bce5b02p-3, 0x1.ee9c7f8458e05p-1, -0x1.665c226d6961p-3, - -0x1.1344bb7391703p-2, 0x1.42aca8b0081b9p-3, 0x1.c32d9381d7c03p-4, - -0x1.13e970672e246p-3, -0x1.181ed934dd733p-5, 0x1.bad81ea190c08p-4}, - {0x1.f5b75f92c80ddp-3, 0x1.e1e1e1e1e1e2cp-1, -0x1.c5894d10d363dp-3, - -0x1.ce6de025f9f5ep-3, 0x1.78a3a07c8dd7fp-3, 0x1.dd5f5180f386ep-5, - -0x1.1b1f513c4536bp-3, 0x1.0df852e58c43cp-6, 0x1.722e7a7e42505p-4}, - {0x1.362773707ebccp-2, 0x1.d272ca3fc5b2ep-1, -0x1.0997e8aeca8fbp-2, - -0x1.6cf6666e5e693p-3, 0x1.8dd1e907e88adp-3, 0x1.24849ac0caa5dp-7, - -0x1.f496be486229dp-4, 0x1.b7d54b8e759ecp-5, 0x1.d39c0d39c3922p-5}, - {0x1.6f61941e4def1p-2, 0x1.c0e070381c0f2p-1, -0x1.2726dd135d9eep-2, - -0x1.09f37b39b70e4p-3, 0x1.85eacdaadd712p-3, -0x1.04d66340d5b9p-5, - -0x1.8056b15a22b98p-4, 0x1.29baf494ad3ddp-4, 0x1.52d5881322a7ap-6}, - {0x1.a64eec3cc23fdp-2, 0x1.adbe87f94906ap-1, -0x1.3b9d8eab55addp-2, - -0x1.57c09646eb7p-4, 0x1.6795319e3b8dfp-3, -0x1.f2d89b5ef31bep-5, - -0x1.f38aac26203cap-5, 0x1.3262802235e3fp-4, -0x1.2afd6b9a57d66p-7}, - {0x1.dac670561bb4fp-2, 0x1.99999999999ap-1, -0x1.47ae147adff11p-2, - -0x1.5d867c40188b7p-5, 0x1.3a92a2df85e7ap-3, -0x1.3ec457c46e851p-4, - -0x1.ec1b9777e2e5bp-6, 0x1.0a542992a821ep-4, -0x1.ccffbe2f0d945p-6}, - {0x1.0657e94db30dp-1, 0x1.84f00c2780615p-1, -0x1.4c62cb562defap-2, - -0x1.e6495b3c14e03p-8, 0x1.063c2fa617bfcp-3, -0x1.58b782d9907aap-4, - -0x1.41e6ff524b7fp-8, 0x1.937dfff3205a7p-5, -0x1.0fb1fd1c729dp-5}, - {0x1.1e00babdefeb4p-1, 0x1.702e05c0b816ep-1, -0x1.4af2b78215fbep-2, - 0x1.5d0b7e9f36997p-6, 0x1.a1247cb978debp-4, -0x1.519e1457734cap-4, - 0x1.a755cf86b5bfbp-7, 0x1.096d174284564p-5, -0x1.081adf539ad58p-5}, - {0x1.345f01cce37bbp-1, 0x1.5babcc647fa8ep-1, -0x1.449db09426a6dp-2, - 0x1.655caac5896dap-5, 0x1.3bbbd22d05a61p-4, -0x1.34a2febee042fp-4, - 0x1.84df9c8269e34p-6, 0x1.200e8176c899ap-6, -0x1.c00b23c3ce222p-6}, - {0x1.4978fa3269ee1p-1, 0x1.47ae147ae1477p-1, -0x1.3a92a3055231ap-2, - 0x1.ec21b515a4a2p-5, 0x1.c2f8b81f9a0d2p-5, -0x1.0ba9964125453p-4, - 0x1.d7b5614777a05p-6, 0x1.971e91ed73595p-8, -0x1.3fc375a78dc74p-6}, - {0x1.5d58987169b18p-1, 0x1.34679ace01343p-1, -0x1.2ddfb039136e5p-2, - 0x1.2491307b9fb73p-4, 0x1.29c7e4886dc22p-5, -0x1.bca78bcca83ap-5, - 0x1.e63efd7cbe1ddp-6, -0x1.8ea6c4f03b42dp-10, -0x1.9385b5c3a6997p-7}, - {0x1.700a7c5784634p-1, 0x1.21fb78121fb76p-1, -0x1.1f6a8499e5d1ap-2, - 0x1.41b15e5e29423p-4, 0x1.59bc953163345p-6, -0x1.63b54b13184ddp-5, - 0x1.c9086666d213p-6, -0x1.90c3b4ad8d4bcp-8, -0x1.80f08ed9f6f57p-8}, - {0x1.819d0b7158a4dp-1, 0x1.107fbbe01107ep-1, -0x1.0feeb4089670ep-2, - 0x1.50e5afb93f5cbp-4, 0x1.2a7c2adffeffbp-7, -0x1.12bd29b4f1b43p-5, - 0x1.93f71f0eb00eap-6, -0x1.10ece5ad30e28p-7, -0x1.db1a76bcd2b9cp-10}, - {0x1.921fb54442d18p-1, 0x1.ffffffffffffep-2, -0x1.fffffffffc51cp-3, - 0x1.555555557002ep-4, -0x1.a88260c338e75p-30, -0x1.99999f9a7614fp-6, - 0x1.555e31a1e15e9p-6, -0x1.245240d65e629p-7, -0x1.fa9ba66478903p-11}, -}; - -} // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/inv_trigf_utils.h b/libc/src/math/generic/inv_trigf_utils.h deleted file mode 100644 index 8b47aba..0000000 --- a/libc/src/math/generic/inv_trigf_utils.h +++ /dev/null @@ -1,110 +0,0 @@ -//===-- Single-precision general inverse trigonometric functions ----------===// -// -// 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_SRC_MATH_GENERIC_INV_TRIGF_UTILS_H -#define LLVM_LIBC_SRC_MATH_GENERIC_INV_TRIGF_UTILS_H - -#include "src/__support/FPUtil/PolyEval.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/common.h" -#include "src/__support/macros/config.h" - -namespace LIBC_NAMESPACE_DECL { - -// PI and PI / 2 -static constexpr double M_MATH_PI = 0x1.921fb54442d18p+1; -static constexpr double M_MATH_PI_2 = 0x1.921fb54442d18p+0; - -extern double ATAN_COEFFS[17][9]; - -// Look-up table for atan(k/16) with k = 0..16. -static constexpr double ATAN_K_OVER_16[17] = { - 0.0, - 0x1.ff55bb72cfdeap-5, - 0x1.fd5ba9aac2f6ep-4, - 0x1.7b97b4bce5b02p-3, - 0x1.f5b75f92c80ddp-3, - 0x1.362773707ebccp-2, - 0x1.6f61941e4def1p-2, - 0x1.a64eec3cc23fdp-2, - 0x1.dac670561bb4fp-2, - 0x1.0657e94db30dp-1, - 0x1.1e00babdefeb4p-1, - 0x1.345f01cce37bbp-1, - 0x1.4978fa3269ee1p-1, - 0x1.5d58987169b18p-1, - 0x1.700a7c5784634p-1, - 0x1.819d0b7158a4dp-1, - 0x1.921fb54442d18p-1, -}; - -// For |x| <= 1/32 and 0 <= i <= 16, return Q(x) such that: -// Q(x) ~ (atan(x + i/16) - atan(i/16)) / x. -LIBC_INLINE static double atan_eval(double x, unsigned i) { - double x2 = x * x; - - double c0 = fputil::multiply_add(x, ATAN_COEFFS[i][2], ATAN_COEFFS[i][1]); - double c1 = fputil::multiply_add(x, ATAN_COEFFS[i][4], ATAN_COEFFS[i][3]); - double c2 = fputil::multiply_add(x, ATAN_COEFFS[i][6], ATAN_COEFFS[i][5]); - double c3 = fputil::multiply_add(x, ATAN_COEFFS[i][8], ATAN_COEFFS[i][7]); - - double x4 = x2 * x2; - double d1 = fputil::multiply_add(x2, c1, c0); - double d2 = fputil::multiply_add(x2, c3, c2); - double p = fputil::multiply_add(x4, d2, d1); - return p; -} - -// Evaluate atan without big lookup table. -// atan(n/d) - atan(k/16) = atan((n/d - k/16) / (1 + (n/d) * (k/16))) -// = atan((n - d * k/16)) / (d + n * k/16)) -// So we let q = (n - d * k/16) / (d + n * k/16), -// and approximate with Taylor polynomial: -// atan(q) ~ q - q^3/3 + q^5/5 - q^7/7 + q^9/9 -LIBC_INLINE static double atan_eval_no_table(double num, double den, - double k_over_16) { - double num_r = fputil::multiply_add(den, -k_over_16, num); - double den_r = fputil::multiply_add(num, k_over_16, den); - double q = num_r / den_r; - - constexpr double ATAN_TAYLOR[] = { - -0x1.5555555555555p-2, - 0x1.999999999999ap-3, - -0x1.2492492492492p-3, - 0x1.c71c71c71c71cp-4, - }; - double q2 = q * q; - double q3 = q2 * q; - double q4 = q2 * q2; - double c0 = fputil::multiply_add(q2, ATAN_TAYLOR[1], ATAN_TAYLOR[0]); - double c1 = fputil::multiply_add(q2, ATAN_TAYLOR[3], ATAN_TAYLOR[2]); - double d = fputil::multiply_add(q4, c1, c0); - return fputil::multiply_add(q3, d, q); -} - -// > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|], -// [|1, D...|], [0, 0.5]); -static constexpr double ASIN_COEFFS[10] = { - 0x1.5555555540fa1p-3, 0x1.333333512edc2p-4, 0x1.6db6cc1541b31p-5, - 0x1.f1caff324770ep-6, 0x1.6e43899f5f4f4p-6, 0x1.1f847cf652577p-6, - 0x1.9b60f47f87146p-7, 0x1.259e2634c494fp-6, -0x1.df946fa875ddp-8, - 0x1.02311ecf99c28p-5}; - -// Evaluate P(x^2) - 1, where P(x^2) ~ asin(x)/x -LIBC_INLINE static double asin_eval(double xsq) { - double x4 = xsq * xsq; - double r1 = fputil::polyeval(x4, ASIN_COEFFS[0], ASIN_COEFFS[2], - ASIN_COEFFS[4], ASIN_COEFFS[6], ASIN_COEFFS[8]); - double r2 = fputil::polyeval(x4, ASIN_COEFFS[1], ASIN_COEFFS[3], - ASIN_COEFFS[5], ASIN_COEFFS[7], ASIN_COEFFS[9]); - return fputil::multiply_add(xsq, r2, r1); -} - -} // namespace LIBC_NAMESPACE_DECL - -#endif // LLVM_LIBC_SRC_MATH_GENERIC_INV_TRIGF_UTILS_H diff --git a/libc/src/math/generic/log1pf.cpp b/libc/src/math/generic/log1pf.cpp index 7f61429..16b1b34 100644 --- a/libc/src/math/generic/log1pf.cpp +++ b/libc/src/math/generic/log1pf.cpp @@ -37,6 +37,7 @@ namespace internal { // We don't need to treat denormal and 0 LIBC_INLINE float log(double x) { + using namespace acoshf_internal; constexpr double LOG_2 = 0x1.62e42fefa39efp-1; using FPBits = typename fputil::FPBits<double>; |