106 lines · cpp
1//===-- Implementation of expk 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-exception6//7//===----------------------------------------------------------------------===//8 9#include "expk.h"10#include "src/__support/CPP/bit.h"11#include "src/__support/common.h"12#include "src/__support/fixed_point/fx_bits.h"13#include "src/__support/macros/config.h"14 15namespace LIBC_NAMESPACE_DECL {16 17namespace {18 19// Look up tables for exp(hi) and exp(mid).20// Generated with Sollya:21// > for i from 0 to 23 do {22// hi = i - 11;23// e_hi = nearestint(exp(hi) * 2^15) * 2^-15;24// print(e_hi, "k,");25// };26static constexpr accum EXP_HI[24] = {27 0x1p-15k, 0x1p-15k, 0x1p-13k, 0x1.6p-12k,28 0x1.ep-11k, 0x1.44p-9k, 0x1.bap-8k, 0x1.2cp-6k,29 0x1.97cp-5k, 0x1.153p-3k, 0x1.78b8p-2k, 0x1p0k,30 0x1.5bf1p1k, 0x1.d8e68p2k, 0x1.415e6p4k, 0x1.b4c9p5k,31 0x1.28d388p7k, 0x1.936dc6p8k, 0x1.1228858p10k, 0x1.749ea7cp11k,32 0x1.fa7157cp12k, 0x1.5829dcf8p14k, 0x1.d3c4489p15k, ACCUM_MAX,33};34 35// Generated with Sollya:36// > for i from 0 to 15 do {37// m = i/16 - 0.0625;38// e_m = nearestint(exp(m) * 2^15) * 2^-15;39// print(e_m, "k,");40// };41static constexpr accum EXP_MID[16] = {42 0x1.e0fcp-1k, 0x1p0k, 0x1.1082p0k, 0x1.2216p0k,43 0x1.34ccp0k, 0x1.48b6p0k, 0x1.5deap0k, 0x1.747ap0k,44 0x1.8c8p0k, 0x1.a612p0k, 0x1.c14cp0k, 0x1.de46p0k,45 0x1.fd1ep0k, 0x1.0efap1k, 0x1.2074p1k, 0x1.330ep1k,46};47 48} // anonymous namespace49 50LLVM_LIBC_FUNCTION(accum, expk, (accum x)) {51 using FXRep = fixed_point::FXRep<accum>;52 using StorageType = typename FXRep::StorageType;53 // Output overflow54 // > floor(log(2^16) * 2^15) * 2^-1555 if (LIBC_UNLIKELY(x >= 0x1.62e4p3k))56 return FXRep::MAX();57 // Lower bound where exp(x) -> 0:58 // floor(log(2^-16) * 2^15) * 2^-1559 if (LIBC_UNLIKELY(x <= -0x1.62e44p3k))60 return FXRep::ZERO();61 62 // Current range of x:63 // -0x1.62e4p3 <= x <= 0x1.62e3cp364 // Range reduction:65 // x = hi + mid + lo,66 // where:67 // hi is an integer68 // mid * 2^4 is an integer69 // |lo| <= 2^-5.70 // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo)71 // ~ exp(hi) * exp(mid) * (1 + lo + lo^2 / 2)72 // with relative errors < |lo|^3/2 <= 2^-16.73 // exp(hi) and exp(mid) are extracted from small lookup tables.74 75 // Round-to-nearest 1/16, tie-to-(+Int):76 constexpr accum ONE_THIRTY_SECOND = 0x1.0p-5k;77 // x_rounded = floor(x + 1/16).78 accum x_rounded = ((x + ONE_THIRTY_SECOND) >> (FXRep::FRACTION_LEN - 4))79 << (FXRep::FRACTION_LEN - 4);80 accum lo = x - x_rounded;81 82 // Range of x_rounded:83 // x_rounded >= floor((-0x1.62e4p3 + 0x1.0p-5) * 2^4) * 2^-484 // = -0x1.62p3 = -11.062585 // To get the indices, we shift the values so that it start with 0.86 // Range of indices: 0 <= indices <= 355.87 StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.62p3k) >>88 (FXRep::FRACTION_LEN - 4));89 // So we have the following relation:90 // indices = (hi + mid + 177/16) * 1691 // That implies:92 // hi + mid = indices/16 - 11.062593 // So for lookup tables, we can use the upper 4 bits to get:94 // exp( floor(indices / 16) - 11 )95 // and lower 4 bits for:96 // exp( (indices - floor(indices)) - 0.0625 )97 accum exp_hi = EXP_HI[indices >> 4];98 accum exp_mid = EXP_MID[indices & 0xf];99 // exp(x) ~ exp(hi) * exp(mid) * (1 + lo);100 accum l1 = 0x1.0p0k + (lo >> 1); // = 1 + lo / 2101 accum l2 = 0x1.0p0k + lo * l1; // = 1 + lo * (1 + lo / 2) = 1 + lo + lo^2/2102 return (exp_hi * (exp_mid * l2));103}104 105} // namespace LIBC_NAMESPACE_DECL106