libc/src/math/generic/exp10.cpp

   1 //===-- Double-precision 10^x function ------------------------------------===//
   2 //
   3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
   4 // See https://llvm.org/LICENSE.txt for license information.
   5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
   6 //
   7 //===----------------------------------------------------------------------===//
   8
   9 #include "src/math/exp10.h"
  10 #include "common_constants.h" // Lookup tables EXP2_MID1 and EXP_M2.
  11 #include "explogxf.h"         // ziv_test_denorm.
  12 #include "src/__support/CPP/bit.h"
  13 #include "src/__support/CPP/optional.h"
  14 #include "src/__support/FPUtil/FEnvImpl.h"
  15 #include "src/__support/FPUtil/FPBits.h"
  16 #include "src/__support/FPUtil/PolyEval.h"
  17 #include "src/__support/FPUtil/double_double.h"
  18 #include "src/__support/FPUtil/dyadic_float.h"
  19 #include "src/__support/FPUtil/multiply_add.h"
  20 #include "src/__support/FPUtil/nearest_integer.h"
  21 #include "src/__support/FPUtil/rounding_mode.h"
  22 #include "src/__support/FPUtil/triple_double.h"
  23 #include "src/__support/common.h"
  24 #include "src/__support/integer_literals.h"
  25 #include "src/__support/macros/config.h"
  26 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
  27
  28 namespace LIBC_NAMESPACE_DECL {
  29
  30 using fputil::DoubleDouble;
  31 using fputil::TripleDouble;
  32 using Float128 = typename fputil::DyadicFloat<128>;
  33
  34 using LIBC_NAMESPACE::operator""_u128;
  35
  36 // log2(10)
  37 constexpr double LOG2_10 = 0x1.a934f0979a371p+1;
  38
  39 // -2^-12 * log10(2)
  40 // > a = -2^-12 * log10(2);
  41 // > b = round(a, 32, RN);
  42 // > c = round(a - b, 32, RN);
  43 // > d = round(a - b - c, D, RN);
  44 // Errors < 1.5 * 2^-144
  45 constexpr double MLOG10_2_EXP2_M12_HI = -0x1.3441350ap-14;
  46 constexpr double MLOG10_2_EXP2_M12_MID = 0x1.0c0219dc1da99p-51;
  47 constexpr double MLOG10_2_EXP2_M12_MID_32 = 0x1.0c0219dcp-51;
  48 constexpr double MLOG10_2_EXP2_M12_LO = 0x1.da994fd20dba2p-87;
  49
  50 // Error bounds:
  51 // Errors when using double precision.
  52 constexpr double ERR_D = 0x1.8p-63;
  53
  54 // Errors when using double-double precision.
  55 constexpr double ERR_DD = 0x1.8p-99;
  56
  57 namespace {
  58
  59 // Polynomial approximations with double precision.  Generated by Sollya with:
  60 // > P = fpminimax((10^x - 1)/x, 3, [|D...|], [-2^-14, 2^-14]);
  61 // > P;
  62 // Error bounds:
  63 //   | output - (10^dx - 1) / dx | < 2^-52.
  64 LIBC_INLINE double poly_approx_d(double dx) {
  65   // dx^2
  66   double dx2 = dx * dx;
  67   double c0 =
  68       fputil::multiply_add(dx, 0x1.53524c73cea6ap+1, 0x1.26bb1bbb55516p+1);
  69   double c1 =
  70       fputil::multiply_add(dx, 0x1.2bd75cc6afc65p+0, 0x1.0470587aa264cp+1);
  71   double p = fputil::multiply_add(dx2, c1, c0);
  72   return p;
  73 }
  74
  75 // Polynomial approximation with double-double precision.  Generated by Solya
  76 // with:
  77 // > P = fpminimax((10^x - 1)/x, 5, [|DD...|], [-2^-14, 2^-14]);
  78 // Error bounds:
  79 //   | output - 10^(dx) | < 2^-101
  80 DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
  81   // Taylor polynomial.
  82   constexpr DoubleDouble COEFFS[] = {
  83       {0, 0x1p0},
  84       {-0x1.f48ad494e927bp-53, 0x1.26bb1bbb55516p1},
  85       {-0x1.e2bfab3191cd2p-53, 0x1.53524c73cea69p1},
  86       {0x1.80fb65ec3b503p-53, 0x1.0470591de2ca4p1},
  87       {0x1.338fc05e21e55p-54, 0x1.2bd7609fd98c4p0},
  88       {0x1.d4ea116818fbp-56, 0x1.1429ffd519865p-1},
  89       {-0x1.872a8ff352077p-57, 0x1.a7ed70847c8b3p-3},
  90
  91   };
  92
  93   DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2],
  94                                     COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]);
  95   return p;
  96 }
  97
  98 // Polynomial approximation with 128-bit precision:
  99 // Return exp(dx) ~ 1 + a0 * dx + a1 * dx^2 + ... + a6 * dx^7
 100 // For |dx| < 2^-14:
 101 //   | output - 10^dx | < 1.5 * 2^-124.
 102 Float128 poly_approx_f128(const Float128 &dx) {
 103   constexpr Float128 COEFFS_128[]{
 104       {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
 105       {Sign::POS, -126, 0x935d8ddd'aaa8ac16'ea56d62b'82d30a2d_u128},
 106       {Sign::POS, -126, 0xa9a92639'e753443a'80a99ce7'5f4d5bdb_u128},
 107       {Sign::POS, -126, 0x82382c8e'f1652304'6a4f9d7d'bf6c9635_u128},
 108       {Sign::POS, -124, 0x12bd7609'fd98c44c'34578701'9216c7af_u128},
 109       {Sign::POS, -127, 0x450a7ff4'7535d889'cc41ed7e'0d27aee5_u128},
 110       {Sign::POS, -130, 0xd3f6b844'702d636b'8326bb91'a6e7601d_u128},
 111       {Sign::POS, -130, 0x45b937f0'd05bb1cd'fa7b46df'314112a9_u128},
 112   };
 113
 114   Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2],
 115                                 COEFFS_128[3], COEFFS_128[4], COEFFS_128[5],
 116                                 COEFFS_128[6], COEFFS_128[7]);
 117   return p;
 118 }
 119
 120 // Compute 10^(x) using 128-bit precision.
 121 // TODO(lntue): investigate triple-double precision implementation for this
 122 // step.
 123 Float128 exp10_f128(double x, double kd, int idx1, int idx2) {
 124   double t1 = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
 125   double t2 = kd * MLOG10_2_EXP2_M12_MID_32;                     // exact
 126   double t3 = kd * MLOG10_2_EXP2_M12_LO; // Error < 2^-144
 127
 128   Float128 dx = fputil::quick_add(
 129       Float128(t1), fputil::quick_add(Float128(t2), Float128(t3)));
 130
 131   // TODO: Skip recalculating exp_mid1 and exp_mid2.
 132   Float128 exp_mid1 =
 133       fputil::quick_add(Float128(EXP2_MID1[idx1].hi),
 134                         fputil::quick_add(Float128(EXP2_MID1[idx1].mid),
 135                                           Float128(EXP2_MID1[idx1].lo)));
 136
 137   Float128 exp_mid2 =
 138       fputil::quick_add(Float128(EXP2_MID2[idx2].hi),
 139                         fputil::quick_add(Float128(EXP2_MID2[idx2].mid),
 140                                           Float128(EXP2_MID2[idx2].lo)));
 141
 142   Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);
 143
 144   Float128 p = poly_approx_f128(dx);
 145
 146   Float128 r = fputil::quick_mul(exp_mid, p);
 147
 148   r.exponent += static_cast<int>(kd) >> 12;
 149
 150   return r;
 151 }
 152
 153 // Compute 10^x with double-double precision.
 154 DoubleDouble exp10_double_double(double x, double kd,
 155                                  const DoubleDouble &exp_mid) {
 156   // Recalculate dx:
 157   //   dx = x - k * 2^-12 * log10(2)
 158   double t1 = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
 159   double t2 = kd * MLOG10_2_EXP2_M12_MID_32;                     // exact
 160   double t3 = kd * MLOG10_2_EXP2_M12_LO; // Error < 2^-140
 161
 162   DoubleDouble dx = fputil::exact_add(t1, t2);
 163   dx.lo += t3;
 164
 165   // Degree-6 polynomial approximation in double-double precision.
 166   // | p - 10^x | < 2^-103.
 167   DoubleDouble p = poly_approx_dd(dx);
 168
 169   // Error bounds: 2^-102.
 170   DoubleDouble r = fputil::quick_mult(exp_mid, p);
 171
 172   return r;
 173 }
 174
 175 // When output is denormal.
 176 double exp10_denorm(double x) {
 177   // Range reduction.
 178   double tmp = fputil::multiply_add(x, LOG2_10, 0x1.8000'0000'4p21);
 179   int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
 180   double kd = static_cast<double>(k);
 181
 182   uint32_t idx1 = (k >> 6) & 0x3f;
 183   uint32_t idx2 = k & 0x3f;
 184
 185   int hi = k >> 12;
 186
 187   DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
 188   DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
 189   DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
 190
 191   // |dx| < 1.5 * 2^-15 + 2^-31 < 2^-14
 192   double lo_h = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
 193   double dx = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_MID, lo_h);
 194
 195   double mid_lo = dx * exp_mid.hi;
 196
 197   // Approximate (10^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.
 198   double p = poly_approx_d(dx);
 199
 200   double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
 201
 202   if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
 203       LIBC_LIKELY(r.has_value()))
 204     return r.value();
 205
 206   // Use double-double
 207   DoubleDouble r_dd = exp10_double_double(x, kd, exp_mid);
 208
 209   if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
 210       LIBC_LIKELY(r.has_value()))
 211     return r.value();
 212
 213   // Use 128-bit precision
 214   Float128 r_f128 = exp10_f128(x, kd, idx1, idx2);
 215
 216   return static_cast<double>(r_f128);
 217 }
 218
 219 // Check for exceptional cases when:
 220 //  * log10(1 - 2^-54) < x < log10(1 + 2^-53)
 221 //  * x >= log10(2^1024)
 222 //  * x <= log10(2^-1022)
 223 //  * x is inf or nan
 224 double set_exceptional(double x) {
 225   using FPBits = typename fputil::FPBits<double>;
 226   FPBits xbits(x);
 227
 228   uint64_t x_u = xbits.uintval();
 229   uint64_t x_abs = xbits.abs().uintval();
 230
 231   // |x| < log10(1 + 2^-53)
 232   if (x_abs <= 0x3c8bcb7b1526e50e) {
 233     // 10^(x) ~ 1 + x/2
 234     return fputil::multiply_add(x, 0.5, 1.0);
 235   }
 236
 237   // x <= log10(2^-1022) || x >= log10(2^1024) or inf/nan.
 238   if (x_u >= 0xc0733a7146f72a42) {
 239     // x <= log10(2^-1075) or -inf/nan
 240     if (x_u > 0xc07439b746e36b52) {
 241       // exp(-Inf) = 0
 242       if (xbits.is_inf())
 243         return 0.0;
 244
 245       // exp(nan) = nan
 246       if (xbits.is_nan())
 247         return x;
 248
 249       if (fputil::quick_get_round() == FE_UPWARD)
 250         return FPBits::min_subnormal().get_val();
 251       fputil::set_errno_if_required(ERANGE);
 252       fputil::raise_except_if_required(FE_UNDERFLOW);
 253       return 0.0;
 254     }
 255
 256     return exp10_denorm(x);
 257   }
 258
 259   // x >= log10(2^1024) or +inf/nan
 260   // x is finite
 261   if (x_u < 0x7ff0'0000'0000'0000ULL) {
 262     int rounding = fputil::quick_get_round();
 263     if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
 264       return FPBits::max_normal().get_val();
 265
 266     fputil::set_errno_if_required(ERANGE);
 267     fputil::raise_except_if_required(FE_OVERFLOW);
 268   }
 269   // x is +inf or nan
 270   return x + FPBits::inf().get_val();
 271 }
 272
 273 } // namespace
 274
 275 LLVM_LIBC_FUNCTION(double, exp10, (double x)) {
 276   using FPBits = typename fputil::FPBits<double>;
 277   FPBits xbits(x);
 278
 279   uint64_t x_u = xbits.uintval();
 280
 281   // x <= log10(2^-1022) or x >= log10(2^1024) or
 282   // log10(1 - 2^-54) < x < log10(1 + 2^-53).
 283   if (LIBC_UNLIKELY(x_u >= 0xc0733a7146f72a42 ||
 284                     (x_u <= 0xbc7bcb7b1526e50e && x_u >= 0x40734413509f79ff) ||
 285                     x_u < 0x3c8bcb7b1526e50e)) {
 286     return set_exceptional(x);
 287   }
 288
 289   // Now log10(2^-1075) < x <= log10(1 - 2^-54) or
 290   //     log10(1 + 2^-53) < x < log10(2^1024)
 291
 292   // Range reduction:
 293   // Let x = log10(2) * (hi + mid1 + mid2) + lo
 294   // in which:
 295   //   hi is an integer
 296   //   mid1 * 2^6 is an integer
 297   //   mid2 * 2^12 is an integer
 298   // then:
 299   //   10^(x) = 2^hi * 2^(mid1) * 2^(mid2) * 10^(lo).
 300   // With this formula:
 301   //   - multiplying by 2^hi is exact and cheap, simply by adding the exponent
 302   //     field.
 303   //   - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
 304   //   - 10^(lo) ~ 1 + a0*lo + a1 * lo^2 + ...
 305   //
 306   // We compute (hi + mid1 + mid2) together by perform the rounding on
 307   //   x * log2(10) * 2^12.
 308   // Since |x| < |log10(2^-1075)| < 2^9,
 309   //   |x * 2^12| < 2^9 * 2^12 < 2^21,
 310   // So we can fit the rounded result round(x * 2^12) in int32_t.
 311   // Thus, the goal is to be able to use an additional addition and fixed width
 312   // shift to get an int32_t representing round(x * 2^12).
 313   //
 314   // Assuming int32_t using 2-complement representation, since the mantissa part
 315   // of a double precision is unsigned with the leading bit hidden, if we add an
 316   // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^23 to the product, the
 317   // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
 318   // considered as a proper 2-complement representations of x*2^12.
 319   //
 320   // One small problem with this approach is that the sum (x*2^12 + C) in
 321   // double precision is rounded to the least significant bit of the dorminant
 322   // factor C.  In order to minimize the rounding errors from this addition, we
 323   // want to minimize e1.  Another constraint that we want is that after
 324   // shifting the mantissa so that the least significant bit of int32_t
 325   // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
 326   // any adjustment.  So combining these 2 requirements, we can choose
 327   //   C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
 328   // after right shifting the mantissa, the resulting int32_t has correct sign.
 329   // With this choice of C, the number of mantissa bits we need to shift to the
 330   // right is: 52 - 33 = 19.
 331   //
 332   // Moreover, since the integer right shifts are equivalent to rounding down,
 333   // we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
 334   // +infinity.  So in particular, we can compute:
 335   //   hmm = x * 2^12 + C,
 336   // where C = 2^33 + 2^32 + 2^-1, then if
 337   //   k = int32_t(lower 51 bits of double(x * 2^12 + C) >> 19),
 338   // the reduced argument:
 339   //   lo = x - log10(2) * 2^-12 * k is bounded by:
 340   //   |lo|  = |x - log10(2) * 2^-12 * k|
 341   //         = log10(2) * 2^-12 * | x * log2(10) * 2^12 - k |
 342   //        <= log10(2) * 2^-12 * (2^-1 + 2^-19)
 343   //         < 1.5 * 2^-2 * (2^-13 + 2^-31)
 344   //         = 1.5 * (2^-15 * 2^-31)
 345   //
 346   // Finally, notice that k only uses the mantissa of x * 2^12, so the
 347   // exponent 2^12 is not needed.  So we can simply define
 348   //   C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
 349   //   k = int32_t(lower 51 bits of double(x + C) >> 19).
 350
 351   // Rounding errors <= 2^-31.
 352   double tmp = fputil::multiply_add(x, LOG2_10, 0x1.8000'0000'4p21);
 353   int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
 354   double kd = static_cast<double>(k);
 355
 356   uint32_t idx1 = (k >> 6) & 0x3f;
 357   uint32_t idx2 = k & 0x3f;
 358
 359   int hi = k >> 12;
 360
 361   DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
 362   DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
 363   DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
 364
 365   // |dx| < 1.5 * 2^-15 + 2^-31 < 2^-14
 366   double lo_h = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
 367   double dx = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_MID, lo_h);
 368
 369   // We use the degree-4 polynomial to approximate 10^(lo):
 370   //   10^(lo) ~ 1 + a0 * lo + a1 * lo^2 + a2 * lo^3 + a3 * lo^4
 371   //           = 1 + lo * P(lo)
 372   // So that the errors are bounded by:
 373   //   |P(lo) - (10^lo - 1)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
 374   // Let P_ be an evaluation of P where all intermediate computations are in
 375   // double precision.  Using either Horner's or Estrin's schemes, the evaluated
 376   // errors can be bounded by:
 377   //      |P_(lo) - P(lo)| < 2^-51
 378   //   => |lo * P_(lo) - (2^lo - 1) | < 2^-65
 379   //   => 2^(mid1 + mid2) * |lo * P_(lo) - expm1(lo)| < 2^-64.
 380   // Since we approximate
 381   //   2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
 382   // We use the expression:
 383   //    (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
 384   //  ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
 385   // with errors bounded by 2^-64.
 386
 387   double mid_lo = dx * exp_mid.hi;
 388
 389   // Approximate (10^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.
 390   double p = poly_approx_d(dx);
 391
 392   double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
 393
 394   double upper = exp_mid.hi + (lo + ERR_D);
 395   double lower = exp_mid.hi + (lo - ERR_D);
 396
 397   if (LIBC_LIKELY(upper == lower)) {
 398     // To multiply by 2^hi, a fast way is to simply add hi to the exponent
 399     // field.
 400     int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
 401     double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));
 402     return r;
 403   }
 404
 405   // Exact outputs when x = 1, 2, ..., 22 + hard to round with x = 23.
 406   // Quick check mask: 0x800f'ffffU = ~(bits of 1.0 | ... | bits of 23.0)
 407   if (LIBC_UNLIKELY((x_u & 0x8000'ffff'ffff'ffffULL) == 0ULL)) {
 408     switch (x_u) {
 409     case 0x3ff0000000000000: // x = 1.0
 410       return 10.0;
 411     case 0x4000000000000000: // x = 2.0
 412       return 100.0;
 413     case 0x4008000000000000: // x = 3.0
 414       return 1'000.0;
 415     case 0x4010000000000000: // x = 4.0
 416       return 10'000.0;
 417     case 0x4014000000000000: // x = 5.0
 418       return 100'000.0;
 419     case 0x4018000000000000: // x = 6.0
 420       return 1'000'000.0;
 421     case 0x401c000000000000: // x = 7.0
 422       return 10'000'000.0;
 423     case 0x4020000000000000: // x = 8.0
 424       return 100'000'000.0;
 425     case 0x4022000000000000: // x = 9.0
 426       return 1'000'000'000.0;
 427     case 0x4024000000000000: // x = 10.0
 428       return 10'000'000'000.0;
 429     case 0x4026000000000000: // x = 11.0
 430       return 100'000'000'000.0;
 431     case 0x4028000000000000: // x = 12.0
 432       return 1'000'000'000'000.0;
 433     case 0x402a000000000000: // x = 13.0
 434       return 10'000'000'000'000.0;
 435     case 0x402c000000000000: // x = 14.0
 436       return 100'000'000'000'000.0;
 437     case 0x402e000000000000: // x = 15.0
 438       return 1'000'000'000'000'000.0;
 439     case 0x4030000000000000: // x = 16.0
 440       return 10'000'000'000'000'000.0;
 441     case 0x4031000000000000: // x = 17.0
 442       return 100'000'000'000'000'000.0;
 443     case 0x4032000000000000: // x = 18.0
 444       return 1'000'000'000'000'000'000.0;
 445     case 0x4033000000000000: // x = 19.0
 446       return 10'000'000'000'000'000'000.0;
 447     case 0x4034000000000000: // x = 20.0
 448       return 100'000'000'000'000'000'000.0;
 449     case 0x4035000000000000: // x = 21.0
 450       return 1'000'000'000'000'000'000'000.0;
 451     case 0x4036000000000000: // x = 22.0
 452       return 10'000'000'000'000'000'000'000.0;
 453     case 0x4037000000000000: // x = 23.0
 454       return 0x1.52d02c7e14af6p76 + x;
 455     }
 456   }
 457
 458   // Use double-double
 459   DoubleDouble r_dd = exp10_double_double(x, kd, exp_mid);
 460
 461   double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);
 462   double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);
 463
 464   if (LIBC_LIKELY(upper_dd == lower_dd)) {
 465     // To multiply by 2^hi, a fast way is to simply add hi to the exponent
 466     // field.
 467     int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
 468     double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));
 469     return r;
 470   }
 471
 472   // Use 128-bit precision
 473   Float128 r_f128 = exp10_f128(x, kd, idx1, idx2);
 474
 475   return static_cast<double>(r_f128);
 476 }
 477
 478 } // namespace LIBC_NAMESPACE_DECL