diff --git a/libc/config/darwin/arm/entrypoints.txt b/libc/config/darwin/arm/entrypoints.txt --- a/libc/config/darwin/arm/entrypoints.txt +++ b/libc/config/darwin/arm/entrypoints.txt @@ -129,6 +129,7 @@ libc.src.math.coshf libc.src.math.cosf libc.src.math.erff + libc.src.math.exp libc.src.math.expf libc.src.math.exp10f libc.src.math.exp2f diff --git a/libc/config/linux/aarch64/entrypoints.txt b/libc/config/linux/aarch64/entrypoints.txt --- a/libc/config/linux/aarch64/entrypoints.txt +++ b/libc/config/linux/aarch64/entrypoints.txt @@ -243,6 +243,7 @@ libc.src.math.coshf libc.src.math.cosf libc.src.math.erff + libc.src.math.exp libc.src.math.expf libc.src.math.exp10f libc.src.math.exp2f diff --git a/libc/config/linux/riscv64/entrypoints.txt b/libc/config/linux/riscv64/entrypoints.txt --- a/libc/config/linux/riscv64/entrypoints.txt +++ b/libc/config/linux/riscv64/entrypoints.txt @@ -252,6 +252,7 @@ libc.src.math.coshf libc.src.math.cosf libc.src.math.erff + libc.src.math.exp libc.src.math.expf libc.src.math.exp10f libc.src.math.exp2f diff --git a/libc/config/linux/x86_64/entrypoints.txt b/libc/config/linux/x86_64/entrypoints.txt --- a/libc/config/linux/x86_64/entrypoints.txt +++ b/libc/config/linux/x86_64/entrypoints.txt @@ -256,6 +256,7 @@ libc.src.math.coshf libc.src.math.cosf libc.src.math.erff + libc.src.math.exp libc.src.math.expf libc.src.math.exp10f libc.src.math.exp2f diff --git a/libc/config/windows/entrypoints.txt b/libc/config/windows/entrypoints.txt --- a/libc/config/windows/entrypoints.txt +++ b/libc/config/windows/entrypoints.txt @@ -128,6 +128,7 @@ libc.src.math.cosf libc.src.math.coshf libc.src.math.erff + libc.src.math.exp libc.src.math.expf libc.src.math.exp10f libc.src.math.exp2f diff --git a/libc/docs/math/index.rst b/libc/docs/math/index.rst --- a/libc/docs/math/index.rst +++ b/libc/docs/math/index.rst @@ -352,7 +352,7 @@ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ | erfcl | | | | | | | | | | | | | +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ -| exp | | | | | | | | | | | | | +| exp | |check| | |check| | | |check| | |check| | | | |check| | | | | | +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ | expf | |check| | |check| | | |check| | |check| | | | |check| | | | | | +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ @@ -483,7 +483,7 @@ cos |check| large cosh |check| erf |check| -exp |check| +exp |check| |check| exp10 |check| exp2 |check| expm1 |check| diff --git a/libc/spec/stdc.td b/libc/spec/stdc.td --- a/libc/spec/stdc.td +++ b/libc/spec/stdc.td @@ -434,7 +434,9 @@ FunctionSpec<"erff", RetValSpec, [ArgSpec]>, + FunctionSpec<"exp", RetValSpec, [ArgSpec]>, FunctionSpec<"expf", RetValSpec, [ArgSpec]>, + FunctionSpec<"exp2f", RetValSpec, [ArgSpec]>, FunctionSpec<"expm1f", RetValSpec, [ArgSpec]>, diff --git a/libc/src/__support/FPUtil/PolyEval.h b/libc/src/__support/FPUtil/PolyEval.h --- a/libc/src/__support/FPUtil/PolyEval.h +++ b/libc/src/__support/FPUtil/PolyEval.h @@ -10,6 +10,7 @@ #define LLVM_LIBC_SRC_SUPPORT_FPUTIL_POLYEVAL_H #include "multiply_add.h" +#include "src/__support/CPP/type_traits.h" #include "src/__support/common.h" // Evaluate polynomial using Horner's Scheme: @@ -22,10 +23,12 @@ namespace __llvm_libc { namespace fputil { -template LIBC_INLINE T polyeval(T, T a0) { return a0; } +template LIBC_INLINE T polyeval(const T &, const T &a0) { + return a0; +} template -LIBC_INLINE T polyeval(T x, T a0, Ts... a) { +LIBC_INLINE T polyeval(const T &x, const T &a0, const Ts &...a) { return multiply_add(x, polyeval(x, a...), a0); } diff --git a/libc/src/__support/FPUtil/double_double.h b/libc/src/__support/FPUtil/double_double.h --- a/libc/src/__support/FPUtil/double_double.h +++ b/libc/src/__support/FPUtil/double_double.h @@ -31,14 +31,15 @@ } // Assumption: |a.hi| >= |b.hi| -LIBC_INLINE constexpr DoubleDouble add(DoubleDouble a, DoubleDouble b) { +LIBC_INLINE constexpr DoubleDouble add(const DoubleDouble &a, + const DoubleDouble &b) { DoubleDouble r = exact_add(a.hi, b.hi); double lo = a.lo + b.lo; return exact_add(r.hi, r.lo + lo); } // Assumption: |a.hi| >= |b| -LIBC_INLINE constexpr DoubleDouble add(DoubleDouble a, double b) { +LIBC_INLINE constexpr DoubleDouble add(const DoubleDouble &a, double b) { DoubleDouble r = exact_add(a.hi, b); return exact_add(r.hi, r.lo + a.lo); } @@ -75,14 +76,29 @@ return r; } -LIBC_INLINE DoubleDouble quick_mult(DoubleDouble a, DoubleDouble b) { +LIBC_INLINE DoubleDouble quick_mult(double a, const DoubleDouble &b) { + DoubleDouble r = exact_mult(a, b.hi); + r.lo = multiply_add(a, b.lo, r.lo); + return r; +} + +LIBC_INLINE DoubleDouble quick_mult(const DoubleDouble &a, + const DoubleDouble &b) { DoubleDouble r = exact_mult(a.hi, b.hi); - double t1 = fputil::multiply_add(a.hi, b.lo, r.lo); - double t2 = fputil::multiply_add(a.lo, b.hi, t1); + double t1 = multiply_add(a.hi, b.lo, r.lo); + double t2 = multiply_add(a.lo, b.hi, t1); r.lo = t2; return r; } +// Assuming |c| >= |a * b|. +template <> +LIBC_INLINE DoubleDouble multiply_add(const DoubleDouble &a, + const DoubleDouble &b, + const DoubleDouble &c) { + return add(c, quick_mult(a, b)); +} + } // namespace __llvm_libc::fputil #endif // LLVM_LIBC_SRC_SUPPORT_FPUTIL_DOUBLEDOUBLE_H diff --git a/libc/src/__support/FPUtil/dyadic_float.h b/libc/src/__support/FPUtil/dyadic_float.h --- a/libc/src/__support/FPUtil/dyadic_float.h +++ b/libc/src/__support/FPUtil/dyadic_float.h @@ -82,9 +82,9 @@ return *this; } - // Assume that it is already normalized and output is also normal. + // Assume that it is already normalized and output is not underflow. // Output is rounded correctly with respect to the current rounding mode. - // TODO(lntue): Test or add support for denormal output. + // TODO(lntue): Add support for underflow. // TODO(lntue): Test or add specialization for x86 long double. template && @@ -99,24 +99,72 @@ constexpr size_t PRECISION = FloatProperties::MANTISSA_WIDTH + 1; using output_bits_t = typename FPBits::UIntType; - MantissaType m_hi(mantissa >> (Bits - PRECISION)); - auto d_hi = FPBits::create_value( - sign, exponent + (Bits - 1) + FloatProperties::EXPONENT_BIAS, - output_bits_t(m_hi) & FloatProperties::MANTISSA_MASK); + int exp_hi = exponent + static_cast((Bits - 1) + + FloatProperties::EXPONENT_BIAS); - const MantissaType round_mask = MantissaType(1) << (Bits - PRECISION - 1); + bool denorm = false; + uint32_t shift = Bits - PRECISION; + if (LIBC_UNLIKELY(exp_hi <= 0)) { + // Output is denormal. + denorm = true; + shift = (Bits - PRECISION) + static_cast(1 - exp_hi); + + exp_hi = FloatProperties::EXPONENT_BIAS; + } + + int exp_lo = exp_hi - PRECISION - 1; + + MantissaType m_hi(mantissa >> shift); + + T d_hi = FPBits::create_value(sign, exp_hi, + output_bits_t(m_hi) & + FloatProperties::MANTISSA_MASK) + .get_val(); + + const MantissaType round_mask = MantissaType(1) << (shift - 1); const MantissaType sticky_mask = round_mask - MantissaType(1); bool round_bit = !(mantissa & round_mask).is_zero(); bool sticky_bit = !(mantissa & sticky_mask).is_zero(); int round_and_sticky = int(round_bit) * 2 + int(sticky_bit); - auto d_lo = FPBits::create_value(sign, - exponent + (Bits - PRECISION - 2) + - FloatProperties::EXPONENT_BIAS, - output_bits_t(0)); + + T d_lo; + if (LIBC_UNLIKELY(exp_lo <= 0)) { + // d_lo is denormal, but the output is normal. + int scale_up_exponent = 2 * PRECISION; + T scale_up_factor = + FPBits::create_value( + sign, FloatProperties::EXPONENT_BIAS + scale_up_exponent, + output_bits_t(0)) + .get_val(); + T scale_down_factor = + FPBits::create_value( + sign, FloatProperties::EXPONENT_BIAS - scale_up_exponent, + output_bits_t(0)) + .get_val(); + + d_lo = FPBits::create_value(sign, exp_lo + scale_up_exponent, + output_bits_t(0)) + .get_val(); + + return multiply_add(d_lo, T(round_and_sticky), d_hi * scale_up_factor) * + scale_down_factor; + } + + d_lo = FPBits::create_value(sign, exp_lo, output_bits_t(0)).get_val(); // Still correct without FMA instructions if `d_lo` is not underflow. - return multiply_add(d_lo.get_val(), T(round_and_sticky), d_hi.get_val()); + T r = multiply_add(d_lo, T(round_and_sticky), d_hi); + + if (LIBC_UNLIKELY(denorm)) { + // Output is denormal, simply clear the exponent field. + output_bits_t clear_exp = output_bits_t(exp_hi) + << FloatProperties::MANTISSA_WIDTH; + output_bits_t r_bits = FPBits(r).uintval() - clear_exp; + return FPBits(r_bits).get_val(); + } + + return r; } explicit operator MantissaType() const { @@ -226,6 +274,14 @@ return result; } +// Simple polynomial approximation. +template +constexpr DyadicFloat multiply_add(const DyadicFloat &a, + const DyadicFloat &b, + const DyadicFloat &c) { + return quick_add(c, quick_mul(a, b)); +} + // Simple exponentiation implementation for printf. Only handles positive // exponents, since division isn't implemented. template diff --git a/libc/src/__support/FPUtil/multiply_add.h b/libc/src/__support/FPUtil/multiply_add.h --- a/libc/src/__support/FPUtil/multiply_add.h +++ b/libc/src/__support/FPUtil/multiply_add.h @@ -20,7 +20,8 @@ // multiply_add(x, y, z) = x*y + z // which uses FMA instructions to speed up if available. -template LIBC_INLINE T multiply_add(T x, T y, T z) { +template +LIBC_INLINE T multiply_add(const T &x, const T &y, const T &z) { return x * y + z; } @@ -35,12 +36,11 @@ namespace __llvm_libc { namespace fputil { -template <> LIBC_INLINE float multiply_add(float x, float y, float z) { +LIBC_INLINE float multiply_add(float x, float y, float z) { return fma(x, y, z); } -template <> -LIBC_INLINE double multiply_add(double x, double y, double z) { +LIBC_INLINE double multiply_add(double x, double y, double z) { return fma(x, y, z); } diff --git a/libc/src/math/CMakeLists.txt b/libc/src/math/CMakeLists.txt --- a/libc/src/math/CMakeLists.txt +++ b/libc/src/math/CMakeLists.txt @@ -79,6 +79,7 @@ add_math_entrypoint_object(erff) +add_math_entrypoint_object(exp) add_math_entrypoint_object(expf) add_math_entrypoint_object(exp2f) diff --git a/libc/src/math/exp.h b/libc/src/math/exp.h new file mode 100644 --- /dev/null +++ b/libc/src/math/exp.h @@ -0,0 +1,18 @@ +//===-- Implementation header for exp ---------------------------*- 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_EXP_H +#define LLVM_LIBC_SRC_MATH_EXP_H + +namespace __llvm_libc { + +double exp(double x); + +} // namespace __llvm_libc + +#endif // LLVM_LIBC_SRC_MATH_EXP_H diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt --- a/libc/src/math/generic/CMakeLists.txt +++ b/libc/src/math/generic/CMakeLists.txt @@ -548,6 +548,29 @@ -O3 ) +add_entrypoint_object( + exp + SRCS + exp.cpp + HDRS + ../exp.h + DEPENDS + .common_constants + libc.src.__support.FPUtil.basic_operations + 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.include.errno + libc.src.errno.errno + libc.include.math + COMPILE_OPTIONS + -O3 +) + add_entrypoint_object( expf SRCS diff --git a/libc/src/math/generic/exp.cpp b/libc/src/math/generic/exp.cpp new file mode 100644 --- /dev/null +++ b/libc/src/math/generic/exp.cpp @@ -0,0 +1,595 @@ +//===-- Double-precision e^x function -------------------------------------===// +// +// 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 "src/math/exp.h" +#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2. +#include "src/__support/CPP/bit.h" +#include "src/__support/CPP/optional.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/nearest_integer.h" +#include "src/__support/FPUtil/rounding_mode.h" +#include "src/__support/common.h" +#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY + +#include + +namespace __llvm_libc { + +using fputil::DoubleDouble; +using Float128 = typename fputil::DyadicFloat<128>; + +// 2^12 * log2(e) +constexpr double LOG2_E = 0x1.71547652b82fep+0; + +// Error bounds: +// Errors when using double precision. +constexpr double ERR_D = 0x1.8p-63; +// Errors when using double-double precision. +constexpr double ERR_DD = 0x1.0p-99; + +struct TripleDouble { + double hi = 0.0; + double mid = 0.0; + double lo = 0.0; +}; + +// -2^-12 * log(2) +// > a = -2^-12 * log(2); +// > b = round(a, 30, RN); +// > c = round(a - b, 30, RN); +// > d = round(a - b - c, D, RN); +// Errors < 1.5 * 2^-133 +constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13; +constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47; +constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47; +constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79; + +// 2^(k * 2^-6), for k = 0..63. +constexpr TripleDouble EXP_MID1[64] = { + {0x1p0, 0, 0}, + {0x1.02c9a3e778061p0, -0x1.19083535b085dp-56, -0x1.9085b0a3d74d5p-110}, + {0x1.059b0d3158574p0, 0x1.d73e2a475b465p-55, 0x1.05ff94f8d257ep-110}, + {0x1.0874518759bc8p0, 0x1.186be4bb284ffp-57, 0x1.15820d96b414fp-111}, + {0x1.0b5586cf9890fp0, 0x1.8a62e4adc610bp-54, -0x1.67c9bd6ebf74cp-108}, + {0x1.0e3ec32d3d1a2p0, 0x1.03a1727c57b53p-59, -0x1.5aa76994e9ddbp-113}, + {0x1.11301d0125b51p0, -0x1.6c51039449b3ap-54, 0x1.9d58b988f562dp-109}, + {0x1.1429aaea92dep0, -0x1.32fbf9af1369ep-54, -0x1.2fe7bb4c76416p-108}, + {0x1.172b83c7d517bp0, -0x1.19041b9d78a76p-55, 0x1.4f2406aa13ffp-109}, + {0x1.1a35beb6fcb75p0, 0x1.e5b4c7b4968e4p-55, 0x1.ad36183926ae8p-111}, + {0x1.1d4873168b9aap0, 0x1.e016e00a2643cp-54, 0x1.ea62d0881b918p-110}, + {0x1.2063b88628cd6p0, 0x1.dc775814a8495p-55, -0x1.781dbc16f1ea4p-111}, + {0x1.2387a6e756238p0, 0x1.9b07eb6c70573p-54, -0x1.4d89f9af532ep-109}, + {0x1.26b4565e27cddp0, 0x1.2bd339940e9d9p-55, 0x1.277393a461b77p-110}, + {0x1.29e9df51fdee1p0, 0x1.612e8afad1255p-55, 0x1.de5448560469p-111}, + {0x1.2d285a6e4030bp0, 0x1.0024754db41d5p-54, -0x1.ee9d8f8cb9307p-110}, + {0x1.306fe0a31b715p0, 0x1.6f46ad23182e4p-55, 0x1.7b7b2f09cd0d9p-110}, + {0x1.33c08b26416ffp0, 0x1.32721843659a6p-54, -0x1.406a2ea6cfc6bp-108}, + {0x1.371a7373aa9cbp0, -0x1.63aeabf42eae2p-54, 0x1.87e3e12516bfap-108}, + {0x1.3a7db34e59ff7p0, -0x1.5e436d661f5e3p-56, 0x1.9b0b1ff17c296p-111}, + {0x1.3dea64c123422p0, 0x1.ada0911f09ebcp-55, -0x1.808ba68fa8fb7p-109}, + {0x1.4160a21f72e2ap0, -0x1.ef3691c309278p-58, -0x1.32b43eafc6518p-114}, + {0x1.44e086061892dp0, 0x1.89b7a04ef80dp-59, -0x1.0ac312de3d922p-114}, + {0x1.486a2b5c13cdp0, 0x1.3c1a3b69062fp-56, 0x1.e1eebae743acp-111}, + {0x1.4bfdad5362a27p0, 0x1.d4397afec42e2p-56, 0x1.c06c7745c2b39p-113}, + {0x1.4f9b2769d2ca7p0, -0x1.4b309d25957e3p-54, -0x1.1aa1fd7b685cdp-112}, + {0x1.5342b569d4f82p0, -0x1.07abe1db13cadp-55, 0x1.fa733951f214cp-111}, + {0x1.56f4736b527dap0, 0x1.9bb2c011d93adp-54, -0x1.ff86852a613ffp-111}, + {0x1.5ab07dd485429p0, 0x1.6324c054647adp-54, -0x1.744ee506fdafep-109}, + {0x1.5e76f15ad2148p0, 0x1.ba6f93080e65ep-54, -0x1.95f9ab75fa7d6p-108}, + {0x1.6247eb03a5585p0, -0x1.383c17e40b497p-54, 0x1.5d8e757cfb991p-111}, + {0x1.6623882552225p0, -0x1.bb60987591c34p-54, 0x1.4a337f4dc0a3bp-108}, + {0x1.6a09e667f3bcdp0, -0x1.bdd3413b26456p-54, 0x1.57d3e3adec175p-108}, + {0x1.6dfb23c651a2fp0, -0x1.bbe3a683c88abp-57, 0x1.a59f88abbe778p-115}, + {0x1.71f75e8ec5f74p0, -0x1.16e4786887a99p-55, -0x1.269796953a4c3p-109}, + {0x1.75feb564267c9p0, -0x1.0245957316dd3p-54, -0x1.8f8e7fa19e5e8p-108}, + {0x1.7a11473eb0187p0, -0x1.41577ee04992fp-55, -0x1.4217a932d10d4p-113}, + {0x1.7e2f336cf4e62p0, 0x1.05d02ba15797ep-56, 0x1.70a1427f8fcdfp-112}, + {0x1.82589994cce13p0, -0x1.d4c1dd41532d8p-54, 0x1.0f6ad65cbbac1p-112}, + {0x1.868d99b4492edp0, -0x1.fc6f89bd4f6bap-54, -0x1.f16f65181d921p-109}, + {0x1.8ace5422aa0dbp0, 0x1.6e9f156864b27p-54, -0x1.30644a7836333p-110}, + {0x1.8f1ae99157736p0, 0x1.5cc13a2e3976cp-55, 0x1.3bf26d2b85163p-114}, + {0x1.93737b0cdc5e5p0, -0x1.75fc781b57ebcp-57, 0x1.697e257ac0db2p-111}, + {0x1.97d829fde4e5p0, -0x1.d185b7c1b85d1p-54, 0x1.7edb9d7144b6fp-108}, + {0x1.9c49182a3f09p0, 0x1.c7c46b071f2bep-56, 0x1.6376b7943085cp-110}, + {0x1.a0c667b5de565p0, -0x1.359495d1cd533p-54, 0x1.354084551b4fbp-109}, + {0x1.a5503b23e255dp0, -0x1.d2f6edb8d41e1p-54, -0x1.bfd7adfd63f48p-111}, + {0x1.a9e6b5579fdbfp0, 0x1.0fac90ef7fd31p-54, 0x1.8b16ae39e8cb9p-109}, + {0x1.ae89f995ad3adp0, 0x1.7a1cd345dcc81p-54, 0x1.a7fbc3ae675eap-108}, + {0x1.b33a2b84f15fbp0, -0x1.2805e3084d708p-57, 0x1.2babc0edda4d9p-111}, + {0x1.b7f76f2fb5e47p0, -0x1.5584f7e54ac3bp-56, 0x1.aa64481e1ab72p-111}, + {0x1.bcc1e904bc1d2p0, 0x1.23dd07a2d9e84p-55, 0x1.9a164050e1258p-109}, + {0x1.c199bdd85529cp0, 0x1.11065895048ddp-55, 0x1.99e51125928dap-110}, + {0x1.c67f12e57d14bp0, 0x1.2884dff483cadp-54, -0x1.fc44c329d5cb2p-109}, + {0x1.cb720dcef9069p0, 0x1.503cbd1e949dbp-56, 0x1.d8765566b032ep-110}, + {0x1.d072d4a07897cp0, -0x1.cbc3743797a9cp-54, -0x1.e7044039da0f6p-108}, + {0x1.d5818dcfba487p0, 0x1.2ed02d75b3707p-55, -0x1.ab053b05531fcp-111}, + {0x1.da9e603db3285p0, 0x1.c2300696db532p-54, 0x1.7f6246f0ec615p-108}, + {0x1.dfc97337b9b5fp0, -0x1.1a5cd4f184b5cp-54, 0x1.b7225a944efd6p-108}, + {0x1.e502ee78b3ff6p0, 0x1.39e8980a9cc8fp-55, 0x1.1e92cb3c2d278p-109}, + {0x1.ea4afa2a490dap0, -0x1.e9c23179c2893p-54, -0x1.fc0f242bbf3dep-109}, + {0x1.efa1bee615a27p0, 0x1.dc7f486a4b6bp-54, 0x1.f6dd5d229ff69p-108}, + {0x1.f50765b6e454p0, 0x1.9d3e12dd8a18bp-54, -0x1.4019bffc80ef3p-110}, + {0x1.fa7c1819e90d8p0, 0x1.74853f3a5931ep-55, 0x1.dc060c36f7651p-112}, +}; + +// 2^(k * 2^-12), for k = 0..63. +constexpr TripleDouble EXP_MID2[64] = { + {0x1p0, 0, 0}, + {0x1.000b175effdc7p0, 0x1.ae8e38c59c72ap-54, 0x1.39726694630e3p-108}, + {0x1.00162f3904052p0, -0x1.7b5d0d58ea8f4p-58, 0x1.e5e06ddd31156p-112}, + {0x1.0021478e11ce6p0, 0x1.4115cb6b16a8ep-54, 0x1.5a0768b51f609p-111}, + {0x1.002c605e2e8cfp0, -0x1.d7c96f201bb2fp-55, 0x1.d008403605217p-111}, + {0x1.003779a95f959p0, 0x1.84711d4c35e9fp-54, 0x1.89bc16f765708p-109}, + {0x1.0042936faa3d8p0, -0x1.0484245243777p-55, -0x1.4535b7f8c1e2dp-109}, + {0x1.004dadb113dap0, -0x1.4b237da2025f9p-54, -0x1.8ba92f6b25456p-108}, + {0x1.0058c86da1c0ap0, -0x1.5e00e62d6b30dp-56, -0x1.30c72e81f4294p-113}, + {0x1.0063e3a559473p0, 0x1.a1d6cedbb9481p-54, -0x1.34a5384e6f0b9p-110}, + {0x1.006eff583fc3dp0, -0x1.4acf197a00142p-54, 0x1.f8d0580865d2ep-108}, + {0x1.007a1b865a8cap0, -0x1.eaf2ea42391a5p-57, -0x1.002bcb3ae9a99p-111}, + {0x1.0085382faef83p0, 0x1.da93f90835f75p-56, 0x1.c3c5aedee9851p-111}, + {0x1.00905554425d4p0, -0x1.6a79084ab093cp-55, 0x1.7217851d1ec6ep-109}, + {0x1.009b72f41a12bp0, 0x1.86364f8fbe8f8p-54, -0x1.80cbca335a7c3p-110}, + {0x1.00a6910f3b6fdp0, -0x1.82e8e14e3110ep-55, -0x1.706bd4eb22595p-110}, + {0x1.00b1afa5abcbfp0, -0x1.4f6b2a7609f71p-55, -0x1.b55dd523f3c08p-111}, + {0x1.00bcceb7707ecp0, -0x1.e1a258ea8f71bp-56, 0x1.90a1e207cced1p-110}, + {0x1.00c7ee448ee02p0, 0x1.4362ca5bc26f1p-56, 0x1.78d0472db37c5p-110}, + {0x1.00d30e4d0c483p0, 0x1.095a56c919d02p-54, -0x1.bcd4db3cb52fep-109}, + {0x1.00de2ed0ee0f5p0, -0x1.406ac4e81a645p-57, -0x1.cf1b131575ec2p-112}, + {0x1.00e94fd0398ep0, 0x1.b5a6902767e09p-54, -0x1.6aaa1fa7ff913p-112}, + {0x1.00f4714af41d3p0, -0x1.91b2060859321p-54, 0x1.68f236dff3218p-110}, + {0x1.00ff93412315cp0, 0x1.427068ab22306p-55, -0x1.e8bb58067e60ap-109}, + {0x1.010ab5b2cbd11p0, 0x1.c1d0660524e08p-54, 0x1.d4cd5e1d71fdfp-108}, + {0x1.0115d89ff3a8bp0, -0x1.e7bdfb3204be8p-54, 0x1.e4ecf350ebe88p-108}, + {0x1.0120fc089ff63p0, 0x1.843aa8b9cbbc6p-55, 0x1.6a2aa2c89c4f8p-109}, + {0x1.012c1fecd613bp0, -0x1.34104ee7edae9p-56, 0x1.1ca368a20ed05p-110}, + {0x1.0137444c9b5b5p0, -0x1.2b6aeb6176892p-56, 0x1.edb1095d925cfp-114}, + {0x1.01426927f5278p0, 0x1.a8cd33b8a1bb3p-56, -0x1.488c78eded75fp-111}, + {0x1.014d8e7ee8d2fp0, 0x1.2edc08e5da99ap-56, -0x1.7480f5ea1b3c9p-113}, + {0x1.0158b4517bb88p0, 0x1.57ba2dc7e0c73p-55, -0x1.ae45989a04dd5p-111}, + {0x1.0163da9fb3335p0, 0x1.b61299ab8cdb7p-54, 0x1.bf48007d80987p-109}, + {0x1.016f0169949edp0, -0x1.90565902c5f44p-54, 0x1.1aa91a059292cp-109}, + {0x1.017a28af25567p0, 0x1.70fc41c5c2d53p-55, 0x1.b6663292855f5p-110}, + {0x1.018550706ab62p0, 0x1.4b9a6e145d76cp-54, 0x1.e7fbca6793d94p-108}, + {0x1.019078ad6a19fp0, -0x1.008eff5142bf9p-56, -0x1.5b9f5c7de3b93p-110}, + {0x1.019ba16628de2p0, -0x1.77669f033c7dep-54, 0x1.4638bf2f6acabp-110}, + {0x1.01a6ca9aac5f3p0, -0x1.09bb78eeead0ap-54, -0x1.ab237b9a069c5p-109}, + {0x1.01b1f44af9f9ep0, 0x1.371231477ece5p-54, 0x1.3ab358be97cefp-108}, + {0x1.01bd1e77170b4p0, 0x1.5e7626621eb5bp-56, -0x1.4027b2294bb64p-110}, + {0x1.01c8491f08f08p0, -0x1.bc72b100828a5p-54, 0x1.656394426c99p-111}, + {0x1.01d37442d507p0, -0x1.ce39cbbab8bbep-57, 0x1.bf9785189bdd8p-111}, + {0x1.01de9fe280ac8p0, 0x1.16996709da2e2p-55, 0x1.7c12f86114fe3p-109}, + {0x1.01e9cbfe113efp0, -0x1.c11f5239bf535p-55, -0x1.653d5d24b5d28p-109}, + {0x1.01f4f8958c1c6p0, 0x1.e1d4eb5edc6b3p-55, 0x1.04a0cdc1d86d7p-109}, + {0x1.020025a8f6a35p0, -0x1.afb99946ee3fp-54, 0x1.c678c46149782p-109}, + {0x1.020b533856324p0, -0x1.8f06d8a148a32p-54, 0x1.48524e1e9df7p-108}, + {0x1.02168143b0281p0, -0x1.2bf310fc54eb6p-55, 0x1.9953ea727ff0bp-109}, + {0x1.0221afcb09e3ep0, -0x1.c95a035eb4175p-54, -0x1.ccfbbec22d28ep-108}, + {0x1.022cdece68c4fp0, -0x1.491793e46834dp-54, 0x1.9e2bb6e181de1p-108}, + {0x1.02380e4dd22adp0, -0x1.3e8d0d9c49091p-56, 0x1.f17609ae29308p-110}, + {0x1.02433e494b755p0, -0x1.314aa16278aa3p-54, -0x1.c7dc2c476bfb8p-110}, + {0x1.024e6ec0da046p0, 0x1.48daf888e9651p-55, -0x1.fab994971d4a3p-109}, + {0x1.02599fb483385p0, 0x1.56dc8046821f4p-55, 0x1.848b62cbdd0afp-109}, + {0x1.0264d1244c719p0, 0x1.45b42356b9d47p-54, -0x1.bf603ba715d0cp-109}, + {0x1.027003103b10ep0, -0x1.082ef51b61d7ep-56, 0x1.89434e751e1aap-110}, + {0x1.027b357854772p0, 0x1.2106ed0920a34p-56, -0x1.03b54fd64e8acp-110}, + {0x1.0286685c9e059p0, -0x1.fd4cf26ea5d0fp-54, 0x1.7785ea0acc486p-109}, + {0x1.02919bbd1d1d8p0, -0x1.09f8775e78084p-54, -0x1.ce447fdb35ff9p-109}, + {0x1.029ccf99d720ap0, 0x1.64cbba902ca27p-58, 0x1.5b884aab5642ap-112}, + {0x1.02a803f2d170dp0, 0x1.4383ef231d207p-54, -0x1.cfb3e46d7c1cp-108}, + {0x1.02b338c811703p0, 0x1.4a47a505b3a47p-54, -0x1.0d40cee4b81afp-112}, + {0x1.02be6e199c811p0, 0x1.e47120223467fp-54, 0x1.6ae7d36d7c1f7p-109}, +}; + +// Polynomial approximations with double precision: +// Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. +// For |dx| < 2^-13 + 2^-30: +// | output - expm1(dx) / dx | < 2^-51. +LIBC_INLINE double poly_approx_d(double dx) { + // dx^2 + double dx2 = dx * dx; + // c0 = 1 + dx / 2 + double c0 = fputil::multiply_add(dx, 0.5, 1.0); + // c1 = 1/6 + dx / 24 + double c1 = + fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3); + // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24 + double p = fputil::multiply_add(dx2, c1, c0); + return p; +} + +// Polynomial approximation with double-double precision: +// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720 +// For |dx| < 2^-13 + 2^-30: +// | output - exp(dx) | < 2^-101 +DoubleDouble poly_approx_dd(const DoubleDouble &dx) { + // Taylor polynomial. + constexpr DoubleDouble COEFFS[] = { + {0, 0x1p0}, // 1 + {0, 0x1p0}, // 1 + {0, 0x1p-1}, // 1/2 + {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6 + {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24 + {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120 + {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720 + }; + + DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2], + COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]); + return p; +} + +// Polynomial approximation with 128-bit precision: +// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^7 / 5040 +// For |dx| < 2^-13 + 2^-30: +// | output - exp(dx) | < 2^-126. +Float128 poly_approx_f128(const Float128 &dx) { + using MType = typename Float128::MantissaType; + + constexpr Float128 COEFFS_128[]{ + {false, -127, MType({0, 0x8000000000000000})}, // 1.0 + {false, -127, MType({0, 0x8000000000000000})}, // 1.0 + {false, -128, MType({0, 0x8000000000000000})}, // 0.5 + {false, -130, MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/6 + {false, -132, MType({0xaaaaaaaaaaaaaaab, 0xaaaaaaaaaaaaaaaa})}, // 1/24 + {false, -134, MType({0x8888888888888889, 0x8888888888888888})}, // 1/120 + {false, -137, MType({0x60b60b60b60b60b6, 0xb60b60b60b60b60b})}, // 1/720 + {false, -140, MType({0x00b00b00b00b00b0, 0xb00b00b00b00b00b})}, // 1/5040 + }; + + Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2], + COEFFS_128[3], COEFFS_128[4], COEFFS_128[5], + COEFFS_128[6], COEFFS_128[7]); + return p; +} + +// Compute exp(x) using 128-bit precision. +// TODO(lntue): investigate triple-double precision implementation for this +// step. +Float128 exp_f128(double x, double kd, int idx1, int idx2) { + // Recalculate dx: + + double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact + double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact + double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133 + + Float128 dx = fputil::quick_add( + Float128(t1), fputil::quick_add(Float128(t2), Float128(t3))); + + // TODO: Skip recalculating exp_mid1 and exp_mid2. + Float128 exp_mid1 = + fputil::quick_add(Float128(EXP_MID1[idx1].hi), + fputil::quick_add(Float128(EXP_MID1[idx1].mid), + Float128(EXP_MID1[idx1].lo))); + + Float128 exp_mid2 = + fputil::quick_add(Float128(EXP_MID2[idx2].hi), + fputil::quick_add(Float128(EXP_MID2[idx2].mid), + Float128(EXP_MID2[idx2].lo))); + + Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2); + + Float128 p = poly_approx_f128(dx); + + Float128 r = fputil::quick_mul(exp_mid, p); + + r.exponent += static_cast(kd) >> 12; + + return r; +} + +// Compute exp(x) with double-double precision. +DoubleDouble exp_double_double(double x, double kd, + const DoubleDouble &exp_mid) { + // Recalculate dx: + // dx = x - k * 2^-12 * log(2) + double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact + double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact + double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130 + + DoubleDouble dx = fputil::exact_add(t1, t2); + dx.lo += t3; + + // Degree-6 Taylor polynomial approximation in double-double precision. + // | p - exp(x) | < 2^-100. + DoubleDouble p = poly_approx_dd(dx); + + // Error bounds: 2^-99. + DoubleDouble r = fputil::quick_mult(exp_mid, p); + + return r; +} + +// Rounding tests when the output might be denormal. +cpp::optional ziv_test_denorm(int hi, double mid, double lo, + double err) { + using FloatProp = typename fputil::FloatProperties; + + // Scaling factor = 1/(min normal number) = 2^1022 + int64_t exp_hi = static_cast(hi + 1022) << FloatProp::MANTISSA_WIDTH; + double mid_hi = cpp::bit_cast(exp_hi + cpp::bit_cast(mid)); + + // Extra errors from another rounding step. + err += 0x1.0p-52; + + double lo_u = lo + err; + double lo_l = lo - err; + double mid_lo_u = + cpp::bit_cast(exp_hi + cpp::bit_cast(lo_u)); + double mid_lo_l = + cpp::bit_cast(exp_hi + cpp::bit_cast(lo_l)); + + // By adding 2^-511, the results will have similar rounding points as denormal + // outputs. + double upper = (mid_hi + mid_lo_u); + double lower = (mid_hi + mid_lo_l); + + uint64_t scale_down = 0; + + if (upper < 1.0) { + // Upper bound is in denormal range, need extra rounding. + upper += 1.0; + lower += 1.0; + scale_down = 0x3FF0'0000'0000'0000; // 1.0 + } + + if (LIBC_LIKELY(upper == lower)) { + return cpp::bit_cast(cpp::bit_cast(upper) - scale_down); + } + + return cpp::nullopt; +} + +// Check for exceptional cases when +// |x| < 2^-53 +double set_exceptional(double x) { + using FPBits = typename fputil::FPBits; + using FloatProp = typename fputil::FloatProperties; + FPBits xbits(x); + + uint64_t x_u = xbits.uintval(); + uint64_t x_abs = x_u & FloatProp::EXP_MANT_MASK; + + // |x| < 2^-53 + if (x_abs <= 0x3ca0'0000'0000'0000ULL) { + // exp(x) ~ 1 + x + return 1 + x; + } + + // x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan. + + // x <= log(2^-1075) or -inf/nan + if (x_u >= 0xc087'4910'd52d'3052ULL) { + // exp(-Inf) = 0 + if (xbits.is_inf()) + return 0.0; + + // exp(nan) = nan + if (xbits.is_nan()) + return x; + + if (fputil::quick_get_round() == FE_UPWARD) + return static_cast(FPBits(FPBits::MIN_SUBNORMAL)); + fputil::set_errno_if_required(ERANGE); + fputil::raise_except_if_required(FE_UNDERFLOW); + return 0.0; + } + + // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan + // x is finite + if (x_u < 0x7ff0'0000'0000'0000ULL) { + int rounding = fputil::quick_get_round(); + if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) + return static_cast(FPBits(FPBits::MAX_NORMAL)); + + fputil::set_errno_if_required(ERANGE); + fputil::raise_except_if_required(FE_OVERFLOW); + } + // x is +inf or nan + return x + static_cast(FPBits::inf()); +} + +LLVM_LIBC_FUNCTION(double, exp, (double x)) { + using FPBits = typename fputil::FPBits; + using FloatProp = typename fputil::FloatProperties; + FPBits xbits(x); + + uint64_t x_u = xbits.uintval(); + + // Upper bound: max normal number = 2^1023 * (2 - 2^-52) + // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9 + // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9 + // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9 + // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty + + // Lower bound: min denormal number / 2 = 2^-1075 + // > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9 + + // Another lower bound: min normal number = 2^-1022 + // > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9 + + // x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53. + if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 || + (x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) || + x_u < 0x3ca0000000000000)) { + return set_exceptional(x); + } + + // Now log(2^-1022) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52)) + + // Range reduction: + // Let x = log(2) * (hi + mid1 + mid2) + lo + // in which: + // hi is an integer + // mid1 * 2^6 is an integer + // mid2 * 2^12 is an integer + // then: + // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo). + // With this formula: + // - multiplying by 2^hi is exact and cheap, simply by adding the exponent + // field. + // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables. + // - exp(lo) ~ 1 + lo + a0 * lo^2 + ... + // + // They can be defined by: + // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x) + // If we store L2E = round(log2(e), D, RN), then: + // log2(e) - L2E ~ 1.5 * 2^(-56) + // So the errors when computing in double precision is: + // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <= + // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | + + // + | x * 2^12 * L2E - D(x * 2^12 * L2E) | + // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN + // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes. + // So if: + // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely + // in double precision, the reduced argument: + // lo = x - log(2) * (hi + mid1 + mid2) is bounded by: + // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x)) + // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52)) + // < 2^-13 + 2^-41 + // + + // The following trick computes the round(x * L2E) more efficiently + // than using the rounding instructions, with the tradeoff for less accuracy, + // and hence a slightly larger range for the reduced argument `lo`. + // + // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9, + // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23, + // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t. + // Thus, the goal is to be able to use an additional addition and fixed width + // shift to get an int32_t representing round(x * 2^12 * L2E). + // + // Assuming int32_t using 2-complement representation, since the mantissa part + // of a double precision is unsigned with the leading bit hidden, if we add an + // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the + // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be + // considered as a proper 2-complement representations of x*2^12*L2E. + // + // One small problem with this approach is that the sum (x*2^12*L2E + C) in + // double precision is rounded to the least significant bit of the dorminant + // factor C. In order to minimize the rounding errors from this addition, we + // want to minimize e1. Another constraint that we want is that after + // shifting the mantissa so that the least significant bit of int32_t + // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without + // any adjustment. So combining these 2 requirements, we can choose + // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence + // after right shifting the mantissa, the resulting int32_t has correct sign. + // With this choice of C, the number of mantissa bits we need to shift to the + // right is: 52 - 33 = 19. + // + // Moreover, since the integer right shifts are equivalent to rounding down, + // we can add an extra 0.5 so that it will become round-to-nearest, tie-to- + // +infinity. So in particular, we can compute: + // hmm = x * 2^12 * L2E + C, + // where C = 2^33 + 2^32 + 2^-1, then if + // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19), + // the reduced argument: + // lo = x - log(2) * 2^-12 * k is bounded by: + // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19 + // = 2^-13 + 2^-31 + 2^-41. + // + // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the + // exponent 2^12 is not needed. So we can simply define + // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and + // k = int32_t(lower 51 bits of double(x * L2E + C) >> 19). + + // Rounding errors <= 2^-31 + 2^-41. + double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21); + int k = static_cast(cpp::bit_cast(tmp) >> 19); + double kd = static_cast(k); + + uint32_t idx1 = (k >> 6) & 0x3f; + uint32_t idx2 = k & 0x3f; + int hi = k >> 12; + + bool denorm = (hi <= -1022); + + DoubleDouble exp_mid1{EXP_MID1[idx1].mid, EXP_MID1[idx1].hi}; + DoubleDouble exp_mid2{EXP_MID2[idx2].mid, EXP_MID2[idx2].hi}; + + DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2); + + // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo) + // = 2^11 * 2^-13 * 2^-52 + // = 2^-54. + // |dx| < 2^-13 + 2^-30. + double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact + double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h); + + // We use the degree-4 Taylor polynomial to approximate exp(lo): + // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo) + // So that the errors are bounded by: + // |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58 + // Let P_ be an evaluation of P where all intermediate computations are in + // double precision. Using either Horner's or Estrin's schemes, the evaluated + // errors can be bounded by: + // |P_(dx) - P(dx)| < 2^-51 + // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64 + // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63. + // Since we approximate + // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo, + // We use the expression: + // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~ + // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo) + // with errors bounded by 1.5 * 2^-63. + + double mid_lo = dx * exp_mid.hi; + + // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. + double p = poly_approx_d(dx); + + double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo); + + if (LIBC_UNLIKELY(denorm)) { + if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D); + LIBC_LIKELY(r.has_value())) + return r.value(); + } else { + double upper = exp_mid.hi + (lo + ERR_D); + double lower = exp_mid.hi + (lo - ERR_D); + + if (LIBC_LIKELY(upper == lower)) { + // to multiply by 2^hi, a fast way is to simply add hi to the exponent + // field. + int64_t exp_hi = static_cast(hi) << FloatProp::MANTISSA_WIDTH; + double r = cpp::bit_cast(exp_hi + cpp::bit_cast(upper)); + return r; + } + } + + // Use double-double + DoubleDouble r_dd = exp_double_double(x, kd, exp_mid); + + if (LIBC_UNLIKELY(denorm)) { + if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD); + LIBC_LIKELY(r.has_value())) + return r.value(); + } else { + double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD); + double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD); + + if (LIBC_LIKELY(upper_dd == lower_dd)) { + int64_t exp_hi = static_cast(hi) << FloatProp::MANTISSA_WIDTH; + double r = + cpp::bit_cast(exp_hi + cpp::bit_cast(upper_dd)); + return r; + } + } + + // Use 128-bit precision + Float128 r_f128 = exp_f128(x, kd, idx1, idx2); + + return static_cast(r_f128); +} + +} // namespace __llvm_libc diff --git a/libc/test/src/math/CMakeLists.txt b/libc/test/src/math/CMakeLists.txt --- a/libc/test/src/math/CMakeLists.txt +++ b/libc/test/src/math/CMakeLists.txt @@ -591,6 +591,20 @@ libc.src.__support.FPUtil.fp_bits ) +add_fp_unittest( + exp_test + NEED_MPFR + SUITE + libc_math_unittests + SRCS + exp_test.cpp + DEPENDS + libc.src.errno.errno + libc.include.math + libc.src.math.exp + libc.src.__support.FPUtil.fp_bits +) + add_fp_unittest( exp2f_test NEED_MPFR diff --git a/libc/test/src/math/exp_test.cpp b/libc/test/src/math/exp_test.cpp new file mode 100644 --- /dev/null +++ b/libc/test/src/math/exp_test.cpp @@ -0,0 +1,123 @@ +//===-- Unittests for exp -------------------------------------------------===// +// +// 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 "src/__support/FPUtil/FPBits.h" +#include "src/errno/libc_errno.h" +#include "src/math/exp.h" +#include "test/UnitTest/FPMatcher.h" +#include "test/UnitTest/Test.h" +#include "utils/MPFRWrapper/MPFRUtils.h" +#include + +#include +#include + +namespace mpfr = __llvm_libc::testing::mpfr; +using __llvm_libc::testing::tlog; + +DECLARE_SPECIAL_CONSTANTS(double) + +TEST(LlvmLibcExpTest, SpecialNumbers) { + EXPECT_FP_EQ(aNaN, __llvm_libc::exp(aNaN)); + EXPECT_FP_EQ(inf, __llvm_libc::exp(inf)); + EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::exp(neg_inf)); + EXPECT_FP_EQ_WITH_EXCEPTION(zero, __llvm_libc::exp(-0x1.0p20), FE_UNDERFLOW); + EXPECT_FP_EQ_WITH_EXCEPTION(inf, __llvm_libc::exp(0x1.0p20), FE_OVERFLOW); + EXPECT_FP_EQ_ALL_ROUNDING(1.0, __llvm_libc::exp(0.0)); + EXPECT_FP_EQ_ALL_ROUNDING(1.0, __llvm_libc::exp(-0.0)); +} + +TEST(LlvmLibcExpTest, TrickyInputs) { + constexpr int N = 14; + constexpr uint64_t INPUTS[N] = { + 0x3FD79289C6E6A5C0, + 0x3FD05DE80A173EA0, // 0x1.05de80a173eap-2 + 0xbf1eb7a4cb841fcc, // -0x1.eb7a4cb841fccp-14 + 0xbf19a61fb925970d, + 0x3fda7b764e2cf47a, // 0x1.a7b764e2cf47ap-2 + 0xc04757852a4b93aa, // -0x1.757852a4b93aap+5 + 0x4044c19e5712e377, // x=0x1.4c19e5712e377p+5 + 0xbf19a61fb925970d, // x=-0x1.9a61fb925970dp-14 + 0xc039a74cdab36c28, // x=-0x1.9a74cdab36c28p+4 + 0xc085b3e4e2e3bba9, // x=-0x1.5b3e4e2e3bba9p+9 + 0xc086960d591aec34, // x=-0x1.6960d591aec34p+9 + 0xc086232c09d58d91, // x=-0x1.6232c09d58d91p+9 + 0xc0874910d52d3051, // x=-0x1.74910d52d3051p9 + 0xc0867a172ceb0990, // x=-0x1.67a172ceb099p+9 + }; + for (int i = 0; i < N; ++i) { + double x = double(FPBits(INPUTS[i])); + EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp, x, __llvm_libc::exp(x), + 0.5); + } +} + +TEST(LlvmLibcExpTest, InDoubleRange) { + constexpr uint64_t COUNT = 1'231; + uint64_t START = __llvm_libc::fputil::FPBits(0.25).uintval(); + uint64_t STOP = __llvm_libc::fputil::FPBits(4.0).uintval(); + uint64_t STEP = (STOP - START) / COUNT; + + auto test = [&](mpfr::RoundingMode rounding_mode) { + mpfr::ForceRoundingMode __r(rounding_mode); + if (!__r.success) + return; + + uint64_t fails = 0; + uint64_t count = 0; + uint64_t cc = 0; + double mx, mr = 0.0; + double tol = 0.5; + + for (uint64_t i = 0, v = START; i <= COUNT; ++i, v += STEP) { + double x = FPBits(v).get_val(); + if (isnan(x) || isinf(x) || x < 0.0) + continue; + libc_errno = 0; + double result = __llvm_libc::exp(x); + ++cc; + if (isnan(result) || isinf(result)) + continue; + + ++count; + // ASSERT_MPFR_MATCH(mpfr::Operation::Log, x, result, 0.5); + if (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Exp, x, result, + 0.5, rounding_mode)) { + ++fails; + while (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Exp, x, + result, tol, rounding_mode)) { + mx = x; + mr = result; + + if (tol > 1000.0) + break; + + tol *= 2.0; + } + } + } + tlog << " Exp failed: " << fails << "/" << count << "/" << cc + << " tests.\n"; + tlog << " Max ULPs is at most: " << static_cast(tol) << ".\n"; + if (fails) { + EXPECT_MPFR_MATCH(mpfr::Operation::Exp, mx, mr, 0.5, rounding_mode); + } + }; + + tlog << " Test Rounding To Nearest...\n"; + test(mpfr::RoundingMode::Nearest); + + tlog << " Test Rounding Downward...\n"; + test(mpfr::RoundingMode::Downward); + + tlog << " Test Rounding Upward...\n"; + test(mpfr::RoundingMode::Upward); + + tlog << " Test Rounding Toward Zero...\n"; + test(mpfr::RoundingMode::TowardZero); +} diff --git a/libc/test/src/math/log10_test.cpp b/libc/test/src/math/log10_test.cpp --- a/libc/test/src/math/log10_test.cpp +++ b/libc/test/src/math/log10_test.cpp @@ -33,7 +33,7 @@ } TEST(LlvmLibcLog10Test, TrickyInputs) { - constexpr int N = 35; + constexpr int N = 36; constexpr uint64_t INPUTS[N] = { 0x3ff0000000000000, // x = 1.0 0x4024000000000000, // x = 10.0 @@ -61,7 +61,8 @@ 0x3fefffffffef06ad, 0x3fefde0f22c7d0eb, 0x225e7812faadb32f, 0x3fee1076964c2903, 0x3fdfe93fff7fceb0, 0x3ff012631ad8df10, 0x3fefbfdaa448ed98, 0x44b0c9705a25ce02, 0x2c88d301065c7f9b, - 0x30160580e7268a99, 0x5ca04103b7eaa345, 0x19ad77dc4a40093f}; + 0x30160580e7268a99, 0x5ca04103b7eaa345, 0x19ad77dc4a40093f, + 0x0000449fb5c8a96e}; for (int i = 0; i < N; ++i) { double x = double(FPBits(INPUTS[i])); EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log10, x,