libc/src/math/generic/explogxf.h

   1 //===-- Single-precision general exp/log functions ------------------------===//
   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 #ifndef LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H
  10 #define LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H
  11
  12 #include "common_constants.h"
  13 #include "src/__support/CPP/bit.h"
  14 #include "src/__support/CPP/optional.h"
  15 #include "src/__support/FPUtil/FEnvImpl.h"
  16 #include "src/__support/FPUtil/FPBits.h"
  17 #include "src/__support/FPUtil/PolyEval.h"
  18 #include "src/__support/FPUtil/nearest_integer.h"
  19 #include "src/__support/common.h"
  20 #include "src/__support/macros/config.h"
  21 #include "src/__support/macros/properties/cpu_features.h"
  22
  23 namespace LIBC_NAMESPACE_DECL {
  24
  25 struct ExpBase {
  26   // Base = e
  27   static constexpr int MID_BITS = 5;
  28   static constexpr int MID_MASK = (1 << MID_BITS) - 1;
  29   // log2(e) * 2^5
  30   static constexpr double LOG2_B = 0x1.71547652b82fep+0 * (1 << MID_BITS);
  31   // High and low parts of -log(2) * 2^(-5)
  32   static constexpr double M_LOGB_2_HI = -0x1.62e42fefa0000p-1 / (1 << MID_BITS);
  33   static constexpr double M_LOGB_2_LO =
  34       -0x1.cf79abc9e3b3ap-40 / (1 << MID_BITS);
  35   // Look up table for bit fields of 2^(i/32) for i = 0..31, generated by Sollya
  36   // with:
  37   // > for i from 0 to 31 do printdouble(round(2^(i/32), D, RN));
  38   static constexpr int64_t EXP_2_MID[1 << MID_BITS] = {
  39       0x3ff0000000000000, 0x3ff059b0d3158574, 0x3ff0b5586cf9890f,
  40       0x3ff11301d0125b51, 0x3ff172b83c7d517b, 0x3ff1d4873168b9aa,
  41       0x3ff2387a6e756238, 0x3ff29e9df51fdee1, 0x3ff306fe0a31b715,
  42       0x3ff371a7373aa9cb, 0x3ff3dea64c123422, 0x3ff44e086061892d,
  43       0x3ff4bfdad5362a27, 0x3ff5342b569d4f82, 0x3ff5ab07dd485429,
  44       0x3ff6247eb03a5585, 0x3ff6a09e667f3bcd, 0x3ff71f75e8ec5f74,
  45       0x3ff7a11473eb0187, 0x3ff82589994cce13, 0x3ff8ace5422aa0db,
  46       0x3ff93737b0cdc5e5, 0x3ff9c49182a3f090, 0x3ffa5503b23e255d,
  47       0x3ffae89f995ad3ad, 0x3ffb7f76f2fb5e47, 0x3ffc199bdd85529c,
  48       0x3ffcb720dcef9069, 0x3ffd5818dcfba487, 0x3ffdfc97337b9b5f,
  49       0x3ffea4afa2a490da, 0x3fff50765b6e4540,
  50   };
  51
  52   // Approximating e^dx with degree-5 minimax polynomial generated by Sollya:
  53   // > Q = fpminimax(expm1(x)/x, 4, [|1, D...|], [-log(2)/64, log(2)/64]);
  54   // Then:
  55   //   e^dx ~ P(dx) = 1 + dx + COEFFS[0] * dx^2 + ... + COEFFS[3] * dx^5.
  56   static constexpr double COEFFS[4] = {
  57       0x1.ffffffffe5bc8p-2, 0x1.555555555cd67p-3, 0x1.5555c2a9b48b4p-5,
  58       0x1.11112a0e34bdbp-7};
  59
  60   LIBC_INLINE static double powb_lo(double dx) {
  61     using fputil::multiply_add;
  62     double dx2 = dx * dx;
  63     double c0 = 1.0 + dx;
  64     // c1 = COEFFS[0] + COEFFS[1] * dx
  65     double c1 = multiply_add(dx, ExpBase::COEFFS[1], ExpBase::COEFFS[0]);
  66     // c2 = COEFFS[2] + COEFFS[3] * dx
  67     double c2 = multiply_add(dx, ExpBase::COEFFS[3], ExpBase::COEFFS[2]);
  68     // r = c4 + c5 * dx^4
  69     //   = 1 + dx + COEFFS[0] * dx^2 + ... + COEFFS[5] * dx^7
  70     return fputil::polyeval(dx2, c0, c1, c2);
  71   }
  72 };
  73
  74 struct Exp10Base : public ExpBase {
  75   // log2(10) * 2^5
  76   static constexpr double LOG2_B = 0x1.a934f0979a371p1 * (1 << MID_BITS);
  77   // High and low parts of -log10(2) * 2^(-5).
  78   // Notice that since |x * log2(10)| < 150:
  79   //   |k| = |round(x * log2(10) * 2^5)| < 2^8 * 2^5 = 2^13
  80   // So when the FMA instructions are not available, in order for the product
  81   //   k * M_LOGB_2_HI
  82   // to be exact, we only store the high part of log10(2) up to 38 bits
  83   // (= 53 - 15) of precision.
  84   // It is generated by Sollya with:
  85   // > round(log10(2), 44, RN);
  86   static constexpr double M_LOGB_2_HI = -0x1.34413509f8p-2 / (1 << MID_BITS);
  87   // > round(log10(2) - 0x1.34413509f8p-2, D, RN);
  88   static constexpr double M_LOGB_2_LO = 0x1.80433b83b532ap-44 / (1 << MID_BITS);
  89
  90   // Approximating 10^dx with degree-5 minimax polynomial generated by Sollya:
  91   // > Q = fpminimax((10^x - 1)/x, 4, [|D...|], [-log10(2)/2^6, log10(2)/2^6]);
  92   // Then:
  93   //   10^dx ~ P(dx) = 1 + COEFFS[0] * dx + ... + COEFFS[4] * dx^5.
  94   static constexpr double COEFFS[5] = {0x1.26bb1bbb55515p1, 0x1.53524c73bd3eap1,
  95                                        0x1.0470591dff149p1, 0x1.2bd7c0a9fbc4dp0,
  96                                        0x1.1429e74a98f43p-1};
  97
  98   static double powb_lo(double dx) {
  99     using fputil::multiply_add;
 100     double dx2 = dx * dx;
 101     // c0 = 1 + COEFFS[0] * dx
 102     double c0 = multiply_add(dx, Exp10Base::COEFFS[0], 1.0);
 103     // c1 = COEFFS[1] + COEFFS[2] * dx
 104     double c1 = multiply_add(dx, Exp10Base::COEFFS[2], Exp10Base::COEFFS[1]);
 105     // c2 = COEFFS[3] + COEFFS[4] * dx
 106     double c2 = multiply_add(dx, Exp10Base::COEFFS[4], Exp10Base::COEFFS[3]);
 107     // r = c0 + dx^2 * (c1 + c2 * dx^2)
 108     //   = c0 + c1 * dx^2 + c2 * dx^4
 109     //   = 1 + COEFFS[0] * dx + ... + COEFFS[4] * dx^5.
 110     return fputil::polyeval(dx2, c0, c1, c2);
 111   }
 112 };
 113
 114 constexpr int LOG_P1_BITS = 6;
 115 constexpr int LOG_P1_SIZE = 1 << LOG_P1_BITS;
 116
 117 // N[Table[Log[2, 1 + x], {x, 0/64, 63/64, 1/64}], 40]
 118 extern const double LOG_P1_LOG2[LOG_P1_SIZE];
 119
 120 // N[Table[1/(1 + x), {x, 0/64, 63/64, 1/64}], 40]
 121 extern const double LOG_P1_1_OVER[LOG_P1_SIZE];
 122
 123 // Taylor series expansion for Log[2, 1 + x] splitted to EVEN AND ODD numbers
 124 // K_LOG2_ODD starts from x^3
 125 extern const double K_LOG2_ODD[4];
 126 extern const double K_LOG2_EVEN[4];
 127
 128 // Output of range reduction for exp_b: (2^(mid + hi), lo)
 129 // where:
 130 //   b^x = 2^(mid + hi) * b^lo
 131 struct exp_b_reduc_t {
 132   double mh; // 2^(mid + hi)
 133   double lo;
 134 };
 135
 136 // The function correctly calculates b^x value with at least float precision
 137 // in a limited range.
 138 // Range reduction:
 139 //   b^x = 2^(hi + mid) * b^lo
 140 // where:
 141 //   x = (hi + mid) * log_b(2) + lo
 142 //   hi is an integer,
 143 //   0 <= mid * 2^MID_BITS < 2^MID_BITS is an integer
 144 //   -2^(-MID_BITS - 1) <= lo * log2(b) <= 2^(-MID_BITS - 1)
 145 // Base class needs to provide the following constants:
 146 //   - MID_BITS    : number of bits after decimal points used for mid
 147 //   - MID_MASK    : 2^MID_BITS - 1, mask to extract mid bits
 148 //   - LOG2_B      : log2(b) * 2^MID_BITS for scaling
 149 //   - M_LOGB_2_HI : high part of -log_b(2) * 2^(-MID_BITS)
 150 //   - M_LOGB_2_LO : low part of -log_b(2) * 2^(-MID_BITS)
 151 //   - EXP_2_MID   : look up table for bit fields of 2^mid
 152 // Return:
 153 //   { 2^(hi + mid), lo }
 154 template <class Base> LIBC_INLINE exp_b_reduc_t exp_b_range_reduc(float x) {
 155   double xd = static_cast<double>(x);
 156   // kd = round((hi + mid) * log2(b) * 2^MID_BITS)
 157   double kd = fputil::nearest_integer(Base::LOG2_B * xd);
 158   // k = round((hi + mid) * log2(b) * 2^MID_BITS)
 159   int k = static_cast<int>(kd);
 160   // hi = floor(kd * 2^(-MID_BITS))
 161   // exp_hi = shift hi to the exponent field of double precision.
 162   uint64_t exp_hi = static_cast<uint64_t>(k >> Base::MID_BITS)
 163                     << fputil::FPBits<double>::FRACTION_LEN;
 164   // mh = 2^hi * 2^mid
 165   // mh_bits = bit field of mh
 166   uint64_t mh_bits = Base::EXP_2_MID[k & Base::MID_MASK] + exp_hi;
 167   double mh = fputil::FPBits<double>(mh_bits).get_val();
 168   // dx = lo = x - (hi + mid) * log(2)
 169   double dx = fputil::multiply_add(
 170       kd, Base::M_LOGB_2_LO, fputil::multiply_add(kd, Base::M_LOGB_2_HI, xd));
 171   return {mh, dx};
 172 }
 173
 174 // The function correctly calculates sinh(x) and cosh(x) by calculating exp(x)
 175 // and exp(-x) simultaneously.
 176 // To compute e^x, we perform the following range
 177 // reduction: find hi, mid, lo such that:
 178 //   x = (hi + mid) * log(2) + lo, in which
 179 //     hi is an integer,
 180 //     0 <= mid * 2^5 < 32 is an integer
 181 //     -2^(-6) <= lo * log2(e) <= 2^-6.
 182 // In particular,
 183 //   hi + mid = round(x * log2(e) * 2^5) * 2^(-5).
 184 // Then,
 185 //   e^x = 2^(hi + mid) * e^lo = 2^hi * 2^mid * e^lo.
 186 // 2^mid is stored in the lookup table of 32 elements.
 187 // e^lo is computed using a degree-5 minimax polynomial
 188 // generated by Sollya:
 189 //   e^lo ~ P(lo) = 1 + lo + c2 * lo^2 + ... + c5 * lo^5
 190 //        = (1 + c2*lo^2 + c4*lo^4) + lo * (1 + c3*lo^2 + c5*lo^4)
 191 //        = P_even + lo * P_odd
 192 // We perform 2^hi * 2^mid by simply add hi to the exponent field
 193 // of 2^mid.
 194 // To compute e^(-x), notice that:
 195 //   e^(-x) = 2^(-(hi + mid)) * e^(-lo)
 196 //          ~ 2^(-(hi + mid)) * P(-lo)
 197 //          = 2^(-(hi + mid)) * (P_even - lo * P_odd)
 198 // So:
 199 //   sinh(x) = (e^x - e^(-x)) / 2
 200 //           ~ 0.5 * (2^(hi + mid) * (P_even + lo * P_odd) -
 201 //                    2^(-(hi + mid)) * (P_even - lo * P_odd))
 202 //           = 0.5 * (P_even * (2^(hi + mid) - 2^(-(hi + mid))) +
 203 //                    lo * P_odd * (2^(hi + mid) + 2^(-(hi + mid))))
 204 // And similarly:
 205 //   cosh(x) = (e^x + e^(-x)) / 2
 206 //           ~ 0.5 * (P_even * (2^(hi + mid) + 2^(-(hi + mid))) +
 207 //                    lo * P_odd * (2^(hi + mid) - 2^(-(hi + mid))))
 208 // The main point of these formulas is that the expensive part of calculating
 209 // the polynomials approximating lower parts of e^(x) and e^(-x) are shared
 210 // and only done once.
 211 template <bool is_sinh> LIBC_INLINE double exp_pm_eval(float x) {
 212   double xd = static_cast<double>(x);
 213
 214   // kd = round(x * log2(e) * 2^5)
 215   // k_p = round(x * log2(e) * 2^5)
 216   // k_m = round(-x * log2(e) * 2^5)
 217   double kd;
 218   int k_p, k_m;
 219
 220 #ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
 221   kd = fputil::nearest_integer(ExpBase::LOG2_B * xd);
 222   k_p = static_cast<int>(kd);
 223   k_m = -k_p;
 224 #else
 225   constexpr double HALF_WAY[2] = {0.5, -0.5};
 226
 227   k_p = static_cast<int>(
 228       fputil::multiply_add(xd, ExpBase::LOG2_B, HALF_WAY[x < 0.0f]));
 229   k_m = -k_p;
 230   kd = static_cast<double>(k_p);
 231 #endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
 232
 233   // hi = floor(kf * 2^(-5))
 234   // exp_hi = shift hi to the exponent field of double precision.
 235   int64_t exp_hi_p = static_cast<int64_t>((k_p >> ExpBase::MID_BITS))
 236                      << fputil::FPBits<double>::FRACTION_LEN;
 237   int64_t exp_hi_m = static_cast<int64_t>((k_m >> ExpBase::MID_BITS))
 238                      << fputil::FPBits<double>::FRACTION_LEN;
 239   // mh_p = 2^(hi + mid)
 240   // mh_m = 2^(-(hi + mid))
 241   // mh_bits_* = bit field of mh_*
 242   int64_t mh_bits_p = ExpBase::EXP_2_MID[k_p & ExpBase::MID_MASK] + exp_hi_p;
 243   int64_t mh_bits_m = ExpBase::EXP_2_MID[k_m & ExpBase::MID_MASK] + exp_hi_m;
 244   double mh_p = fputil::FPBits<double>(uint64_t(mh_bits_p)).get_val();
 245   double mh_m = fputil::FPBits<double>(uint64_t(mh_bits_m)).get_val();
 246   // mh_sum = 2^(hi + mid) + 2^(-(hi + mid))
 247   double mh_sum = mh_p + mh_m;
 248   // mh_diff = 2^(hi + mid) - 2^(-(hi + mid))
 249   double mh_diff = mh_p - mh_m;
 250
 251   // dx = lo = x - (hi + mid) * log(2)
 252   double dx =
 253       fputil::multiply_add(kd, ExpBase::M_LOGB_2_LO,
 254                            fputil::multiply_add(kd, ExpBase::M_LOGB_2_HI, xd));
 255   double dx2 = dx * dx;
 256
 257   // c0 = 1 + COEFFS[0] * lo^2
 258   // P_even = (1 + COEFFS[0] * lo^2 + COEFFS[2] * lo^4) / 2
 259   double p_even = fputil::polyeval(dx2, 0.5, ExpBase::COEFFS[0] * 0.5,
 260                                    ExpBase::COEFFS[2] * 0.5);
 261   // P_odd = (1 + COEFFS[1] * lo^2 + COEFFS[3] * lo^4) / 2
 262   double p_odd = fputil::polyeval(dx2, 0.5, ExpBase::COEFFS[1] * 0.5,
 263                                   ExpBase::COEFFS[3] * 0.5);
 264
 265   double r;
 266   if constexpr (is_sinh)
 267     r = fputil::multiply_add(dx * mh_sum, p_odd, p_even * mh_diff);
 268   else
 269     r = fputil::multiply_add(dx * mh_diff, p_odd, p_even * mh_sum);
 270   return r;
 271 }
 272
 273 // x should be positive, normal finite value
 274 LIBC_INLINE static double log2_eval(double x) {
 275   using FPB = fputil::FPBits<double>;
 276   FPB bs(x);
 277
 278   double result = 0;
 279   result += bs.get_exponent();
 280
 281   int p1 = (bs.get_mantissa() >> (FPB::FRACTION_LEN - LOG_P1_BITS)) &
 282            (LOG_P1_SIZE - 1);
 283
 284   bs.set_uintval(bs.uintval() & (FPB::FRACTION_MASK >> LOG_P1_BITS));
 285   bs.set_biased_exponent(FPB::EXP_BIAS);
 286   double dx = (bs.get_val() - 1.0) * LOG_P1_1_OVER[p1];
 287
 288   // Taylor series for log(2,1+x)
 289   double c1 = fputil::multiply_add(dx, K_LOG2_ODD[0], K_LOG2_EVEN[0]);
 290   double c2 = fputil::multiply_add(dx, K_LOG2_ODD[1], K_LOG2_EVEN[1]);
 291   double c3 = fputil::multiply_add(dx, K_LOG2_ODD[2], K_LOG2_EVEN[2]);
 292   double c4 = fputil::multiply_add(dx, K_LOG2_ODD[3], K_LOG2_EVEN[3]);
 293
 294   // c0 = dx * (1.0 / ln(2)) + LOG_P1_LOG2[p1]
 295   double c0 = fputil::multiply_add(dx, 0x1.71547652b82fep+0, LOG_P1_LOG2[p1]);
 296   result += LIBC_NAMESPACE::fputil::polyeval(dx * dx, c0, c1, c2, c3, c4);
 297   return result;
 298 }
 299
 300 // x should be positive, normal finite value
 301 LIBC_INLINE static double log_eval(double x) {
 302   // For x = 2^ex * (1 + mx)
 303   //   log(x) = ex * log(2) + log(1 + mx)
 304   using FPB = fputil::FPBits<double>;
 305   FPB bs(x);
 306
 307   double ex = static_cast<double>(bs.get_exponent());
 308
 309   // p1 is the leading 7 bits of mx, i.e.
 310   // p1 * 2^(-7) <= m_x < (p1 + 1) * 2^(-7).
 311   int p1 = static_cast<int>(bs.get_mantissa() >> (FPB::FRACTION_LEN - 7));
 312
 313   // Set bs to (1 + (mx - p1*2^(-7))
 314   bs.set_uintval(bs.uintval() & (FPB::FRACTION_MASK >> 7));
 315   bs.set_biased_exponent(FPB::EXP_BIAS);
 316   // dx = (mx - p1*2^(-7)) / (1 + p1*2^(-7)).
 317   double dx = (bs.get_val() - 1.0) * ONE_OVER_F[p1];
 318
 319   // Minimax polynomial of log(1 + dx) generated by Sollya with:
 320   // > P = fpminimax(log(1 + x)/x, 6, [|D...|], [0, 2^-7]);
 321   const double COEFFS[6] = {-0x1.ffffffffffffcp-2, 0x1.5555555552ddep-2,
 322                             -0x1.ffffffefe562dp-3, 0x1.9999817d3a50fp-3,
 323                             -0x1.554317b3f67a5p-3, 0x1.1dc5c45e09c18p-3};
 324   double dx2 = dx * dx;
 325   double c1 = fputil::multiply_add(dx, COEFFS[1], COEFFS[0]);
 326   double c2 = fputil::multiply_add(dx, COEFFS[3], COEFFS[2]);
 327   double c3 = fputil::multiply_add(dx, COEFFS[5], COEFFS[4]);
 328
 329   double p = fputil::polyeval(dx2, dx, c1, c2, c3);
 330   double result =
 331       fputil::multiply_add(ex, /*log(2)*/ 0x1.62e42fefa39efp-1, LOG_F[p1] + p);
 332   return result;
 333 }
 334
 335 // Rounding tests for 2^hi * (mid + lo) when the output might be denormal. We
 336 // assume further that 1 <= mid < 2, mid + lo < 2, and |lo| << mid.
 337 // Notice that, if 0 < x < 2^-1022,
 338 //   double(2^-1022 + x) - 2^-1022 = double(x).
 339 // So if we scale x up by 2^1022, we can use
 340 //   double(1.0 + 2^1022 * x) - 1.0 to test how x is rounded in denormal range.
 341 LIBC_INLINE cpp::optional<double> ziv_test_denorm(int hi, double mid, double lo,
 342                                                   double err) {
 343   using FPBits = typename fputil::FPBits<double>;
 344
 345   // Scaling factor = 1/(min normal number) = 2^1022
 346   int64_t exp_hi = static_cast<int64_t>(hi + 1022) << FPBits::FRACTION_LEN;
 347   double mid_hi = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(mid));
 348   double lo_scaled =
 349       (lo != 0.0) ? cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(lo))
 350                   : 0.0;
 351
 352   double extra_factor = 0.0;
 353   uint64_t scale_down = 0x3FE0'0000'0000'0000; // 1022 in the exponent field.
 354
 355   // Result is denormal if (mid_hi + lo_scale < 1.0).
 356   if ((1.0 - mid_hi) > lo_scaled) {
 357     // Extra rounding step is needed, which adds more rounding errors.
 358     err += 0x1.0p-52;
 359     extra_factor = 1.0;
 360     scale_down = 0x3FF0'0000'0000'0000; // 1023 in the exponent field.
 361   }
 362
 363   double err_scaled =
 364       cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(err));
 365
 366   double lo_u = lo_scaled + err_scaled;
 367   double lo_l = lo_scaled - err_scaled;
 368
 369   // By adding 1.0, the results will have similar rounding points as denormal
 370   // outputs.
 371   double upper = extra_factor + (mid_hi + lo_u);
 372   double lower = extra_factor + (mid_hi + lo_l);
 373
 374   if (LIBC_LIKELY(upper == lower)) {
 375     return cpp::bit_cast<double>(cpp::bit_cast<uint64_t>(upper) - scale_down);
 376   }
 377
 378   return cpp::nullopt;
 379 }
 380
 381 } // namespace LIBC_NAMESPACE_DECL
 382
 383 #endif // LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H