//===-- Double-precision 10^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/exp10.h" #include "common_constants.h" // Lookup tables EXP2_MID1 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(10) constexpr double LOG2_10 = 0x1.a934f0979a371p+1; // -2^-12 * log10(2) // > a = -2^-12 * log10(2); // > b = round(a, 32, RN); // > c = round(a - b, 32, RN); // > d = round(a - b - c, D, RN); // Errors < 1.5 * 2^-144 constexpr double MLOG10_2_EXP2_M12_HI = -0x1.3441350ap-14; constexpr double MLOG10_2_EXP2_M12_MID = 0x1.0c0219dc1da99p-51; constexpr double MLOG10_2_EXP2_M12_MID_32 = 0x1.0c0219dcp-51; constexpr double MLOG10_2_EXP2_M12_LO = 0x1.da994fd20dba2p-87; // Error bounds: // Errors when using double precision. constexpr double ERR_D = 0x1.8p-63; // Errors when using double-double precision. constexpr double ERR_DD = 0x1.8p-99; namespace { // Polynomial approximations with double precision. Generated by Sollya with: // > P = fpminimax((10^x - 1)/x, 3, [|D...|], [-2^-14, 2^-14]); // > P; // Error bounds: // | output - (10^dx - 1) / dx | < 2^-52. LIBC_INLINE double poly_approx_d(double dx) { // dx^2 double dx2 = dx * dx; double c0 = fputil::multiply_add(dx, 0x1.53524c73cea6ap+1, 0x1.26bb1bbb55516p+1); double c1 = fputil::multiply_add(dx, 0x1.2bd75cc6afc65p+0, 0x1.0470587aa264cp+1); double p = fputil::multiply_add(dx2, c1, c0); return p; } // Polynomial approximation with double-double precision. Generated by Solya // with: // > P = fpminimax((10^x - 1)/x, 5, [|DD...|], [-2^-14, 2^-14]); // Error bounds: // | output - 10^(dx) | < 2^-101 DoubleDouble poly_approx_dd(const DoubleDouble &dx) { // Taylor polynomial. constexpr DoubleDouble COEFFS[] = { {0, 0x1p0}, {-0x1.f48ad494e927bp-53, 0x1.26bb1bbb55516p1}, {-0x1.e2bfab3191cd2p-53, 0x1.53524c73cea69p1}, {0x1.80fb65ec3b503p-53, 0x1.0470591de2ca4p1}, {0x1.338fc05e21e55p-54, 0x1.2bd7609fd98c4p0}, {0x1.d4ea116818fbp-56, 0x1.1429ffd519865p-1}, {-0x1.872a8ff352077p-57, 0x1.a7ed70847c8b3p-3}, }; 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 + a0 * dx + a1 * dx^2 + ... + a6 * dx^7 // For |dx| < 2^-14: // | output - 10^dx | < 1.5 * 2^-124. Float128 poly_approx_f128(const Float128 &dx) { constexpr Float128 COEFFS_128[]{ {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0 {Sign::POS, -126, 0x935d8ddd'aaa8ac16'ea56d62b'82d30a2d_u128}, {Sign::POS, -126, 0xa9a92639'e753443a'80a99ce7'5f4d5bdb_u128}, {Sign::POS, -126, 0x82382c8e'f1652304'6a4f9d7d'bf6c9635_u128}, {Sign::POS, -124, 0x12bd7609'fd98c44c'34578701'9216c7af_u128}, {Sign::POS, -127, 0x450a7ff4'7535d889'cc41ed7e'0d27aee5_u128}, {Sign::POS, -130, 0xd3f6b844'702d636b'8326bb91'a6e7601d_u128}, {Sign::POS, -130, 0x45b937f0'd05bb1cd'fa7b46df'314112a9_u128}, }; 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 10^(x) using 128-bit precision. // TODO(lntue): investigate triple-double precision implementation for this // step. Float128 exp10_f128(double x, double kd, int idx1, int idx2) { double t1 = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact double t2 = kd * MLOG10_2_EXP2_M12_MID_32; // exact double t3 = kd * MLOG10_2_EXP2_M12_LO; // Error < 2^-144 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 10^x with double-double precision. DoubleDouble exp10_double_double(double x, double kd, const DoubleDouble &exp_mid) { // Recalculate dx: // dx = x - k * 2^-12 * log10(2) double t1 = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact double t2 = kd * MLOG10_2_EXP2_M12_MID_32; // exact double t3 = kd * MLOG10_2_EXP2_M12_LO; // Error < 2^-140 DoubleDouble dx = fputil::exact_add(t1, t2); dx.lo += t3; // Degree-6 polynomial approximation in double-double precision. // | p - 10^x | < 2^-103. DoubleDouble p = poly_approx_dd(dx); // Error bounds: 2^-102. DoubleDouble r = fputil::quick_mult(exp_mid, p); return r; } // When output is denormal. double exp10_denorm(double x) { // Range reduction. double tmp = fputil::multiply_add(x, LOG2_10, 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; 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); // |dx| < 1.5 * 2^-15 + 2^-31 < 2^-14 double lo_h = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact double dx = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_MID, lo_h); double mid_lo = dx * exp_mid.hi; // Approximate (10^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4. double p = poly_approx_d(dx); double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo); if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D); LIBC_LIKELY(r.has_value())) return r.value(); // Use double-double DoubleDouble r_dd = exp10_double_double(x, kd, exp_mid); if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD); LIBC_LIKELY(r.has_value())) return r.value(); // Use 128-bit precision Float128 r_f128 = exp10_f128(x, kd, idx1, idx2); return static_cast(r_f128); } // Check for exceptional cases when: // * log10(1 - 2^-54) < x < log10(1 + 2^-53) // * x >= log10(2^1024) // * x <= log10(2^-1022) // * x is inf or nan 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| < log10(1 + 2^-53) if (x_abs <= 0x3c8bcb7b1526e50e) { // 10^(x) ~ 1 + x/2 return fputil::multiply_add(x, 0.5, 1.0); } // x <= log10(2^-1022) || x >= log10(2^1024) or inf/nan. if (x_u >= 0xc0733a7146f72a42) { // x <= log10(2^-1075) or -inf/nan if (x_u > 0xc07439b746e36b52) { // 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; } return exp10_denorm(x); } // x >= log10(2^1024) 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, exp10, (double x)) { using FPBits = typename fputil::FPBits; FPBits xbits(x); uint64_t x_u = xbits.uintval(); // x <= log10(2^-1022) or x >= log10(2^1024) or // log10(1 - 2^-54) < x < log10(1 + 2^-53). if (LIBC_UNLIKELY(x_u >= 0xc0733a7146f72a42 || (x_u <= 0xbc7bcb7b1526e50e && x_u >= 0x40734413509f79ff) || x_u < 0x3c8bcb7b1526e50e)) { return set_exceptional(x); } // Now log10(2^-1075) < x <= log10(1 - 2^-54) or // log10(1 + 2^-53) < x < log10(2^1024) // Range reduction: // Let x = log10(2) * (hi + mid1 + mid2) + lo // in which: // hi is an integer // mid1 * 2^6 is an integer // mid2 * 2^12 is an integer // then: // 10^(x) = 2^hi * 2^(mid1) * 2^(mid2) * 10^(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. // - 10^(lo) ~ 1 + a0*lo + a1 * lo^2 + ... // // We compute (hi + mid1 + mid2) together by perform the rounding on // x * log2(10) * 2^12. // Since |x| < |log10(2^-1075)| < 2^9, // |x * 2^12| < 2^9 * 2^12 < 2^21, // So we can fit the rounded result round(x * 2^12) 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). // // 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 >= 2^23 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. // // One small problem with this approach is that the sum (x*2^12 + 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 + C, // where C = 2^33 + 2^32 + 2^-1, then if // k = int32_t(lower 51 bits of double(x * 2^12 + C) >> 19), // the reduced argument: // lo = x - log10(2) * 2^-12 * k is bounded by: // |lo| = |x - log10(2) * 2^-12 * k| // = log10(2) * 2^-12 * | x * log2(10) * 2^12 - k | // <= log10(2) * 2^-12 * (2^-1 + 2^-19) // < 1.5 * 2^-2 * (2^-13 + 2^-31) // = 1.5 * (2^-15 * 2^-31) // // Finally, notice that k only uses the mantissa of x * 2^12, 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 + C) >> 19). // Rounding errors <= 2^-31. double tmp = fputil::multiply_add(x, LOG2_10, 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; 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); // |dx| < 1.5 * 2^-15 + 2^-31 < 2^-14 double lo_h = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact double dx = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_MID, lo_h); // We use the degree-4 polynomial to approximate 10^(lo): // 10^(lo) ~ 1 + a0 * lo + a1 * lo^2 + a2 * lo^3 + a3 * lo^4 // = 1 + lo * P(lo) // So that the errors are bounded by: // |P(lo) - (10^lo - 1)/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_(lo) - P(lo)| < 2^-51 // => |lo * P_(lo) - (2^lo - 1) | < 2^-65 // => 2^(mid1 + mid2) * |lo * P_(lo) - expm1(lo)| < 2^-64. // 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 2^-64. double mid_lo = dx * exp_mid.hi; // Approximate (10^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4. double p = poly_approx_d(dx); double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo); 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; } // Exact outputs when x = 1, 2, ..., 22 + hard to round with x = 23. // Quick check mask: 0x800f'ffffU = ~(bits of 1.0 | ... | bits of 23.0) if (LIBC_UNLIKELY((x_u & 0x8000'ffff'ffff'ffffULL) == 0ULL)) { switch (x_u) { case 0x3ff0000000000000: // x = 1.0 return 10.0; case 0x4000000000000000: // x = 2.0 return 100.0; case 0x4008000000000000: // x = 3.0 return 1'000.0; case 0x4010000000000000: // x = 4.0 return 10'000.0; case 0x4014000000000000: // x = 5.0 return 100'000.0; case 0x4018000000000000: // x = 6.0 return 1'000'000.0; case 0x401c000000000000: // x = 7.0 return 10'000'000.0; case 0x4020000000000000: // x = 8.0 return 100'000'000.0; case 0x4022000000000000: // x = 9.0 return 1'000'000'000.0; case 0x4024000000000000: // x = 10.0 return 10'000'000'000.0; case 0x4026000000000000: // x = 11.0 return 100'000'000'000.0; case 0x4028000000000000: // x = 12.0 return 1'000'000'000'000.0; case 0x402a000000000000: // x = 13.0 return 10'000'000'000'000.0; case 0x402c000000000000: // x = 14.0 return 100'000'000'000'000.0; case 0x402e000000000000: // x = 15.0 return 1'000'000'000'000'000.0; case 0x4030000000000000: // x = 16.0 return 10'000'000'000'000'000.0; case 0x4031000000000000: // x = 17.0 return 100'000'000'000'000'000.0; case 0x4032000000000000: // x = 18.0 return 1'000'000'000'000'000'000.0; case 0x4033000000000000: // x = 19.0 return 10'000'000'000'000'000'000.0; case 0x4034000000000000: // x = 20.0 return 100'000'000'000'000'000'000.0; case 0x4035000000000000: // x = 21.0 return 1'000'000'000'000'000'000'000.0; case 0x4036000000000000: // x = 22.0 return 10'000'000'000'000'000'000'000.0; case 0x4037000000000000: // x = 23.0 return 0x1.52d02c7e14af6p76 + x; } } // Use double-double DoubleDouble r_dd = exp10_double_double(x, kd, exp_mid); 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)) { // 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_dd)); return r; } // Use 128-bit precision Float128 r_f128 = exp10_f128(x, kd, idx1, idx2); return static_cast(r_f128); } } // namespace LIBC_NAMESPACE_DECL