llvm-project

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//===-- Implementation of exphk function ----------------------------------===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#include "exphk.h"
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#include "src/__support/CPP/bit.h"
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#include "src/__support/common.h"
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#include "src/__support/fixed_point/fx_bits.h"
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namespace LIBC_NAMESPACE {
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namespace {
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// Look up tables for exp(hi) and exp(mid).
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// Generated with Sollya:
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// > for i from 0 to 89 do {
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//     hi = floor(i/8) - 5;
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//     m = i/8 - floor(i/8) - 0.5;
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//     e_hi = nearestint(exp(hi) * 2^7) * 2^-7;
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//     e_mid = nearestint(exp(m) * 2^7) * 2^-7;
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//     print(hi, e_hi, m, e_mid);
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//   };
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// Notice that when i = 88 and 89, e_hi will overflow short accum range.
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static constexpr short accum EXP_HI[12] = {
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    0x1.0p-7hk, 0x1.0p-6hk, 0x1.8p-5hk,  0x1.1p-3hk,  0x1.78p-2hk,  0x1.0p0hk,
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    0x1.5cp1hk, 0x1.d9p2hk, 0x1.416p4hk, 0x1.b4dp5hk, 0x1.28d4p7hk, SACCUM_MAX,
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};
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static constexpr short accum EXP_MID[8] = {
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    0x1.38p-1hk, 0x1.6p-1hk, 0x1.9p-1hk, 0x1.c4p-1hk,
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    0x1.0p0hk,   0x1.22p0hk, 0x1.48p0hk, 0x1.74p0hk,
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};
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} // anonymous namespace
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LLVM_LIBC_FUNCTION(short accum, exphk, (short accum x)) {
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  using FXRep = fixed_point::FXRep<short accum>;
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  using StorageType = typename FXRep::StorageType;
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  // Output overflow
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  if (LIBC_UNLIKELY(x >= 0x1.64p2hk))
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    return FXRep::MAX();
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  // Lower bound where exp(x) -> 0:
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  //   floor(log(2^-8) * 2^7) * 2^-7
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  if (LIBC_UNLIKELY(x <= -0x1.63p2hk))
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    return FXRep::ZERO();
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  // Current range of x:
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  //   -0x1.628p2 <= x <= 0x1.638p2
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  // Range reduction:
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  //   x = hi + mid + lo,
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  // where:
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  //   hi is an integer
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  //   mid * 2^3 is an integer
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  //   |lo| <= 2^-4.
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  // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo)
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  //             ~ exp(hi) * exp(mid) * (1 + lo)
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  // with relative errors < |lo|^2 <= 2^-8.
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  //   exp(hi) and exp(mid) are extracted from small lookup tables.
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  // Round-to-nearest 1/8, tie-to-(+Int):
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  constexpr short accum ONE_SIXTEENTH = 0x1.0p-4hk;
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  // x_rounded = floor(x + 1/16).
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  short accum x_rounded = ((x + ONE_SIXTEENTH) >> (FXRep::FRACTION_LEN - 3))
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                          << (FXRep::FRACTION_LEN - 3);
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  short accum lo = x - x_rounded;
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  // Range of x_rounded:
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  //   x_rounded >= floor((-0x1.628p2 + 0x1.0p-4) * 2^3) * 2^-3
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  //              = -0x1.6p2 = -5.5
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  // To get the indices, we shift the values so that it start with 0.
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  // Range of indices:  0 <= indices <= 89
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  StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.6p2hk) >>
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                                                   (FXRep::FRACTION_LEN - 3));
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  // So we have the following relation:
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  //   indices = (hi + mid + 44/8) * 8
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  // That implies:
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  //   hi + mid = indices/8 - 5.5
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  // So for lookup tables, we can use the upper 4 bits to get:
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  //   exp( floor(indices / 8) - 5 )
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  // and lower 3 bits for:
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  //   exp( (indices - floor(indices)) - 0.5 )
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  short accum exp_hi = EXP_HI[indices >> 3];
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  short accum exp_mid = EXP_MID[indices & 0x7];
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  // exp(x) ~ exp(hi) * exp(mid) * (1 + lo);
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  return (exp_hi * (exp_mid * (0x1.0p0hk + lo)));
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}
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} // namespace LIBC_NAMESPACE
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