94 lines · cpp
1//===-- Implementation of exphk 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 "exphk.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 89 do {22// hi = floor(i/8) - 5;23// m = i/8 - floor(i/8) - 0.5;24// e_hi = nearestint(exp(hi) * 2^7) * 2^-7;25// e_mid = nearestint(exp(m) * 2^7) * 2^-7;26// print(hi, e_hi, m, e_mid);27// };28// Notice that when i = 88 and 89, e_hi will overflow short accum range.29static constexpr short accum EXP_HI[12] = {30 0x1.0p-7hk, 0x1.0p-6hk, 0x1.8p-5hk, 0x1.1p-3hk, 0x1.78p-2hk, 0x1.0p0hk,31 0x1.5cp1hk, 0x1.d9p2hk, 0x1.416p4hk, 0x1.b4dp5hk, 0x1.28d4p7hk, SACCUM_MAX,32};33 34static constexpr short accum EXP_MID[8] = {35 0x1.38p-1hk, 0x1.6p-1hk, 0x1.9p-1hk, 0x1.c4p-1hk,36 0x1.0p0hk, 0x1.22p0hk, 0x1.48p0hk, 0x1.74p0hk,37};38 39} // anonymous namespace40 41LLVM_LIBC_FUNCTION(short accum, exphk, (short accum x)) {42 using FXRep = fixed_point::FXRep<short accum>;43 using StorageType = typename FXRep::StorageType;44 // Output overflow45 if (LIBC_UNLIKELY(x >= 0x1.64p2hk))46 return FXRep::MAX();47 // Lower bound where exp(x) -> 0:48 // floor(log(2^-8) * 2^7) * 2^-749 if (LIBC_UNLIKELY(x <= -0x1.63p2hk))50 return FXRep::ZERO();51 52 // Current range of x:53 // -0x1.628p2 <= x <= 0x1.638p254 // Range reduction:55 // x = hi + mid + lo,56 // where:57 // hi is an integer58 // mid * 2^3 is an integer59 // |lo| <= 2^-4.60 // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo)61 // ~ exp(hi) * exp(mid) * (1 + lo)62 // with relative errors < |lo|^2 <= 2^-8.63 // exp(hi) and exp(mid) are extracted from small lookup tables.64 65 // Round-to-nearest 1/8, tie-to-(+Int):66 constexpr short accum ONE_SIXTEENTH = 0x1.0p-4hk;67 // x_rounded = floor(x + 1/16).68 short accum x_rounded = ((x + ONE_SIXTEENTH) >> (FXRep::FRACTION_LEN - 3))69 << (FXRep::FRACTION_LEN - 3);70 short accum lo = x - x_rounded;71 72 // Range of x_rounded:73 // x_rounded >= floor((-0x1.628p2 + 0x1.0p-4) * 2^3) * 2^-374 // = -0x1.6p2 = -5.575 // To get the indices, we shift the values so that it start with 0.76 // Range of indices: 0 <= indices <= 8977 StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.6p2hk) >>78 (FXRep::FRACTION_LEN - 3));79 // So we have the following relation:80 // indices = (hi + mid + 44/8) * 881 // That implies:82 // hi + mid = indices/8 - 5.583 // So for lookup tables, we can use the upper 4 bits to get:84 // exp( floor(indices / 8) - 5 )85 // and lower 3 bits for:86 // exp( (indices - floor(indices)) - 0.5 )87 short accum exp_hi = EXP_HI[indices >> 3];88 short accum exp_mid = EXP_MID[indices & 0x7];89 // exp(x) ~ exp(hi) * exp(mid) * (1 + lo);90 return (exp_hi * (exp_mid * (0x1.0p0hk + lo)));91}92 93} // namespace LIBC_NAMESPACE_DECL94