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 479