https://github.com/bassiounix updated https://github.com/llvm/llvm-project/pull/162127
>From 079611c5ba5a5e00f6283198954e3b0a7cee8179 Mon Sep 17 00:00:00 2001 From: bassiounix <[email protected]> Date: Mon, 6 Oct 2025 20:41:07 +0300 Subject: [PATCH] [libc][math] Refactor expm1 implementation to header-only in src/__support/math folder. --- libc/shared/math.h | 1 + libc/shared/math/expm1.h | 23 + libc/src/__support/math/CMakeLists.txt | 20 + libc/src/__support/math/expm1.h | 518 ++++++++++++++++++ libc/src/math/generic/CMakeLists.txt | 13 +- libc/src/math/generic/expm1.cpp | 492 +---------------- libc/test/shared/CMakeLists.txt | 1 + libc/test/shared/shared_math_test.cpp | 1 + .../llvm-project-overlay/libc/BUILD.bazel | 28 +- 9 files changed, 586 insertions(+), 511 deletions(-) create mode 100644 libc/shared/math/expm1.h create mode 100644 libc/src/__support/math/expm1.h diff --git a/libc/shared/math.h b/libc/shared/math.h index bcddb39021a9c..a5c9de4fb7029 100644 --- a/libc/shared/math.h +++ b/libc/shared/math.h @@ -54,6 +54,7 @@ #include "math/exp2m1f16.h" #include "math/expf.h" #include "math/expf16.h" +#include "math/expm1.h" #include "math/frexpf.h" #include "math/frexpf128.h" #include "math/frexpf16.h" diff --git a/libc/shared/math/expm1.h b/libc/shared/math/expm1.h new file mode 100644 index 0000000000000..4c8dbdc013a11 --- /dev/null +++ b/libc/shared/math/expm1.h @@ -0,0 +1,23 @@ +//===-- Shared expm1 function -----------------------------------*- C++ -*-===// +// +// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. +// See https://llvm.org/LICENSE.txt for license information. +// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception +// +//===----------------------------------------------------------------------===// + +#ifndef LLVM_LIBC_SHARED_MATH_EXPM1_H +#define LLVM_LIBC_SHARED_MATH_EXPM1_H + +#include "shared/libc_common.h" +#include "src/__support/math/expm1.h" + +namespace LIBC_NAMESPACE_DECL { +namespace shared { + +using math::expm1; + +} // namespace shared +} // namespace LIBC_NAMESPACE_DECL + +#endif // LLVM_LIBC_SHARED_MATH_EXPM1_H diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt index aaace44d04d3b..74f17b9fd8099 100644 --- a/libc/src/__support/math/CMakeLists.txt +++ b/libc/src/__support/math/CMakeLists.txt @@ -886,6 +886,26 @@ add_header_library( libc.src.__support.macros.properties.cpu_features ) +add_header_library( + expm1 + HDRS + expm1.h + DEPENDS + .common_constants + .exp_constants + libc.src.__support.CPP.bit + 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.rounding_mode + libc.src.__support.FPUtil.triple_double + libc.src.__support.integer_literals + libc.src.__support.macros.optimization + libc.src.errno.errno +) + add_header_library( range_reduction_double HDRS diff --git a/libc/src/__support/math/expm1.h b/libc/src/__support/math/expm1.h new file mode 100644 index 0000000000000..2d8b3eacae4f6 --- /dev/null +++ b/libc/src/__support/math/expm1.h @@ -0,0 +1,518 @@ +//===-- Implementation header for expm1 -------------------------*- 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___SUPPORT_MATH_EXPM1_H +#define LLVM_LIBC_SRC___SUPPORT_MATH_EXPM1_H + +#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2. +#include "exp_constants.h" +#include "src/__support/CPP/bit.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/except_value_utils.h" +#include "src/__support/FPUtil/multiply_add.h" +#include "src/__support/FPUtil/rounding_mode.h" +#include "src/__support/FPUtil/triple_double.h" +#include "src/__support/common.h" +#include "src/__support/integer_literals.h" +#include "src/__support/macros/config.h" +#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY + +namespace LIBC_NAMESPACE_DECL { + +namespace math { + +namespace expm1_internal { + +#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0) +#define LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS +#endif + +using fputil::DoubleDouble; +using fputil::TripleDouble; +using Float128 = typename fputil::DyadicFloat<128>; + +using LIBC_NAMESPACE::operator""_u128; + +// log2(e) +static constexpr double LOG2_E = 0x1.71547652b82fep+0; + +// Error bounds: +// Errors when using double precision. +// 0x1.8p-63; +static constexpr uint64_t ERR_D = 0x3c08000000000000; +// Errors when using double-double precision. +// 0x1.0p-99 +[[maybe_unused]] static constexpr uint64_t ERR_DD = 0x39c0000000000000; + +// -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 +static constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13; +static constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47; +static constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47; +static constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79; + +using namespace common_constants_internal; + +// 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 static constexpr 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 expm1(dx) / dx ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 +// For |dx| < 2^-13 + 2^-30: +// | output - expm1(dx) | < 2^-101 +LIBC_INLINE static constexpr DoubleDouble +poly_approx_dd(const DoubleDouble &dx) { + // Taylor polynomial. + constexpr DoubleDouble COEFFS[] = { + {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 + {0x1.a01a01a01a01ap-73, 0x1.a01a01a01a01ap-13}, // 1/5040 + }; + + 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 ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 +// For |dx| < 2^-13 + 2^-30: +// | output - exp(dx) | < 2^-126. +[[maybe_unused]] LIBC_INLINE static constexpr Float128 +poly_approx_f128(const Float128 &dx) { + constexpr Float128 COEFFS_128[]{ + {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0 + {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5 + {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6 + {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24 + {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120 + {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720 + {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 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]); + return p; +} + +#ifdef DEBUGDEBUG +std::ostream &operator<<(std::ostream &OS, const Float128 &r) { + OS << (r.sign == Sign::NEG ? "-(" : "(") << r.mantissa.val[0] << " + " + << r.mantissa.val[1] << " * 2^64) * 2^" << r.exponent << "\n"; + return OS; +} + +std::ostream &operator<<(std::ostream &OS, const DoubleDouble &r) { + OS << std::hexfloat << "(" << r.hi << " + " << r.lo << ")" + << std::defaultfloat << "\n"; + return OS; +} +#endif + +// Compute exp(x) - 1 using 128-bit precision. +// TODO(lntue): investigate triple-double precision implementation for this +// step. +[[maybe_unused]] LIBC_INLINE static constexpr Float128 +expm1_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(EXP2_MID1[idx1].hi), + fputil::quick_add(Float128(EXP2_MID1[idx1].mid), + Float128(EXP2_MID1[idx1].lo))); + + Float128 exp_mid2 = + fputil::quick_add(Float128(EXP2_MID2[idx2].hi), + fputil::quick_add(Float128(EXP2_MID2[idx2].mid), + Float128(EXP2_MID2[idx2].lo))); + + Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2); + + int hi = static_cast<int>(kd) >> 12; + Float128 minus_one{Sign::NEG, -127 - hi, + 0x80000000'00000000'00000000'00000000_u128}; + + Float128 exp_mid_m1 = fputil::quick_add(exp_mid, minus_one); + + Float128 p = poly_approx_f128(dx); + + // r = exp_mid * (1 + dx * P) - 1 + // = (exp_mid - 1) + (dx * exp_mid) * P + Float128 r = + fputil::multiply_add(fputil::quick_mul(exp_mid, dx), p, exp_mid_m1); + + r.exponent += hi; + +#ifdef DEBUGDEBUG + std::cout << "=== VERY SLOW PASS ===\n" + << " kd: " << kd << "\n" + << " hi: " << hi << "\n" + << " minus_one: " << minus_one << " dx: " << dx + << "exp_mid_m1: " << exp_mid_m1 << " exp_mid: " << exp_mid + << " p: " << p << " r: " << r << std::endl; +#endif + + return r; +} + +// Compute exp(x) - 1 with double-double precision. +LIBC_INLINE static constexpr DoubleDouble +exp_double_double(double x, double kd, const DoubleDouble &exp_mid, + const DoubleDouble &hi_part) { + // 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::multiply_add(fputil::quick_mult(exp_mid, dx), p, hi_part); + +#ifdef DEBUGDEBUG + std::cout << "=== SLOW PASS ===\n" + << " dx: " << dx << " p: " << p << " r: " << r << std::endl; +#endif + + return r; +} + +// Check for exceptional cases when +// |x| <= 2^-53 or x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 +LIBC_INLINE static constexpr double set_exceptional(double x) { + using FPBits = typename fputil::FPBits<double>; + FPBits xbits(x); + + uint64_t x_u = xbits.uintval(); + uint64_t x_abs = xbits.abs().uintval(); + + // |x| <= 2^-53. + if (x_abs <= 0x3ca0'0000'0000'0000ULL) { + // expm1(x) ~ x. + + if (LIBC_UNLIKELY(x_abs <= 0x0370'0000'0000'0000ULL)) { + if (LIBC_UNLIKELY(x_abs == 0)) + return x; + // |x| <= 2^-968, need to scale up a bit before rounding, then scale it + // back down. + return 0x1.0p-200 * fputil::multiply_add(x, 0x1.0p+200, 0x1.0p-1022); + } + + // 2^-968 < |x| <= 2^-53. + return fputil::round_result_slightly_up(x); + } + + // x < log(2^-54) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan. + + // x < log(2^-54) or -inf/nan + if (x_u >= 0xc042'b708'8723'20e2ULL) { + // expm1(-Inf) = -1 + if (xbits.is_inf()) + return -1.0; + + // exp(nan) = nan + if (xbits.is_nan()) + return x; + + return fputil::round_result_slightly_up(-1.0); + } + + // 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 FPBits::max_normal().get_val(); + + fputil::set_errno_if_required(ERANGE); + fputil::raise_except_if_required(FE_OVERFLOW); + } + // x is +inf or nan + return x + FPBits::inf().get_val(); +} + +} // namespace expm1_internal + +LIBC_INLINE static constexpr double expm1(double x) { + using namespace expm1_internal; + + using FPBits = typename fputil::FPBits<double>; + + FPBits xbits(x); + + bool x_is_neg = xbits.is_neg(); + 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: log(2^-54) = -0x1.2b708872320e2p5 + // > round(log(2^-54), D, RN) = -0x1.2b708872320e2p5 + + // x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 or |x| <= 2^-53. + + if (LIBC_UNLIKELY(x_u >= 0xc042b708872320e2 || + (x_u <= 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) || + x_u <= 0x3ca0000000000000)) { + return set_exceptional(x); + } + + // Now log(2^-54) <= 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<int>(cpp::bit_cast<uint64_t>(tmp) >> 19); + double kd = static_cast<double>(k); + + uint32_t idx1 = (k >> 6) & 0x3f; + uint32_t idx2 = k & 0x3f; + int hi = k >> 12; + + DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi}; + DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi}; + + DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2); + + // -2^(-hi) + double one_scaled = + FPBits::create_value(Sign::NEG, FPBits::EXP_BIAS - hi, 0).get_val(); + + // 2^(mid1 + mid2) - 2^(-hi) + DoubleDouble hi_part = x_is_neg ? fputil::exact_add(one_scaled, exp_mid.hi) + : fputil::exact_add(exp_mid.hi, one_scaled); + + hi_part.lo += exp_mid.lo; + + // |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. + + // Finally, we have the following approximation formula: + // expm1(x) = 2^hi * 2^(mid1 + mid2) * exp(lo) - 1 + // = 2^hi * ( 2^(mid1 + mid2) * exp(lo) - 2^(-hi) ) + // ~ 2^hi * ( (exp_mid.hi - 2^-hi) + + // + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)) + + 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, hi_part.lo); + + // TODO: The following line leaks encoding abstraction. Use FPBits methods + // instead. + uint64_t err = x_is_neg ? (static_cast<uint64_t>(-hi) << 52) : 0; + + double err_d = cpp::bit_cast<double>(ERR_D + err); + + double upper = hi_part.hi + (lo + err_d); + double lower = hi_part.hi + (lo - err_d); + +#ifdef DEBUGDEBUG + std::cout << "=== FAST PASS ===\n" + << " x: " << std::hexfloat << x << std::defaultfloat << "\n" + << " k: " << k << "\n" + << " idx1: " << idx1 << "\n" + << " idx2: " << idx2 << "\n" + << " hi: " << hi << "\n" + << " dx: " << std::hexfloat << dx << std::defaultfloat << "\n" + << "exp_mid: " << exp_mid << "hi_part: " << hi_part + << " mid_lo: " << std::hexfloat << mid_lo << std::defaultfloat + << "\n" + << " p: " << std::hexfloat << p << std::defaultfloat << "\n" + << " lo: " << std::hexfloat << lo << std::defaultfloat << "\n" + << " upper: " << std::hexfloat << upper << std::defaultfloat + << "\n" + << " lower: " << std::hexfloat << lower << std::defaultfloat + << "\n" + << std::endl; +#endif + + 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<int64_t>(hi) << FPBits::FRACTION_LEN; + double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper)); + return r; + } + + // Use double-double + DoubleDouble r_dd = exp_double_double(x, kd, exp_mid, hi_part); + +#ifdef LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS + int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; + double r = + cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(r_dd.hi + r_dd.lo)); + return r; +#else + double err_dd = cpp::bit_cast<double>(ERR_DD + err); + + 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<int64_t>(hi) << FPBits::FRACTION_LEN; + double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd)); + return r; + } + + // Use 128-bit precision + Float128 r_f128 = expm1_f128(x, kd, idx1, idx2); + + return static_cast<double>(r_f128); +#endif // LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS +} + +} // namespace math + +} // namespace LIBC_NAMESPACE_DECL + +#endif // LLVM_LIBC_SRC___SUPPORT_MATH_EXPM1_H diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt index aca53da4ea9b1..d81b29396c425 100644 --- a/libc/src/math/generic/CMakeLists.txt +++ b/libc/src/math/generic/CMakeLists.txt @@ -1551,18 +1551,7 @@ add_entrypoint_object( HDRS ../expm1.h DEPENDS - libc.src.__support.CPP.bit - 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.rounding_mode - libc.src.__support.FPUtil.triple_double - libc.src.__support.integer_literals - libc.src.__support.macros.optimization - libc.src.__support.math.common_constants - libc.src.errno.errno + libc.src.__support.math.expm1 ) add_entrypoint_object( diff --git a/libc/src/math/generic/expm1.cpp b/libc/src/math/generic/expm1.cpp index a3d0c1aa5261c..c410ae0a33a2a 100644 --- a/libc/src/math/generic/expm1.cpp +++ b/libc/src/math/generic/expm1.cpp @@ -7,498 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/expm1.h" -#include "src/__support/CPP/bit.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/except_value_utils.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/FPUtil/rounding_mode.h" -#include "src/__support/FPUtil/triple_double.h" -#include "src/__support/common.h" -#include "src/__support/integer_literals.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY -#include "src/__support/math/common_constants.h" // Lookup tables EXP_M1 and EXP_M2. -#include "src/__support/math/exp_constants.h" - -#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0) -#define LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS -#endif +#include "src/__support/math/expm1.h" namespace LIBC_NAMESPACE_DECL { -using fputil::DoubleDouble; -using fputil::TripleDouble; -using Float128 = typename fputil::DyadicFloat<128>; - -using LIBC_NAMESPACE::operator""_u128; - -// log2(e) -constexpr double LOG2_E = 0x1.71547652b82fep+0; - -// Error bounds: -// Errors when using double precision. -// 0x1.8p-63; -constexpr uint64_t ERR_D = 0x3c08000000000000; -// Errors when using double-double precision. -// 0x1.0p-99 -[[maybe_unused]] constexpr uint64_t ERR_DD = 0x39c0000000000000; - -// -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; - -namespace { - -using namespace common_constants_internal; - -// 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 expm1(dx) / dx ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 -// For |dx| < 2^-13 + 2^-30: -// | output - expm1(dx) | < 2^-101 -DoubleDouble poly_approx_dd(const DoubleDouble &dx) { - // Taylor polynomial. - constexpr DoubleDouble COEFFS[] = { - {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 - {0x1.a01a01a01a01ap-73, 0x1.a01a01a01a01ap-13}, // 1/5040 - }; - - 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 ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040 -// For |dx| < 2^-13 + 2^-30: -// | output - exp(dx) | < 2^-126. -[[maybe_unused]] Float128 poly_approx_f128(const Float128 &dx) { - constexpr Float128 COEFFS_128[]{ - {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0 - {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5 - {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6 - {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24 - {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120 - {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720 - {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 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]); - return p; -} - -#ifdef DEBUGDEBUG -std::ostream &operator<<(std::ostream &OS, const Float128 &r) { - OS << (r.sign == Sign::NEG ? "-(" : "(") << r.mantissa.val[0] << " + " - << r.mantissa.val[1] << " * 2^64) * 2^" << r.exponent << "\n"; - return OS; -} - -std::ostream &operator<<(std::ostream &OS, const DoubleDouble &r) { - OS << std::hexfloat << "(" << r.hi << " + " << r.lo << ")" - << std::defaultfloat << "\n"; - return OS; -} -#endif - -// Compute exp(x) - 1 using 128-bit precision. -// TODO(lntue): investigate triple-double precision implementation for this -// step. -[[maybe_unused]] Float128 expm1_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(EXP2_MID1[idx1].hi), - fputil::quick_add(Float128(EXP2_MID1[idx1].mid), - Float128(EXP2_MID1[idx1].lo))); - - Float128 exp_mid2 = - fputil::quick_add(Float128(EXP2_MID2[idx2].hi), - fputil::quick_add(Float128(EXP2_MID2[idx2].mid), - Float128(EXP2_MID2[idx2].lo))); - - Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2); - - int hi = static_cast<int>(kd) >> 12; - Float128 minus_one{Sign::NEG, -127 - hi, - 0x80000000'00000000'00000000'00000000_u128}; - - Float128 exp_mid_m1 = fputil::quick_add(exp_mid, minus_one); - - Float128 p = poly_approx_f128(dx); - - // r = exp_mid * (1 + dx * P) - 1 - // = (exp_mid - 1) + (dx * exp_mid) * P - Float128 r = - fputil::multiply_add(fputil::quick_mul(exp_mid, dx), p, exp_mid_m1); - - r.exponent += hi; - -#ifdef DEBUGDEBUG - std::cout << "=== VERY SLOW PASS ===\n" - << " kd: " << kd << "\n" - << " hi: " << hi << "\n" - << " minus_one: " << minus_one << " dx: " << dx - << "exp_mid_m1: " << exp_mid_m1 << " exp_mid: " << exp_mid - << " p: " << p << " r: " << r << std::endl; -#endif - - return r; -} - -// Compute exp(x) - 1 with double-double precision. -DoubleDouble exp_double_double(double x, double kd, const DoubleDouble &exp_mid, - const DoubleDouble &hi_part) { - // 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::multiply_add(fputil::quick_mult(exp_mid, dx), p, hi_part); - -#ifdef DEBUGDEBUG - std::cout << "=== SLOW PASS ===\n" - << " dx: " << dx << " p: " << p << " r: " << r << std::endl; -#endif - - return r; -} - -// Check for exceptional cases when -// |x| <= 2^-53 or x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 -double set_exceptional(double x) { - using FPBits = typename fputil::FPBits<double>; - FPBits xbits(x); - - uint64_t x_u = xbits.uintval(); - uint64_t x_abs = xbits.abs().uintval(); - - // |x| <= 2^-53. - if (x_abs <= 0x3ca0'0000'0000'0000ULL) { - // expm1(x) ~ x. - - if (LIBC_UNLIKELY(x_abs <= 0x0370'0000'0000'0000ULL)) { - if (LIBC_UNLIKELY(x_abs == 0)) - return x; - // |x| <= 2^-968, need to scale up a bit before rounding, then scale it - // back down. - return 0x1.0p-200 * fputil::multiply_add(x, 0x1.0p+200, 0x1.0p-1022); - } - - // 2^-968 < |x| <= 2^-53. - return fputil::round_result_slightly_up(x); - } - - // x < log(2^-54) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan. - - // x < log(2^-54) or -inf/nan - if (x_u >= 0xc042'b708'8723'20e2ULL) { - // expm1(-Inf) = -1 - if (xbits.is_inf()) - return -1.0; - - // exp(nan) = nan - if (xbits.is_nan()) - return x; - - return fputil::round_result_slightly_up(-1.0); - } - - // 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 FPBits::max_normal().get_val(); - - fputil::set_errno_if_required(ERANGE); - fputil::raise_except_if_required(FE_OVERFLOW); - } - // x is +inf or nan - return x + FPBits::inf().get_val(); -} - -} // namespace - -LLVM_LIBC_FUNCTION(double, expm1, (double x)) { - using FPBits = typename fputil::FPBits<double>; - - FPBits xbits(x); - - bool x_is_neg = xbits.is_neg(); - 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: log(2^-54) = -0x1.2b708872320e2p5 - // > round(log(2^-54), D, RN) = -0x1.2b708872320e2p5 - - // x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 or |x| <= 2^-53. - - if (LIBC_UNLIKELY(x_u >= 0xc042b708872320e2 || - (x_u <= 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) || - x_u <= 0x3ca0000000000000)) { - return set_exceptional(x); - } - - // Now log(2^-54) <= 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<int>(cpp::bit_cast<uint64_t>(tmp) >> 19); - double kd = static_cast<double>(k); - - uint32_t idx1 = (k >> 6) & 0x3f; - uint32_t idx2 = k & 0x3f; - int hi = k >> 12; - - DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi}; - DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi}; - - DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2); - - // -2^(-hi) - double one_scaled = - FPBits::create_value(Sign::NEG, FPBits::EXP_BIAS - hi, 0).get_val(); - - // 2^(mid1 + mid2) - 2^(-hi) - DoubleDouble hi_part = x_is_neg ? fputil::exact_add(one_scaled, exp_mid.hi) - : fputil::exact_add(exp_mid.hi, one_scaled); - - hi_part.lo += exp_mid.lo; - - // |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. - - // Finally, we have the following approximation formula: - // expm1(x) = 2^hi * 2^(mid1 + mid2) * exp(lo) - 1 - // = 2^hi * ( 2^(mid1 + mid2) * exp(lo) - 2^(-hi) ) - // ~ 2^hi * ( (exp_mid.hi - 2^-hi) + - // + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)) - - 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, hi_part.lo); - - // TODO: The following line leaks encoding abstraction. Use FPBits methods - // instead. - uint64_t err = x_is_neg ? (static_cast<uint64_t>(-hi) << 52) : 0; - - double err_d = cpp::bit_cast<double>(ERR_D + err); - - double upper = hi_part.hi + (lo + err_d); - double lower = hi_part.hi + (lo - err_d); - -#ifdef DEBUGDEBUG - std::cout << "=== FAST PASS ===\n" - << " x: " << std::hexfloat << x << std::defaultfloat << "\n" - << " k: " << k << "\n" - << " idx1: " << idx1 << "\n" - << " idx2: " << idx2 << "\n" - << " hi: " << hi << "\n" - << " dx: " << std::hexfloat << dx << std::defaultfloat << "\n" - << "exp_mid: " << exp_mid << "hi_part: " << hi_part - << " mid_lo: " << std::hexfloat << mid_lo << std::defaultfloat - << "\n" - << " p: " << std::hexfloat << p << std::defaultfloat << "\n" - << " lo: " << std::hexfloat << lo << std::defaultfloat << "\n" - << " upper: " << std::hexfloat << upper << std::defaultfloat - << "\n" - << " lower: " << std::hexfloat << lower << std::defaultfloat - << "\n" - << std::endl; -#endif - - 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<int64_t>(hi) << FPBits::FRACTION_LEN; - double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper)); - return r; - } - - // Use double-double - DoubleDouble r_dd = exp_double_double(x, kd, exp_mid, hi_part); - -#ifdef LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS - int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN; - double r = - cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(r_dd.hi + r_dd.lo)); - return r; -#else - double err_dd = cpp::bit_cast<double>(ERR_DD + err); - - 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<int64_t>(hi) << FPBits::FRACTION_LEN; - double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd)); - return r; - } - - // Use 128-bit precision - Float128 r_f128 = expm1_f128(x, kd, idx1, idx2); - - return static_cast<double>(r_f128); -#endif // LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS -} +LLVM_LIBC_FUNCTION(double, expm1, (double x)) { return math::expm1(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/test/shared/CMakeLists.txt b/libc/test/shared/CMakeLists.txt index 783ace1ea8351..7d438223db027 100644 --- a/libc/test/shared/CMakeLists.txt +++ b/libc/test/shared/CMakeLists.txt @@ -45,6 +45,7 @@ add_fp_unittest( libc.src.__support.math.exp2f16 libc.src.__support.math.exp2m1f libc.src.__support.math.exp2m1f16 + libc.src.__support.math.expm1 libc.src.__support.math.exp10 libc.src.__support.math.exp10f libc.src.__support.math.exp10f16 diff --git a/libc/test/shared/shared_math_test.cpp b/libc/test/shared/shared_math_test.cpp index 8ecf1fcb20807..40a1c0c7632c6 100644 --- a/libc/test/shared/shared_math_test.cpp +++ b/libc/test/shared/shared_math_test.cpp @@ -85,6 +85,7 @@ TEST(LlvmLibcSharedMathTest, AllDouble) { EXPECT_FP_EQ(0x1p+0, LIBC_NAMESPACE::shared::exp(0.0)); EXPECT_FP_EQ(0x1p+0, LIBC_NAMESPACE::shared::exp2(0.0)); EXPECT_FP_EQ(0x1p+0, LIBC_NAMESPACE::shared::exp10(0.0)); + EXPECT_FP_EQ(0x0p+0, LIBC_NAMESPACE::shared::expm1(0.0)); } #ifdef LIBC_TYPES_HAS_FLOAT128 diff --git a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel index 7893077dfae51..18f8e966cbfad 100644 --- a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel +++ b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel @@ -3020,6 +3020,24 @@ libc_support_library( ], ) +libc_support_library( + name = "__support_math_expm1", + hdrs = ["src/__support/math/expm1.h"], + deps = [ + ":__support_fputil_double_double", + ":__support_fputil_dyadic_float", + ":__support_fputil_except_value_utils", + ":__support_fputil_multiply_add", + ":__support_fputil_polyeval", + ":__support_fputil_rounding_mode", + ":__support_fputil_triple_double", + ":__support_integer_literals", + ":__support_macros_optimization", + ":__support_math_common_constants", + ":__support_math_exp_constants", + ], +) + libc_support_library( name = "__support_range_reduction_double", hdrs = [ @@ -3752,15 +3770,7 @@ libc_math_function( libc_math_function( name = "expm1", additional_deps = [ - ":__support_fputil_double_double", - ":__support_fputil_dyadic_float", - ":__support_fputil_multiply_add", - ":__support_fputil_polyeval", - ":__support_fputil_rounding_mode", - ":__support_fputil_triple_double", - ":__support_integer_literals", - ":__support_macros_optimization", - ":__support_math_common_constants", + ":__support_math_expm1", ], ) _______________________________________________ llvm-branch-commits mailing list [email protected] https://lists.llvm.org/cgi-bin/mailman/listinfo/llvm-branch-commits
