diff --git a/libc/shared/math.h b/libc/shared/math.h index b2f1a03e0940d..3012cbb938816 100644 --- a/libc/shared/math.h +++ b/libc/shared/math.h @@ -11,6 +11,7 @@ #include "libc_common.h" +#include "math/exp.h" #include "math/expf.h" #include "math/expf16.h" #include "math/frexpf.h" diff --git a/libc/shared/math/exp.h b/libc/shared/math/exp.h new file mode 100644 index 0000000000000..7cdd6331e613a --- /dev/null +++ b/libc/shared/math/exp.h @@ -0,0 +1,23 @@ +//===-- Shared exp 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_EXP_H +#define LLVM_LIBC_SHARED_MATH_EXP_H + +#include "shared/libc_common.h" +#include "src/__support/math/exp.h" + +namespace LIBC_NAMESPACE_DECL { +namespace shared { + +using math::exp; + +} // namespace shared +} // namespace LIBC_NAMESPACE_DECL + +#endif // LLVM_LIBC_SHARED_MATH_EXP_H diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt index 900c0ab04d3a3..f7ef9e7694fe6 100644 --- a/libc/src/__support/math/CMakeLists.txt +++ b/libc/src/__support/math/CMakeLists.txt @@ -110,3 +110,42 @@ add_header_library( DEPENDS libc.src.__support.FPUtil.manipulation_functions ) + +add_header_library( + exp_constants + HDRS + exp_constants.h + DEPENDS + libc.src.__support.FPUtil.triple_double +) + +add_header_library( + exp_utils + HDRS + exp_utils.h + DEPENDS + libc.src.__support.CPP.optional + libc.src.__support.CPP.bit + libc.src.__support.FPUtil.fp_bits +) + +add_header_library( + exp + HDRS + exp.h + DEPENDS + .exp_constants + .exp_utils + libc.src.__support.CPP.bit + libc.src.__support.CPP.optional + 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.nearest_integer + 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 +) diff --git a/libc/src/__support/math/exp.h b/libc/src/__support/math/exp.h new file mode 100644 index 0000000000000..5c43e753ea687 --- /dev/null +++ b/libc/src/__support/math/exp.h @@ -0,0 +1,448 @@ +//===-- 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___SUPPORT_MATH_EXP_H +#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP_H + +#include "exp_constants.h" +#include "exp_utils.h" +#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/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 { + +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. +static constexpr double ERR_D = 0x1.8p-63; + +#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS +// Errors when using double-double precision. +static constexpr double ERR_DD = 0x1.0p-99; +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS + +// -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; + +#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS +static constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47; +static constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79; +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS + +namespace { + +// 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. +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; +} + +#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS +// 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 +static constexpr 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. +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, -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], COEFFS_128[7]); + return p; +} + +// Compute exp(x) using 128-bit precision. +// TODO(lntue): investigate triple-double precision implementation for this +// step. +static constexpr 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(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); + + 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. +static constexpr 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; +} +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS + +// Check for exceptional cases when +// |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 +static constexpr double set_exceptional(double x) { + using FPBits = typename fputil::FPBits; + 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) { + // 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 FPBits::min_subnormal().get_val(); + 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 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 + +namespace math { + +static constexpr double exp(double x) { + using FPBits = typename fputil::FPBits; + 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^-1075) <= 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{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); + + // |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); + +#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS + if (LIBC_UNLIKELY(denorm)) { + return ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D) + .value(); + } else { + // to multiply by 2^hi, a fast way is to simply add hi to the exponent + // field. + int64_t exp_hi = static_cast(hi) << FPBits::FRACTION_LEN; + double r = + cpp::bit_cast(exp_hi + cpp::bit_cast(exp_mid.hi + lo)); + return r; + } +#else + 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) << FPBits::FRACTION_LEN; + 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) << FPBits::FRACTION_LEN; + 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); +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS +} + +} // namespace math + +} // namespace LIBC_NAMESPACE_DECL + +#endif // LLVM_LIBC_SRC___SUPPORT_MATH_EXP_H diff --git a/libc/src/__support/math/exp_constants.h b/libc/src/__support/math/exp_constants.h new file mode 100644 index 0000000000000..32976a86a01ad --- /dev/null +++ b/libc/src/__support/math/exp_constants.h @@ -0,0 +1,174 @@ +//===-- Constants for exp 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_SRC___SUPPORT_MATH_EXP_CONSTANTS_H +#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP_CONSTANTS_H + +#include "src/__support/FPUtil/triple_double.h" + +namespace LIBC_NAMESPACE_DECL { + +// Lookup table for 2^(k * 2^-6) with k = 0..63. +// Generated by Sollya with: +// > display=hexadecimal; +// > prec = 500; +// > for i from 0 to 63 do { +// a = 2^(i * 2^-6); +// b = round(a, D, RN); +// c = round(a - b, D, RN); +// d = round(a - b - c, D, RN); +// print("{", d, ",", c, ",", b, "},"); +// }; +alignas(16) static constexpr fputil::TripleDouble EXP2_MID1[64] = { + {0, 0, 0x1p0}, + {-0x1.9085b0a3d74d5p-110, -0x1.19083535b085dp-56, 0x1.02c9a3e778061p0}, + {0x1.05ff94f8d257ep-110, 0x1.d73e2a475b465p-55, 0x1.059b0d3158574p0}, + {0x1.15820d96b414fp-111, 0x1.186be4bb284ffp-57, 0x1.0874518759bc8p0}, + {-0x1.67c9bd6ebf74cp-108, 0x1.8a62e4adc610bp-54, 0x1.0b5586cf9890fp0}, + {-0x1.5aa76994e9ddbp-113, 0x1.03a1727c57b53p-59, 0x1.0e3ec32d3d1a2p0}, + {0x1.9d58b988f562dp-109, -0x1.6c51039449b3ap-54, 0x1.11301d0125b51p0}, + {-0x1.2fe7bb4c76416p-108, -0x1.32fbf9af1369ep-54, 0x1.1429aaea92dep0}, + {0x1.4f2406aa13ffp-109, -0x1.19041b9d78a76p-55, 0x1.172b83c7d517bp0}, + {0x1.ad36183926ae8p-111, 0x1.e5b4c7b4968e4p-55, 0x1.1a35beb6fcb75p0}, + {0x1.ea62d0881b918p-110, 0x1.e016e00a2643cp-54, 0x1.1d4873168b9aap0}, + {-0x1.781dbc16f1ea4p-111, 0x1.dc775814a8495p-55, 0x1.2063b88628cd6p0}, + {-0x1.4d89f9af532ep-109, 0x1.9b07eb6c70573p-54, 0x1.2387a6e756238p0}, + {0x1.277393a461b77p-110, 0x1.2bd339940e9d9p-55, 0x1.26b4565e27cddp0}, + {0x1.de5448560469p-111, 0x1.612e8afad1255p-55, 0x1.29e9df51fdee1p0}, + {-0x1.ee9d8f8cb9307p-110, 0x1.0024754db41d5p-54, 0x1.2d285a6e4030bp0}, + {0x1.7b7b2f09cd0d9p-110, 0x1.6f46ad23182e4p-55, 0x1.306fe0a31b715p0}, + {-0x1.406a2ea6cfc6bp-108, 0x1.32721843659a6p-54, 0x1.33c08b26416ffp0}, + {0x1.87e3e12516bfap-108, -0x1.63aeabf42eae2p-54, 0x1.371a7373aa9cbp0}, + {0x1.9b0b1ff17c296p-111, -0x1.5e436d661f5e3p-56, 0x1.3a7db34e59ff7p0}, + {-0x1.808ba68fa8fb7p-109, 0x1.ada0911f09ebcp-55, 0x1.3dea64c123422p0}, + {-0x1.32b43eafc6518p-114, -0x1.ef3691c309278p-58, 0x1.4160a21f72e2ap0}, + {-0x1.0ac312de3d922p-114, 0x1.89b7a04ef80dp-59, 0x1.44e086061892dp0}, + {0x1.e1eebae743acp-111, 0x1.3c1a3b69062fp-56, 0x1.486a2b5c13cdp0}, + {0x1.c06c7745c2b39p-113, 0x1.d4397afec42e2p-56, 0x1.4bfdad5362a27p0}, + {-0x1.1aa1fd7b685cdp-112, -0x1.4b309d25957e3p-54, 0x1.4f9b2769d2ca7p0}, + {0x1.fa733951f214cp-111, -0x1.07abe1db13cadp-55, 0x1.5342b569d4f82p0}, + {-0x1.ff86852a613ffp-111, 0x1.9bb2c011d93adp-54, 0x1.56f4736b527dap0}, + {-0x1.744ee506fdafep-109, 0x1.6324c054647adp-54, 0x1.5ab07dd485429p0}, + {-0x1.95f9ab75fa7d6p-108, 0x1.ba6f93080e65ep-54, 0x1.5e76f15ad2148p0}, + {0x1.5d8e757cfb991p-111, -0x1.383c17e40b497p-54, 0x1.6247eb03a5585p0}, + {0x1.4a337f4dc0a3bp-108, -0x1.bb60987591c34p-54, 0x1.6623882552225p0}, + {0x1.57d3e3adec175p-108, -0x1.bdd3413b26456p-54, 0x1.6a09e667f3bcdp0}, + {0x1.a59f88abbe778p-115, -0x1.bbe3a683c88abp-57, 0x1.6dfb23c651a2fp0}, + {-0x1.269796953a4c3p-109, -0x1.16e4786887a99p-55, 0x1.71f75e8ec5f74p0}, + {-0x1.8f8e7fa19e5e8p-108, -0x1.0245957316dd3p-54, 0x1.75feb564267c9p0}, + {-0x1.4217a932d10d4p-113, -0x1.41577ee04992fp-55, 0x1.7a11473eb0187p0}, + {0x1.70a1427f8fcdfp-112, 0x1.05d02ba15797ep-56, 0x1.7e2f336cf4e62p0}, + {0x1.0f6ad65cbbac1p-112, -0x1.d4c1dd41532d8p-54, 0x1.82589994cce13p0}, + {-0x1.f16f65181d921p-109, -0x1.fc6f89bd4f6bap-54, 0x1.868d99b4492edp0}, + {-0x1.30644a7836333p-110, 0x1.6e9f156864b27p-54, 0x1.8ace5422aa0dbp0}, + {0x1.3bf26d2b85163p-114, 0x1.5cc13a2e3976cp-55, 0x1.8f1ae99157736p0}, + {0x1.697e257ac0db2p-111, -0x1.75fc781b57ebcp-57, 0x1.93737b0cdc5e5p0}, + {0x1.7edb9d7144b6fp-108, -0x1.d185b7c1b85d1p-54, 0x1.97d829fde4e5p0}, + {0x1.6376b7943085cp-110, 0x1.c7c46b071f2bep-56, 0x1.9c49182a3f09p0}, + {0x1.354084551b4fbp-109, -0x1.359495d1cd533p-54, 0x1.a0c667b5de565p0}, + {-0x1.bfd7adfd63f48p-111, -0x1.d2f6edb8d41e1p-54, 0x1.a5503b23e255dp0}, + {0x1.8b16ae39e8cb9p-109, 0x1.0fac90ef7fd31p-54, 0x1.a9e6b5579fdbfp0}, + {0x1.a7fbc3ae675eap-108, 0x1.7a1cd345dcc81p-54, 0x1.ae89f995ad3adp0}, + {0x1.2babc0edda4d9p-111, -0x1.2805e3084d708p-57, 0x1.b33a2b84f15fbp0}, + {0x1.aa64481e1ab72p-111, -0x1.5584f7e54ac3bp-56, 0x1.b7f76f2fb5e47p0}, + {0x1.9a164050e1258p-109, 0x1.23dd07a2d9e84p-55, 0x1.bcc1e904bc1d2p0}, + {0x1.99e51125928dap-110, 0x1.11065895048ddp-55, 0x1.c199bdd85529cp0}, + {-0x1.fc44c329d5cb2p-109, 0x1.2884dff483cadp-54, 0x1.c67f12e57d14bp0}, + {0x1.d8765566b032ep-110, 0x1.503cbd1e949dbp-56, 0x1.cb720dcef9069p0}, + {-0x1.e7044039da0f6p-108, -0x1.cbc3743797a9cp-54, 0x1.d072d4a07897cp0}, + {-0x1.ab053b05531fcp-111, 0x1.2ed02d75b3707p-55, 0x1.d5818dcfba487p0}, + {0x1.7f6246f0ec615p-108, 0x1.c2300696db532p-54, 0x1.da9e603db3285p0}, + {0x1.b7225a944efd6p-108, -0x1.1a5cd4f184b5cp-54, 0x1.dfc97337b9b5fp0}, + {0x1.1e92cb3c2d278p-109, 0x1.39e8980a9cc8fp-55, 0x1.e502ee78b3ff6p0}, + {-0x1.fc0f242bbf3dep-109, -0x1.e9c23179c2893p-54, 0x1.ea4afa2a490dap0}, + {0x1.f6dd5d229ff69p-108, 0x1.dc7f486a4b6bp-54, 0x1.efa1bee615a27p0}, + {-0x1.4019bffc80ef3p-110, 0x1.9d3e12dd8a18bp-54, 0x1.f50765b6e454p0}, + {0x1.dc060c36f7651p-112, 0x1.74853f3a5931ep-55, 0x1.fa7c1819e90d8p0}, +}; + +// Lookup table for 2^(k * 2^-12) with k = 0..63. +// Generated by Sollya with: +// > display=hexadecimal; +// > prec = 500; +// > for i from 0 to 63 do { +// a = 2^(i * 2^-12); +// b = round(a, D, RN); +// c = round(a - b, D, RN); +// d = round(a - b - c, D, RN); +// print("{", d, ",", c, ",", b, "},"); +// }; +alignas(16) static constexpr fputil::TripleDouble EXP2_MID2[64] = { + {0, 0, 0x1p0}, + {0x1.39726694630e3p-108, 0x1.ae8e38c59c72ap-54, 0x1.000b175effdc7p0}, + {0x1.e5e06ddd31156p-112, -0x1.7b5d0d58ea8f4p-58, 0x1.00162f3904052p0}, + {0x1.5a0768b51f609p-111, 0x1.4115cb6b16a8ep-54, 0x1.0021478e11ce6p0}, + {0x1.d008403605217p-111, -0x1.d7c96f201bb2fp-55, 0x1.002c605e2e8cfp0}, + {0x1.89bc16f765708p-109, 0x1.84711d4c35e9fp-54, 0x1.003779a95f959p0}, + {-0x1.4535b7f8c1e2dp-109, -0x1.0484245243777p-55, 0x1.0042936faa3d8p0}, + {-0x1.8ba92f6b25456p-108, -0x1.4b237da2025f9p-54, 0x1.004dadb113dap0}, + {-0x1.30c72e81f4294p-113, -0x1.5e00e62d6b30dp-56, 0x1.0058c86da1c0ap0}, + {-0x1.34a5384e6f0b9p-110, 0x1.a1d6cedbb9481p-54, 0x1.0063e3a559473p0}, + {0x1.f8d0580865d2ep-108, -0x1.4acf197a00142p-54, 0x1.006eff583fc3dp0}, + {-0x1.002bcb3ae9a99p-111, -0x1.eaf2ea42391a5p-57, 0x1.007a1b865a8cap0}, + {0x1.c3c5aedee9851p-111, 0x1.da93f90835f75p-56, 0x1.0085382faef83p0}, + {0x1.7217851d1ec6ep-109, -0x1.6a79084ab093cp-55, 0x1.00905554425d4p0}, + {-0x1.80cbca335a7c3p-110, 0x1.86364f8fbe8f8p-54, 0x1.009b72f41a12bp0}, + {-0x1.706bd4eb22595p-110, -0x1.82e8e14e3110ep-55, 0x1.00a6910f3b6fdp0}, + {-0x1.b55dd523f3c08p-111, -0x1.4f6b2a7609f71p-55, 0x1.00b1afa5abcbfp0}, + {0x1.90a1e207cced1p-110, -0x1.e1a258ea8f71bp-56, 0x1.00bcceb7707ecp0}, + {0x1.78d0472db37c5p-110, 0x1.4362ca5bc26f1p-56, 0x1.00c7ee448ee02p0}, + {-0x1.bcd4db3cb52fep-109, 0x1.095a56c919d02p-54, 0x1.00d30e4d0c483p0}, + {-0x1.cf1b131575ec2p-112, -0x1.406ac4e81a645p-57, 0x1.00de2ed0ee0f5p0}, + {-0x1.6aaa1fa7ff913p-112, 0x1.b5a6902767e09p-54, 0x1.00e94fd0398ep0}, + {0x1.68f236dff3218p-110, -0x1.91b2060859321p-54, 0x1.00f4714af41d3p0}, + {-0x1.e8bb58067e60ap-109, 0x1.427068ab22306p-55, 0x1.00ff93412315cp0}, + {0x1.d4cd5e1d71fdfp-108, 0x1.c1d0660524e08p-54, 0x1.010ab5b2cbd11p0}, + {0x1.e4ecf350ebe88p-108, -0x1.e7bdfb3204be8p-54, 0x1.0115d89ff3a8bp0}, + {0x1.6a2aa2c89c4f8p-109, 0x1.843aa8b9cbbc6p-55, 0x1.0120fc089ff63p0}, + {0x1.1ca368a20ed05p-110, -0x1.34104ee7edae9p-56, 0x1.012c1fecd613bp0}, + {0x1.edb1095d925cfp-114, -0x1.2b6aeb6176892p-56, 0x1.0137444c9b5b5p0}, + {-0x1.488c78eded75fp-111, 0x1.a8cd33b8a1bb3p-56, 0x1.01426927f5278p0}, + {-0x1.7480f5ea1b3c9p-113, 0x1.2edc08e5da99ap-56, 0x1.014d8e7ee8d2fp0}, + {-0x1.ae45989a04dd5p-111, 0x1.57ba2dc7e0c73p-55, 0x1.0158b4517bb88p0}, + {0x1.bf48007d80987p-109, 0x1.b61299ab8cdb7p-54, 0x1.0163da9fb3335p0}, + {0x1.1aa91a059292cp-109, -0x1.90565902c5f44p-54, 0x1.016f0169949edp0}, + {0x1.b6663292855f5p-110, 0x1.70fc41c5c2d53p-55, 0x1.017a28af25567p0}, + {0x1.e7fbca6793d94p-108, 0x1.4b9a6e145d76cp-54, 0x1.018550706ab62p0}, + {-0x1.5b9f5c7de3b93p-110, -0x1.008eff5142bf9p-56, 0x1.019078ad6a19fp0}, + {0x1.4638bf2f6acabp-110, -0x1.77669f033c7dep-54, 0x1.019ba16628de2p0}, + {-0x1.ab237b9a069c5p-109, -0x1.09bb78eeead0ap-54, 0x1.01a6ca9aac5f3p0}, + {0x1.3ab358be97cefp-108, 0x1.371231477ece5p-54, 0x1.01b1f44af9f9ep0}, + {-0x1.4027b2294bb64p-110, 0x1.5e7626621eb5bp-56, 0x1.01bd1e77170b4p0}, + {0x1.656394426c99p-111, -0x1.bc72b100828a5p-54, 0x1.01c8491f08f08p0}, + {0x1.bf9785189bdd8p-111, -0x1.ce39cbbab8bbep-57, 0x1.01d37442d507p0}, + {0x1.7c12f86114fe3p-109, 0x1.16996709da2e2p-55, 0x1.01de9fe280ac8p0}, + {-0x1.653d5d24b5d28p-109, -0x1.c11f5239bf535p-55, 0x1.01e9cbfe113efp0}, + {0x1.04a0cdc1d86d7p-109, 0x1.e1d4eb5edc6b3p-55, 0x1.01f4f8958c1c6p0}, + {0x1.c678c46149782p-109, -0x1.afb99946ee3fp-54, 0x1.020025a8f6a35p0}, + {0x1.48524e1e9df7p-108, -0x1.8f06d8a148a32p-54, 0x1.020b533856324p0}, + {0x1.9953ea727ff0bp-109, -0x1.2bf310fc54eb6p-55, 0x1.02168143b0281p0}, + {-0x1.ccfbbec22d28ep-108, -0x1.c95a035eb4175p-54, 0x1.0221afcb09e3ep0}, + {0x1.9e2bb6e181de1p-108, -0x1.491793e46834dp-54, 0x1.022cdece68c4fp0}, + {0x1.f17609ae29308p-110, -0x1.3e8d0d9c49091p-56, 0x1.02380e4dd22adp0}, + {-0x1.c7dc2c476bfb8p-110, -0x1.314aa16278aa3p-54, 0x1.02433e494b755p0}, + {-0x1.fab994971d4a3p-109, 0x1.48daf888e9651p-55, 0x1.024e6ec0da046p0}, + {0x1.848b62cbdd0afp-109, 0x1.56dc8046821f4p-55, 0x1.02599fb483385p0}, + {-0x1.bf603ba715d0cp-109, 0x1.45b42356b9d47p-54, 0x1.0264d1244c719p0}, + {0x1.89434e751e1aap-110, -0x1.082ef51b61d7ep-56, 0x1.027003103b10ep0}, + {-0x1.03b54fd64e8acp-110, 0x1.2106ed0920a34p-56, 0x1.027b357854772p0}, + {0x1.7785ea0acc486p-109, -0x1.fd4cf26ea5d0fp-54, 0x1.0286685c9e059p0}, + {-0x1.ce447fdb35ff9p-109, -0x1.09f8775e78084p-54, 0x1.02919bbd1d1d8p0}, + {0x1.5b884aab5642ap-112, 0x1.64cbba902ca27p-58, 0x1.029ccf99d720ap0}, + {-0x1.cfb3e46d7c1cp-108, 0x1.4383ef231d207p-54, 0x1.02a803f2d170dp0}, + {-0x1.0d40cee4b81afp-112, 0x1.4a47a505b3a47p-54, 0x1.02b338c811703p0}, + {0x1.6ae7d36d7c1f7p-109, 0x1.e47120223467fp-54, 0x1.02be6e199c811p0}, +}; + +} // namespace LIBC_NAMESPACE_DECL + +#endif // LLVM_LIBC_SRC___SUPPORT_MATH_EXP_CONSTANTS_H diff --git a/libc/src/__support/math/exp_utils.h b/libc/src/__support/math/exp_utils.h new file mode 100644 index 0000000000000..fc9ab10d76cc4 --- /dev/null +++ b/libc/src/__support/math/exp_utils.h @@ -0,0 +1,72 @@ +//===-- Common utils for exp 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_SRC___SUPPORT_MATH_EXP_UTILS_H +#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP_UTILS_H + +#include "src/__support/CPP/bit.h" +#include "src/__support/CPP/optional.h" +#include "src/__support/FPUtil/FPBits.h" + +namespace LIBC_NAMESPACE_DECL { + +// Rounding tests for 2^hi * (mid + lo) when the output might be denormal. We +// assume further that 1 <= mid < 2, mid + lo < 2, and |lo| << mid. +// Notice that, if 0 < x < 2^-1022, +// double(2^-1022 + x) - 2^-1022 = double(x). +// So if we scale x up by 2^1022, we can use +// double(1.0 + 2^1022 * x) - 1.0 to test how x is rounded in denormal range. +template +static constexpr cpp::optional ziv_test_denorm(int hi, double mid, + double lo, double err) { + using FPBits = typename fputil::FPBits; + + // Scaling factor = 1/(min normal number) = 2^1022 + int64_t exp_hi = static_cast(hi + 1022) << FPBits::FRACTION_LEN; + double mid_hi = cpp::bit_cast(exp_hi + cpp::bit_cast(mid)); + double lo_scaled = + (lo != 0.0) ? cpp::bit_cast(exp_hi + cpp::bit_cast(lo)) + : 0.0; + + double extra_factor = 0.0; + uint64_t scale_down = 0x3FE0'0000'0000'0000; // 1022 in the exponent field. + + // Result is denormal if (mid_hi + lo_scale < 1.0). + if ((1.0 - mid_hi) > lo_scaled) { + // Extra rounding step is needed, which adds more rounding errors. + err += 0x1.0p-52; + extra_factor = 1.0; + scale_down = 0x3FF0'0000'0000'0000; // 1023 in the exponent field. + } + + // By adding 1.0, the results will have similar rounding points as denormal + // outputs. + if constexpr (SKIP_ZIV_TEST) { + double r = extra_factor + (mid_hi + lo_scaled); + return cpp::bit_cast(cpp::bit_cast(r) - scale_down); + } else { + double err_scaled = + cpp::bit_cast(exp_hi + cpp::bit_cast(err)); + + double lo_u = lo_scaled + err_scaled; + double lo_l = lo_scaled - err_scaled; + + double upper = extra_factor + (mid_hi + lo_u); + double lower = extra_factor + (mid_hi + lo_l); + + if (LIBC_LIKELY(upper == lower)) { + return cpp::bit_cast(cpp::bit_cast(upper) - scale_down); + } + + return cpp::nullopt; + } +} + +} // namespace LIBC_NAMESPACE_DECL + +#endif // LLVM_LIBC_SRC___SUPPORT_MATH_EXP_UTILS_H diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt index a9237741788ae..b59beacd94143 100644 --- a/libc/src/math/generic/CMakeLists.txt +++ b/libc/src/math/generic/CMakeLists.txt @@ -1312,20 +1312,7 @@ add_entrypoint_object( HDRS ../exp.h DEPENDS - .common_constants - .explogxf - libc.src.__support.CPP.bit - libc.src.__support.CPP.optional - 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.nearest_integer - 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.exp libc.src.errno.errno ) @@ -1953,8 +1940,8 @@ add_object_library( SRCS common_constants.cpp DEPENDS + libc.src.__support.math.exp_constants libc.src.__support.number_pair - libc.src.__support.FPUtil.triple_double ) add_header_library( @@ -3818,16 +3805,13 @@ add_object_library( explogxf.cpp DEPENDS .common_constants - libc.src.__support.CPP.bit - libc.src.__support.CPP.optional - libc.src.__support.FPUtil.basic_operations 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.common + libc.src.__support.math.exp_utils libc.src.errno.errno ) diff --git a/libc/src/math/generic/common_constants.cpp b/libc/src/math/generic/common_constants.cpp index b2c1293c6326d..4dcf84d00ad50 100644 --- a/libc/src/math/generic/common_constants.cpp +++ b/libc/src/math/generic/common_constants.cpp @@ -7,7 +7,6 @@ //===----------------------------------------------------------------------===// #include "common_constants.h" -#include "src/__support/FPUtil/triple_double.h" #include "src/__support/macros/config.h" #include "src/__support/number_pair.h" @@ -728,160 +727,4 @@ const double EXP_M2[128] = { 0x1.568bb722dd593p1, 0x1.593b7d72305bbp1, }; -// Lookup table for 2^(k * 2^-6) with k = 0..63. -// Generated by Sollya with: -// > display=hexadecimal; -// > prec = 500; -// > for i from 0 to 63 do { -// a = 2^(i * 2^-6); -// b = round(a, D, RN); -// c = round(a - b, D, RN); -// d = round(a - b - c, D, RN); -// print("{", d, ",", c, ",", b, "},"); -// }; -alignas(16) const fputil::TripleDouble EXP2_MID1[64] = { - {0, 0, 0x1p0}, - {-0x1.9085b0a3d74d5p-110, -0x1.19083535b085dp-56, 0x1.02c9a3e778061p0}, - {0x1.05ff94f8d257ep-110, 0x1.d73e2a475b465p-55, 0x1.059b0d3158574p0}, - {0x1.15820d96b414fp-111, 0x1.186be4bb284ffp-57, 0x1.0874518759bc8p0}, - {-0x1.67c9bd6ebf74cp-108, 0x1.8a62e4adc610bp-54, 0x1.0b5586cf9890fp0}, - {-0x1.5aa76994e9ddbp-113, 0x1.03a1727c57b53p-59, 0x1.0e3ec32d3d1a2p0}, - {0x1.9d58b988f562dp-109, -0x1.6c51039449b3ap-54, 0x1.11301d0125b51p0}, - {-0x1.2fe7bb4c76416p-108, -0x1.32fbf9af1369ep-54, 0x1.1429aaea92dep0}, - {0x1.4f2406aa13ffp-109, -0x1.19041b9d78a76p-55, 0x1.172b83c7d517bp0}, - {0x1.ad36183926ae8p-111, 0x1.e5b4c7b4968e4p-55, 0x1.1a35beb6fcb75p0}, - {0x1.ea62d0881b918p-110, 0x1.e016e00a2643cp-54, 0x1.1d4873168b9aap0}, - {-0x1.781dbc16f1ea4p-111, 0x1.dc775814a8495p-55, 0x1.2063b88628cd6p0}, - {-0x1.4d89f9af532ep-109, 0x1.9b07eb6c70573p-54, 0x1.2387a6e756238p0}, - {0x1.277393a461b77p-110, 0x1.2bd339940e9d9p-55, 0x1.26b4565e27cddp0}, - {0x1.de5448560469p-111, 0x1.612e8afad1255p-55, 0x1.29e9df51fdee1p0}, - {-0x1.ee9d8f8cb9307p-110, 0x1.0024754db41d5p-54, 0x1.2d285a6e4030bp0}, - {0x1.7b7b2f09cd0d9p-110, 0x1.6f46ad23182e4p-55, 0x1.306fe0a31b715p0}, - {-0x1.406a2ea6cfc6bp-108, 0x1.32721843659a6p-54, 0x1.33c08b26416ffp0}, - {0x1.87e3e12516bfap-108, -0x1.63aeabf42eae2p-54, 0x1.371a7373aa9cbp0}, - {0x1.9b0b1ff17c296p-111, -0x1.5e436d661f5e3p-56, 0x1.3a7db34e59ff7p0}, - {-0x1.808ba68fa8fb7p-109, 0x1.ada0911f09ebcp-55, 0x1.3dea64c123422p0}, - {-0x1.32b43eafc6518p-114, -0x1.ef3691c309278p-58, 0x1.4160a21f72e2ap0}, - {-0x1.0ac312de3d922p-114, 0x1.89b7a04ef80dp-59, 0x1.44e086061892dp0}, - {0x1.e1eebae743acp-111, 0x1.3c1a3b69062fp-56, 0x1.486a2b5c13cdp0}, - {0x1.c06c7745c2b39p-113, 0x1.d4397afec42e2p-56, 0x1.4bfdad5362a27p0}, - {-0x1.1aa1fd7b685cdp-112, -0x1.4b309d25957e3p-54, 0x1.4f9b2769d2ca7p0}, - {0x1.fa733951f214cp-111, -0x1.07abe1db13cadp-55, 0x1.5342b569d4f82p0}, - {-0x1.ff86852a613ffp-111, 0x1.9bb2c011d93adp-54, 0x1.56f4736b527dap0}, - {-0x1.744ee506fdafep-109, 0x1.6324c054647adp-54, 0x1.5ab07dd485429p0}, - {-0x1.95f9ab75fa7d6p-108, 0x1.ba6f93080e65ep-54, 0x1.5e76f15ad2148p0}, - {0x1.5d8e757cfb991p-111, -0x1.383c17e40b497p-54, 0x1.6247eb03a5585p0}, - {0x1.4a337f4dc0a3bp-108, -0x1.bb60987591c34p-54, 0x1.6623882552225p0}, - {0x1.57d3e3adec175p-108, -0x1.bdd3413b26456p-54, 0x1.6a09e667f3bcdp0}, - {0x1.a59f88abbe778p-115, -0x1.bbe3a683c88abp-57, 0x1.6dfb23c651a2fp0}, - {-0x1.269796953a4c3p-109, -0x1.16e4786887a99p-55, 0x1.71f75e8ec5f74p0}, - {-0x1.8f8e7fa19e5e8p-108, -0x1.0245957316dd3p-54, 0x1.75feb564267c9p0}, - {-0x1.4217a932d10d4p-113, -0x1.41577ee04992fp-55, 0x1.7a11473eb0187p0}, - {0x1.70a1427f8fcdfp-112, 0x1.05d02ba15797ep-56, 0x1.7e2f336cf4e62p0}, - {0x1.0f6ad65cbbac1p-112, -0x1.d4c1dd41532d8p-54, 0x1.82589994cce13p0}, - {-0x1.f16f65181d921p-109, -0x1.fc6f89bd4f6bap-54, 0x1.868d99b4492edp0}, - {-0x1.30644a7836333p-110, 0x1.6e9f156864b27p-54, 0x1.8ace5422aa0dbp0}, - {0x1.3bf26d2b85163p-114, 0x1.5cc13a2e3976cp-55, 0x1.8f1ae99157736p0}, - {0x1.697e257ac0db2p-111, -0x1.75fc781b57ebcp-57, 0x1.93737b0cdc5e5p0}, - {0x1.7edb9d7144b6fp-108, -0x1.d185b7c1b85d1p-54, 0x1.97d829fde4e5p0}, - {0x1.6376b7943085cp-110, 0x1.c7c46b071f2bep-56, 0x1.9c49182a3f09p0}, - {0x1.354084551b4fbp-109, -0x1.359495d1cd533p-54, 0x1.a0c667b5de565p0}, - {-0x1.bfd7adfd63f48p-111, -0x1.d2f6edb8d41e1p-54, 0x1.a5503b23e255dp0}, - {0x1.8b16ae39e8cb9p-109, 0x1.0fac90ef7fd31p-54, 0x1.a9e6b5579fdbfp0}, - {0x1.a7fbc3ae675eap-108, 0x1.7a1cd345dcc81p-54, 0x1.ae89f995ad3adp0}, - {0x1.2babc0edda4d9p-111, -0x1.2805e3084d708p-57, 0x1.b33a2b84f15fbp0}, - {0x1.aa64481e1ab72p-111, -0x1.5584f7e54ac3bp-56, 0x1.b7f76f2fb5e47p0}, - {0x1.9a164050e1258p-109, 0x1.23dd07a2d9e84p-55, 0x1.bcc1e904bc1d2p0}, - {0x1.99e51125928dap-110, 0x1.11065895048ddp-55, 0x1.c199bdd85529cp0}, - {-0x1.fc44c329d5cb2p-109, 0x1.2884dff483cadp-54, 0x1.c67f12e57d14bp0}, - {0x1.d8765566b032ep-110, 0x1.503cbd1e949dbp-56, 0x1.cb720dcef9069p0}, - {-0x1.e7044039da0f6p-108, -0x1.cbc3743797a9cp-54, 0x1.d072d4a07897cp0}, - {-0x1.ab053b05531fcp-111, 0x1.2ed02d75b3707p-55, 0x1.d5818dcfba487p0}, - {0x1.7f6246f0ec615p-108, 0x1.c2300696db532p-54, 0x1.da9e603db3285p0}, - {0x1.b7225a944efd6p-108, -0x1.1a5cd4f184b5cp-54, 0x1.dfc97337b9b5fp0}, - {0x1.1e92cb3c2d278p-109, 0x1.39e8980a9cc8fp-55, 0x1.e502ee78b3ff6p0}, - {-0x1.fc0f242bbf3dep-109, -0x1.e9c23179c2893p-54, 0x1.ea4afa2a490dap0}, - {0x1.f6dd5d229ff69p-108, 0x1.dc7f486a4b6bp-54, 0x1.efa1bee615a27p0}, - {-0x1.4019bffc80ef3p-110, 0x1.9d3e12dd8a18bp-54, 0x1.f50765b6e454p0}, - {0x1.dc060c36f7651p-112, 0x1.74853f3a5931ep-55, 0x1.fa7c1819e90d8p0}, -}; - -// Lookup table for 2^(k * 2^-12) with k = 0..63. -// Generated by Sollya with: -// > display=hexadecimal; -// > prec = 500; -// > for i from 0 to 63 do { -// a = 2^(i * 2^-12); -// b = round(a, D, RN); -// c = round(a - b, D, RN); -// d = round(a - b - c, D, RN); -// print("{", d, ",", c, ",", b, "},"); -// }; -alignas(16) const fputil::TripleDouble EXP2_MID2[64] = { - {0, 0, 0x1p0}, - {0x1.39726694630e3p-108, 0x1.ae8e38c59c72ap-54, 0x1.000b175effdc7p0}, - {0x1.e5e06ddd31156p-112, -0x1.7b5d0d58ea8f4p-58, 0x1.00162f3904052p0}, - {0x1.5a0768b51f609p-111, 0x1.4115cb6b16a8ep-54, 0x1.0021478e11ce6p0}, - {0x1.d008403605217p-111, -0x1.d7c96f201bb2fp-55, 0x1.002c605e2e8cfp0}, - {0x1.89bc16f765708p-109, 0x1.84711d4c35e9fp-54, 0x1.003779a95f959p0}, - {-0x1.4535b7f8c1e2dp-109, -0x1.0484245243777p-55, 0x1.0042936faa3d8p0}, - {-0x1.8ba92f6b25456p-108, -0x1.4b237da2025f9p-54, 0x1.004dadb113dap0}, - {-0x1.30c72e81f4294p-113, -0x1.5e00e62d6b30dp-56, 0x1.0058c86da1c0ap0}, - {-0x1.34a5384e6f0b9p-110, 0x1.a1d6cedbb9481p-54, 0x1.0063e3a559473p0}, - {0x1.f8d0580865d2ep-108, -0x1.4acf197a00142p-54, 0x1.006eff583fc3dp0}, - {-0x1.002bcb3ae9a99p-111, -0x1.eaf2ea42391a5p-57, 0x1.007a1b865a8cap0}, - {0x1.c3c5aedee9851p-111, 0x1.da93f90835f75p-56, 0x1.0085382faef83p0}, - {0x1.7217851d1ec6ep-109, -0x1.6a79084ab093cp-55, 0x1.00905554425d4p0}, - {-0x1.80cbca335a7c3p-110, 0x1.86364f8fbe8f8p-54, 0x1.009b72f41a12bp0}, - {-0x1.706bd4eb22595p-110, -0x1.82e8e14e3110ep-55, 0x1.00a6910f3b6fdp0}, - {-0x1.b55dd523f3c08p-111, -0x1.4f6b2a7609f71p-55, 0x1.00b1afa5abcbfp0}, - {0x1.90a1e207cced1p-110, -0x1.e1a258ea8f71bp-56, 0x1.00bcceb7707ecp0}, - {0x1.78d0472db37c5p-110, 0x1.4362ca5bc26f1p-56, 0x1.00c7ee448ee02p0}, - {-0x1.bcd4db3cb52fep-109, 0x1.095a56c919d02p-54, 0x1.00d30e4d0c483p0}, - {-0x1.cf1b131575ec2p-112, -0x1.406ac4e81a645p-57, 0x1.00de2ed0ee0f5p0}, - {-0x1.6aaa1fa7ff913p-112, 0x1.b5a6902767e09p-54, 0x1.00e94fd0398ep0}, - {0x1.68f236dff3218p-110, -0x1.91b2060859321p-54, 0x1.00f4714af41d3p0}, - {-0x1.e8bb58067e60ap-109, 0x1.427068ab22306p-55, 0x1.00ff93412315cp0}, - {0x1.d4cd5e1d71fdfp-108, 0x1.c1d0660524e08p-54, 0x1.010ab5b2cbd11p0}, - {0x1.e4ecf350ebe88p-108, -0x1.e7bdfb3204be8p-54, 0x1.0115d89ff3a8bp0}, - {0x1.6a2aa2c89c4f8p-109, 0x1.843aa8b9cbbc6p-55, 0x1.0120fc089ff63p0}, - {0x1.1ca368a20ed05p-110, -0x1.34104ee7edae9p-56, 0x1.012c1fecd613bp0}, - {0x1.edb1095d925cfp-114, -0x1.2b6aeb6176892p-56, 0x1.0137444c9b5b5p0}, - {-0x1.488c78eded75fp-111, 0x1.a8cd33b8a1bb3p-56, 0x1.01426927f5278p0}, - {-0x1.7480f5ea1b3c9p-113, 0x1.2edc08e5da99ap-56, 0x1.014d8e7ee8d2fp0}, - {-0x1.ae45989a04dd5p-111, 0x1.57ba2dc7e0c73p-55, 0x1.0158b4517bb88p0}, - {0x1.bf48007d80987p-109, 0x1.b61299ab8cdb7p-54, 0x1.0163da9fb3335p0}, - {0x1.1aa91a059292cp-109, -0x1.90565902c5f44p-54, 0x1.016f0169949edp0}, - {0x1.b6663292855f5p-110, 0x1.70fc41c5c2d53p-55, 0x1.017a28af25567p0}, - {0x1.e7fbca6793d94p-108, 0x1.4b9a6e145d76cp-54, 0x1.018550706ab62p0}, - {-0x1.5b9f5c7de3b93p-110, -0x1.008eff5142bf9p-56, 0x1.019078ad6a19fp0}, - {0x1.4638bf2f6acabp-110, -0x1.77669f033c7dep-54, 0x1.019ba16628de2p0}, - {-0x1.ab237b9a069c5p-109, -0x1.09bb78eeead0ap-54, 0x1.01a6ca9aac5f3p0}, - {0x1.3ab358be97cefp-108, 0x1.371231477ece5p-54, 0x1.01b1f44af9f9ep0}, - {-0x1.4027b2294bb64p-110, 0x1.5e7626621eb5bp-56, 0x1.01bd1e77170b4p0}, - {0x1.656394426c99p-111, -0x1.bc72b100828a5p-54, 0x1.01c8491f08f08p0}, - {0x1.bf9785189bdd8p-111, -0x1.ce39cbbab8bbep-57, 0x1.01d37442d507p0}, - {0x1.7c12f86114fe3p-109, 0x1.16996709da2e2p-55, 0x1.01de9fe280ac8p0}, - {-0x1.653d5d24b5d28p-109, -0x1.c11f5239bf535p-55, 0x1.01e9cbfe113efp0}, - {0x1.04a0cdc1d86d7p-109, 0x1.e1d4eb5edc6b3p-55, 0x1.01f4f8958c1c6p0}, - {0x1.c678c46149782p-109, -0x1.afb99946ee3fp-54, 0x1.020025a8f6a35p0}, - {0x1.48524e1e9df7p-108, -0x1.8f06d8a148a32p-54, 0x1.020b533856324p0}, - {0x1.9953ea727ff0bp-109, -0x1.2bf310fc54eb6p-55, 0x1.02168143b0281p0}, - {-0x1.ccfbbec22d28ep-108, -0x1.c95a035eb4175p-54, 0x1.0221afcb09e3ep0}, - {0x1.9e2bb6e181de1p-108, -0x1.491793e46834dp-54, 0x1.022cdece68c4fp0}, - {0x1.f17609ae29308p-110, -0x1.3e8d0d9c49091p-56, 0x1.02380e4dd22adp0}, - {-0x1.c7dc2c476bfb8p-110, -0x1.314aa16278aa3p-54, 0x1.02433e494b755p0}, - {-0x1.fab994971d4a3p-109, 0x1.48daf888e9651p-55, 0x1.024e6ec0da046p0}, - {0x1.848b62cbdd0afp-109, 0x1.56dc8046821f4p-55, 0x1.02599fb483385p0}, - {-0x1.bf603ba715d0cp-109, 0x1.45b42356b9d47p-54, 0x1.0264d1244c719p0}, - {0x1.89434e751e1aap-110, -0x1.082ef51b61d7ep-56, 0x1.027003103b10ep0}, - {-0x1.03b54fd64e8acp-110, 0x1.2106ed0920a34p-56, 0x1.027b357854772p0}, - {0x1.7785ea0acc486p-109, -0x1.fd4cf26ea5d0fp-54, 0x1.0286685c9e059p0}, - {-0x1.ce447fdb35ff9p-109, -0x1.09f8775e78084p-54, 0x1.02919bbd1d1d8p0}, - {0x1.5b884aab5642ap-112, 0x1.64cbba902ca27p-58, 0x1.029ccf99d720ap0}, - {-0x1.cfb3e46d7c1cp-108, 0x1.4383ef231d207p-54, 0x1.02a803f2d170dp0}, - {-0x1.0d40cee4b81afp-112, 0x1.4a47a505b3a47p-54, 0x1.02b338c811703p0}, - {0x1.6ae7d36d7c1f7p-109, 0x1.e47120223467fp-54, 0x1.02be6e199c811p0}, -}; - } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/common_constants.h b/libc/src/math/generic/common_constants.h index e65f002845953..291816a7889ad 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/exp_constants.h" #include "src/__support/number_pair.h" namespace LIBC_NAMESPACE_DECL { @@ -80,12 +81,6 @@ extern const double EXP_M1[195]; // > for i from 0 to 127 do { D(exp(i / 128)); }; extern const double EXP_M2[128]; -// Lookup table for 2^(k * 2^-6) with k = 0..63. -extern const fputil::TripleDouble EXP2_MID1[64]; - -// Lookup table for 2^(k * 2^-12) with k = 0..63. -extern const fputil::TripleDouble EXP2_MID2[64]; - } // namespace LIBC_NAMESPACE_DECL #endif // LLVM_LIBC_SRC_MATH_GENERIC_COMMON_CONSTANTS_H diff --git a/libc/src/math/generic/exp.cpp b/libc/src/math/generic/exp.cpp index 143800ca078a6..dc4d2ca480cb8 100644 --- a/libc/src/math/generic/exp.cpp +++ b/libc/src/math/generic/exp.cpp @@ -7,434 +7,9 @@ //===----------------------------------------------------------------------===// #include "src/math/exp.h" -#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2. -#include "explogxf.h" // ziv_test_denorm. -#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/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/exp.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. -constexpr double ERR_D = 0x1.8p-63; - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS -// Errors when using double-double precision. -constexpr double ERR_DD = 0x1.0p-99; -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -// -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; - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS -constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47; -constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79; -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -namespace { - -// 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; -} - -#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS -// 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) { - constexpr Float128 COEFFS_128[]{ - {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0 - {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], 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(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); - - 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; -} -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - -// Check for exceptional cases when -// |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 -double set_exceptional(double x) { - using FPBits = typename fputil::FPBits; - 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) { - // 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 FPBits::min_subnormal().get_val(); - 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 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, exp, (double x)) { - using FPBits = typename fputil::FPBits; - 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^-1075) <= 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{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); - - // |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); - -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - if (LIBC_UNLIKELY(denorm)) { - return ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D) - .value(); - } else { - // to multiply by 2^hi, a fast way is to simply add hi to the exponent - // field. - int64_t exp_hi = static_cast(hi) << FPBits::FRACTION_LEN; - double r = - cpp::bit_cast(exp_hi + cpp::bit_cast(exp_mid.hi + lo)); - return r; - } -#else - 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) << FPBits::FRACTION_LEN; - 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) << FPBits::FRACTION_LEN; - 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); -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS -} +LLVM_LIBC_FUNCTION(double, exp, (double x)) { return math::exp(x); } } // namespace LIBC_NAMESPACE_DECL diff --git a/libc/src/math/generic/explogxf.h b/libc/src/math/generic/explogxf.h index 212ede4758549..5ae1457ca780e 100644 --- a/libc/src/math/generic/explogxf.h +++ b/libc/src/math/generic/explogxf.h @@ -10,16 +10,15 @@ #define LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H #include "common_constants.h" -#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/nearest_integer.h" #include "src/__support/common.h" #include "src/__support/macros/config.h" #include "src/__support/macros/properties/cpu_features.h" +#include "src/__support/math/exp_utils.h" + namespace LIBC_NAMESPACE_DECL { struct ExpBase { @@ -375,58 +374,6 @@ LIBC_INLINE static double log_eval(double x) { return result; } -// Rounding tests for 2^hi * (mid + lo) when the output might be denormal. We -// assume further that 1 <= mid < 2, mid + lo < 2, and |lo| << mid. -// Notice that, if 0 < x < 2^-1022, -// double(2^-1022 + x) - 2^-1022 = double(x). -// So if we scale x up by 2^1022, we can use -// double(1.0 + 2^1022 * x) - 1.0 to test how x is rounded in denormal range. -template -LIBC_INLINE static cpp::optional -ziv_test_denorm(int hi, double mid, double lo, double err) { - using FPBits = typename fputil::FPBits; - - // Scaling factor = 1/(min normal number) = 2^1022 - int64_t exp_hi = static_cast(hi + 1022) << FPBits::FRACTION_LEN; - double mid_hi = cpp::bit_cast(exp_hi + cpp::bit_cast(mid)); - double lo_scaled = - (lo != 0.0) ? cpp::bit_cast(exp_hi + cpp::bit_cast(lo)) - : 0.0; - - double extra_factor = 0.0; - uint64_t scale_down = 0x3FE0'0000'0000'0000; // 1022 in the exponent field. - - // Result is denormal if (mid_hi + lo_scale < 1.0). - if ((1.0 - mid_hi) > lo_scaled) { - // Extra rounding step is needed, which adds more rounding errors. - err += 0x1.0p-52; - extra_factor = 1.0; - scale_down = 0x3FF0'0000'0000'0000; // 1023 in the exponent field. - } - - // By adding 1.0, the results will have similar rounding points as denormal - // outputs. - if constexpr (SKIP_ZIV_TEST) { - double r = extra_factor + (mid_hi + lo_scaled); - return cpp::bit_cast(cpp::bit_cast(r) - scale_down); - } else { - double err_scaled = - cpp::bit_cast(exp_hi + cpp::bit_cast(err)); - - double lo_u = lo_scaled + err_scaled; - double lo_l = lo_scaled - err_scaled; - - double upper = extra_factor + (mid_hi + lo_u); - double lower = extra_factor + (mid_hi + lo_l); - - if (LIBC_LIKELY(upper == lower)) { - return cpp::bit_cast(cpp::bit_cast(upper) - scale_down); - } - - return cpp::nullopt; - } -} - } // namespace LIBC_NAMESPACE_DECL #endif // LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H diff --git a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel index cd608a1352a7a..4ab0126291276 100644 --- a/utils/bazel/llvm-project-overlay/libc/BUILD.bazel +++ b/utils/bazel/llvm-project-overlay/libc/BUILD.bazel @@ -1919,7 +1919,7 @@ libc_support_library( srcs = ["src/math/generic/common_constants.cpp"], hdrs = ["src/math/generic/common_constants.h"], deps = [ - ":__support_fputil_triple_double", + ":__support_math_exp_constants", ":__support_number_pair", ], ) @@ -2002,10 +2002,10 @@ libc_support_library( ":__support_common", ":__support_fputil_fenv_impl", ":__support_fputil_fma", - ":__support_fputil_fp_bits", ":__support_fputil_multiply_add", ":__support_fputil_nearest_integer", ":__support_fputil_polyeval", + ":__support_math_exp_utils", ":common_constants", ], ) @@ -2205,6 +2205,46 @@ libc_support_library( ], ) +libc_support_library( + name = "__support_math_exp_constants", + hdrs = ["src/__support/math/exp_constants.h"], + deps = [ + ":__support_fputil_triple_double", + ], +) + +libc_support_library( + name = "__support_math_exp_utils", + hdrs = ["src/__support/math/exp_utils.h"], + deps = [ + ":__support_cpp_optional", + ":__support_cpp_bit", + ":__support_fputil_fp_bits", + ], +) + +libc_support_library( + name = "__support_math_exp", + hdrs = ["src/__support/math/exp.h"], + deps = [ + ":__support_math_exp_constants", + ":__support_math_exp_utils", + ":__support_cpp_bit", + ":__support_cpp_optional", + ":__support_fputil_dyadic_float", + ":__support_fputil_fenv_impl", + ":__support_fputil_fp_bits", + ":__support_fputil_multiply_add", + ":__support_fputil_nearest_integer", + ":__support_fputil_polyeval", + ":__support_fputil_rounding_mode", + ":__support_fputil_triple_double", + ":__support_fputil_double_double", + ":__support_integer_literals", + ":__support_macros_optimization", + ], +) + ############################### complex targets ################################ libc_function( @@ -2785,17 +2825,8 @@ libc_math_function( libc_math_function( name = "exp", additional_deps = [ - ":__support_fputil_double_double", - ":__support_fputil_dyadic_float", - ":__support_fputil_multiply_add", - ":__support_fputil_nearest_integer", - ":__support_fputil_polyeval", - ":__support_fputil_rounding_mode", - ":__support_fputil_triple_double", - ":__support_integer_literals", - ":__support_macros_optimization", - ":common_constants", - ":explogxf", + ":__support_math_exp", + ":errno", ], )