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1//===-- Implementation header for exp2 --------------------------*- C++ -*-===//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-exception6//7//===----------------------------------------------------------------------===//8 9#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_EXP2_H10#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP2_H11 12#include "common_constants.h" // Lookup tables EXP2_MID1 and EXP_M2.13#include "exp_constants.h"14#include "exp_utils.h" // ziv_test_denorm.15#include "src/__support/CPP/bit.h"16#include "src/__support/CPP/optional.h"17#include "src/__support/FPUtil/FEnvImpl.h"18#include "src/__support/FPUtil/FPBits.h"19#include "src/__support/FPUtil/PolyEval.h"20#include "src/__support/FPUtil/double_double.h"21#include "src/__support/FPUtil/dyadic_float.h"22#include "src/__support/FPUtil/multiply_add.h"23#include "src/__support/FPUtil/nearest_integer.h"24#include "src/__support/FPUtil/rounding_mode.h"25#include "src/__support/FPUtil/triple_double.h"26#include "src/__support/common.h"27#include "src/__support/integer_literals.h"28#include "src/__support/macros/config.h"29#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY30 31namespace LIBC_NAMESPACE_DECL {32 33namespace math {34 35namespace exp2_internal {36 37using namespace common_constants_internal;38 39using fputil::DoubleDouble;40using fputil::TripleDouble;41using Float128 = typename fputil::DyadicFloat<128>;42 43using LIBC_NAMESPACE::operator""_u128;44 45// Error bounds:46// Errors when using double precision.47#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE48constexpr double ERR_D = 0x1.0p-63;49#else50constexpr double ERR_D = 0x1.8p-63;51#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE52 53#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS54// Errors when using double-double precision.55constexpr double ERR_DD = 0x1.0p-100;56#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS57 58// Polynomial approximations with double precision. Generated by Sollya with:59// > P = fpminimax((2^x - 1)/x, 3, [|D...|], [-2^-13 - 2^-30, 2^-13 + 2^-30]);60// > P;61// Error bounds:62// | output - (2^dx - 1) / dx | < 1.5 * 2^-52.63LIBC_INLINE static double poly_approx_d(double dx) {64 // dx^265 double dx2 = dx * dx;66 double c0 =67 fputil::multiply_add(dx, 0x1.ebfbdff82c58ep-3, 0x1.62e42fefa39efp-1);68 double c1 =69 fputil::multiply_add(dx, 0x1.3b2aba7a95a89p-7, 0x1.c6b08e8fc0c0ep-5);70 double p = fputil::multiply_add(dx2, c1, c0);71 return p;72}73 74#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS75// Polynomial approximation with double-double precision. Generated by Solya76// with:77// > P = fpminimax((2^x - 1)/x, 5, [|DD...|], [-2^-13 - 2^-30, 2^-13 + 2^-30]);78// Error bounds:79// | output - 2^(dx) | < 2^-10180LIBC_INLINE static constexpr DoubleDouble81poly_approx_dd(const DoubleDouble &dx) {82 // Taylor polynomial.83 constexpr DoubleDouble COEFFS[] = {84 {0, 0x1p0},85 {0x1.abc9e3b39824p-56, 0x1.62e42fefa39efp-1},86 {-0x1.5e43a53e4527bp-57, 0x1.ebfbdff82c58fp-3},87 {-0x1.d37963a9444eep-59, 0x1.c6b08d704a0cp-5},88 {0x1.4eda1a81133dap-62, 0x1.3b2ab6fba4e77p-7},89 {-0x1.c53fd1ba85d14p-64, 0x1.5d87fe7a265a5p-10},90 {0x1.d89250b013eb8p-70, 0x1.430912f86cb8ep-13},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^7100// For |dx| < 2^-13 + 2^-30:101// | output - exp(dx) | < 2^-126.102LIBC_INLINE static constexpr Float128 poly_approx_f128(const Float128 &dx) {103 constexpr Float128 COEFFS_128[]{104 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0105 {Sign::POS, -128, 0xb17217f7'd1cf79ab'c9e3b398'03f2f6af_u128},106 {Sign::POS, -128, 0x3d7f7bff'058b1d50'de2d60dd'9c9a1d9f_u128},107 {Sign::POS, -132, 0xe35846b8'2505fc59'9d3b15d9'e7fb6897_u128},108 {Sign::POS, -134, 0x9d955b7d'd273b94e'184462f6'bcd2b9e7_u128},109 {Sign::POS, -137, 0xaec3ff3c'53398883'39ea1bb9'64c51a89_u128},110 {Sign::POS, -138, 0x2861225f'345c396a'842c5341'8fa8ae61_u128},111 {Sign::POS, -144, 0xffe5fe2d'109a319d'7abeb5ab'd5ad2079_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 2^(x) using 128-bit precision.121// TODO(lntue): investigate triple-double precision implementation for this122// step.123LIBC_INLINE static constexpr Float128 exp2_f128(double x, int hi, int idx1,124 int idx2) {125 Float128 dx = Float128(x);126 127 // TODO: Skip recalculating exp_mid1 and exp_mid2.128 Float128 exp_mid1 =129 fputil::quick_add(Float128(EXP2_MID1[idx1].hi),130 fputil::quick_add(Float128(EXP2_MID1[idx1].mid),131 Float128(EXP2_MID1[idx1].lo)));132 133 Float128 exp_mid2 =134 fputil::quick_add(Float128(EXP2_MID2[idx2].hi),135 fputil::quick_add(Float128(EXP2_MID2[idx2].mid),136 Float128(EXP2_MID2[idx2].lo)));137 138 Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);139 140 Float128 p = poly_approx_f128(dx);141 142 Float128 r = fputil::quick_mul(exp_mid, p);143 144 r.exponent += hi;145 146 return r;147}148 149// Compute 2^x with double-double precision.150LIBC_INLINE static DoubleDouble151exp2_double_double(double x, const DoubleDouble &exp_mid) {152 DoubleDouble dx({0, x});153 154 // Degree-6 polynomial approximation in double-double precision.155 // | p - 2^x | < 2^-103.156 DoubleDouble p = poly_approx_dd(dx);157 158 // Error bounds: 2^-102.159 DoubleDouble r = fputil::quick_mult(exp_mid, p);160 161 return r;162}163#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS164 165// When output is denormal.166LIBC_INLINE static double exp2_denorm(double x) {167 // Range reduction.168 int k =169 static_cast<int>(cpp::bit_cast<uint64_t>(x + 0x1.8000'0000'4p21) >> 19);170 double kd = static_cast<double>(k);171 172 uint32_t idx1 = (k >> 6) & 0x3f;173 uint32_t idx2 = k & 0x3f;174 175 int hi = k >> 12;176 177 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};178 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};179 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);180 181 // |dx| < 2^-13 + 2^-30.182 double dx = fputil::multiply_add(kd, -0x1.0p-12, x); // exact183 184 double mid_lo = dx * exp_mid.hi;185 186 // Approximate (2^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.187 double p = poly_approx_d(dx);188 189 double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);190 191#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS192 return ziv_test_denorm</*SKIP_ZIV_TEST=*/true>(hi, exp_mid.hi, lo, ERR_D)193 .value();194#else195 if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);196 LIBC_LIKELY(r.has_value()))197 return r.value();198 199 // Use double-double200 DoubleDouble r_dd = exp2_double_double(dx, exp_mid);201 202 if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);203 LIBC_LIKELY(r.has_value()))204 return r.value();205 206 // Use 128-bit precision207 Float128 r_f128 = exp2_f128(dx, hi, idx1, idx2);208 209 return static_cast<double>(r_f128);210#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS211}212 213// Check for exceptional cases when:214// * log2(1 - 2^-54) < x < log2(1 + 2^-53)215// * x >= 1024216// * x <= -1022217// * x is inf or nan218LIBC_INLINE static constexpr double set_exceptional(double x) {219 using FPBits = typename fputil::FPBits<double>;220 FPBits xbits(x);221 222 uint64_t x_u = xbits.uintval();223 uint64_t x_abs = xbits.abs().uintval();224 225 // |x| < log2(1 + 2^-53)226 if (x_abs <= 0x3ca71547652b82fd) {227 // 2^(x) ~ 1 + x/2228 return fputil::multiply_add(x, 0.5, 1.0);229 }230 231 // x <= -1022 || x >= 1024 or inf/nan.232 if (x_u > 0xc08ff00000000000) {233 // x <= -1075 or -inf/nan234 if (x_u >= 0xc090cc0000000000) {235 // exp(-Inf) = 0236 if (xbits.is_inf())237 return 0.0;238 239 // exp(nan) = nan240 if (xbits.is_nan())241 return x;242 243 if (fputil::quick_get_round() == FE_UPWARD)244 return FPBits::min_subnormal().get_val();245 fputil::set_errno_if_required(ERANGE);246 fputil::raise_except_if_required(FE_UNDERFLOW);247 return 0.0;248 }249 250 return exp2_denorm(x);251 }252 253 // x >= 1024 or +inf/nan254 // x is finite255 if (x_u < 0x7ff0'0000'0000'0000ULL) {256 int rounding = fputil::quick_get_round();257 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)258 return FPBits::max_normal().get_val();259 260 fputil::set_errno_if_required(ERANGE);261 fputil::raise_except_if_required(FE_OVERFLOW);262 }263 // x is +inf or nan264 return x + FPBits::inf().get_val();265}266 267} // namespace exp2_internal268 269LIBC_INLINE static constexpr double exp2(double x) {270 using namespace exp2_internal;271 using FPBits = typename fputil::FPBits<double>;272 FPBits xbits(x);273 274 uint64_t x_u = xbits.uintval();275 276 // x < -1022 or x >= 1024 or log2(1 - 2^-54) < x < log2(1 + 2^-53).277 if (LIBC_UNLIKELY(x_u > 0xc08ff00000000000 ||278 (x_u <= 0xbc971547652b82fe && x_u >= 0x4090000000000000) ||279 x_u <= 0x3ca71547652b82fd)) {280 return set_exceptional(x);281 }282 283 // Now -1075 < x <= log2(1 - 2^-54) or log2(1 + 2^-53) < x < 1024284 285 // Range reduction:286 // Let x = (hi + mid1 + mid2) + lo287 // in which:288 // hi is an integer289 // mid1 * 2^6 is an integer290 // mid2 * 2^12 is an integer291 // then:292 // 2^(x) = 2^hi * 2^(mid1) * 2^(mid2) * 2^(lo).293 // With this formula:294 // - multiplying by 2^hi is exact and cheap, simply by adding the exponent295 // field.296 // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.297 // - 2^(lo) ~ 1 + a0*lo + a1 * lo^2 + ...298 //299 // We compute (hi + mid1 + mid2) together by perform the rounding on x * 2^12.300 // Since |x| < |-1075)| < 2^11,301 // |x * 2^12| < 2^11 * 2^12 < 2^23,302 // So we can fit the rounded result round(x * 2^12) in int32_t.303 // Thus, the goal is to be able to use an additional addition and fixed width304 // shift to get an int32_t representing round(x * 2^12).305 //306 // Assuming int32_t using 2-complement representation, since the mantissa part307 // of a double precision is unsigned with the leading bit hidden, if we add an308 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the309 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be310 // considered as a proper 2-complement representations of x*2^12.311 //312 // One small problem with this approach is that the sum (x*2^12 + C) in313 // double precision is rounded to the least significant bit of the dorminant314 // factor C. In order to minimize the rounding errors from this addition, we315 // want to minimize e1. Another constraint that we want is that after316 // shifting the mantissa so that the least significant bit of int32_t317 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without318 // any adjustment. So combining these 2 requirements, we can choose319 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence320 // after right shifting the mantissa, the resulting int32_t has correct sign.321 // With this choice of C, the number of mantissa bits we need to shift to the322 // right is: 52 - 33 = 19.323 //324 // Moreover, since the integer right shifts are equivalent to rounding down,325 // we can add an extra 0.5 so that it will become round-to-nearest, tie-to-326 // +infinity. So in particular, we can compute:327 // hmm = x * 2^12 + C,328 // where C = 2^33 + 2^32 + 2^-1, then if329 // k = int32_t(lower 51 bits of double(x * 2^12 + C) >> 19),330 // the reduced argument:331 // lo = x - 2^-12 * k is bounded by:332 // |lo| <= 2^-13 + 2^-12*2^-19333 // = 2^-13 + 2^-31.334 //335 // Finally, notice that k only uses the mantissa of x * 2^12, so the336 // exponent 2^12 is not needed. So we can simply define337 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and338 // k = int32_t(lower 51 bits of double(x + C) >> 19).339 340 // Rounding errors <= 2^-31.341 int k =342 static_cast<int>(cpp::bit_cast<uint64_t>(x + 0x1.8000'0000'4p21) >> 19);343 double kd = static_cast<double>(k);344 345 uint32_t idx1 = (k >> 6) & 0x3f;346 uint32_t idx2 = k & 0x3f;347 348 int hi = k >> 12;349 350 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};351 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};352 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);353 354 // |dx| < 2^-13 + 2^-30.355 double dx = fputil::multiply_add(kd, -0x1.0p-12, x); // exact356 357 // We use the degree-4 polynomial to approximate 2^(lo):358 // 2^(lo) ~ 1 + a0 * lo + a1 * lo^2 + a2 * lo^3 + a3 * lo^4 = 1 + lo * P(lo)359 // So that the errors are bounded by:360 // |P(lo) - (2^lo - 1)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58361 // Let P_ be an evaluation of P where all intermediate computations are in362 // double precision. Using either Horner's or Estrin's schemes, the evaluated363 // errors can be bounded by:364 // |P_(lo) - P(lo)| < 2^-51365 // => |lo * P_(lo) - (2^lo - 1) | < 2^-64366 // => 2^(mid1 + mid2) * |lo * P_(lo) - expm1(lo)| < 2^-63.367 // Since we approximate368 // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,369 // We use the expression:370 // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~371 // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)372 // with errors bounded by 2^-63.373 374 double mid_lo = dx * exp_mid.hi;375 376 // Approximate (2^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.377 double p = poly_approx_d(dx);378 379 double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);380 381#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS382 // To multiply by 2^hi, a fast way is to simply add hi to the exponent383 // field.384 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;385 double r =386 cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(exp_mid.hi + lo));387 return r;388#else389 double upper = exp_mid.hi + (lo + ERR_D);390 double lower = exp_mid.hi + (lo - ERR_D);391 392 if (LIBC_LIKELY(upper == lower)) {393 // To multiply by 2^hi, a fast way is to simply add hi to the exponent394 // field.395 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;396 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));397 return r;398 }399 400 // Use double-double401 DoubleDouble r_dd = exp2_double_double(dx, exp_mid);402 403 double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);404 double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);405 406 if (LIBC_LIKELY(upper_dd == lower_dd)) {407 // To multiply by 2^hi, a fast way is to simply add hi to the exponent408 // field.409 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;410 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));411 return r;412 }413 414 // Use 128-bit precision415 Float128 r_f128 = exp2_f128(dx, hi, idx1, idx2);416 417 return static_cast<double>(r_f128);418#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS419}420 421} // namespace math422 423} // namespace LIBC_NAMESPACE_DECL424 425#endif // LLVM_LIBC_SRC___SUPPORT_MATH_EXP2_H426