//===-- Double-precision e^x function -------------------------------------===// // // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. // See https://llvm.org/LICENSE.txt for license information. // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception // //===----------------------------------------------------------------------===// #include "src/math/exp.h" #include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2. #include "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 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; // Errors when using double-double precision. constexpr double ERR_DD = 0x1.0p-99; // -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 { // Polynomial approximations with double precision: // Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24. // For |dx| < 2^-13 + 2^-30: // | output - expm1(dx) / dx | < 2^-51. LIBC_INLINE double poly_approx_d(double dx) { // dx^2 double dx2 = dx * dx; // c0 = 1 + dx / 2 double c0 = fputil::multiply_add(dx, 0.5, 1.0); // c1 = 1/6 + dx / 24 double c1 = fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3); // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24 double p = fputil::multiply_add(dx2, c1, c0); return p; } // Polynomial approximation with double-double precision: // Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720 // For |dx| < 2^-13 + 2^-30: // | output - exp(dx) | < 2^-101 DoubleDouble poly_approx_dd(const DoubleDouble &dx) { // Taylor polynomial. constexpr DoubleDouble COEFFS[] = { {0, 0x1p0}, // 1 {0, 0x1p0}, // 1 {0, 0x1p-1}, // 1/2 {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6 {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24 {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120 {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720 }; DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]); return p; } // Polynomial approximation with 128-bit precision: // Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^7 / 5040 // For |dx| < 2^-13 + 2^-30: // | output - exp(dx) | < 2^-126. Float128 poly_approx_f128(const Float128 &dx) { 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; } // 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); 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); } } // namespace LIBC_NAMESPACE_DECL