llvm-project
92 строки · 3.3 Кб
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-exception
6//
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
14namespace LIBC_NAMESPACE {
15
16namespace {
17
18// Look up tables for exp(hi) and exp(mid).
19// Generated with Sollya:
20// > for i from 0 to 89 do {
21// hi = floor(i/8) - 5;
22// m = i/8 - floor(i/8) - 0.5;
23// e_hi = nearestint(exp(hi) * 2^7) * 2^-7;
24// e_mid = nearestint(exp(m) * 2^7) * 2^-7;
25// print(hi, e_hi, m, e_mid);
26// };
27// Notice that when i = 88 and 89, e_hi will overflow short accum range.
28static constexpr short accum EXP_HI[12] = {
290x1.0p-7hk, 0x1.0p-6hk, 0x1.8p-5hk, 0x1.1p-3hk, 0x1.78p-2hk, 0x1.0p0hk,
300x1.5cp1hk, 0x1.d9p2hk, 0x1.416p4hk, 0x1.b4dp5hk, 0x1.28d4p7hk, SACCUM_MAX,
31};
32
33static constexpr short accum EXP_MID[8] = {
340x1.38p-1hk, 0x1.6p-1hk, 0x1.9p-1hk, 0x1.c4p-1hk,
350x1.0p0hk, 0x1.22p0hk, 0x1.48p0hk, 0x1.74p0hk,
36};
37
38} // anonymous namespace
39
40LLVM_LIBC_FUNCTION(short accum, exphk, (short accum x)) {
41using FXRep = fixed_point::FXRep<short accum>;
42using StorageType = typename FXRep::StorageType;
43// Output overflow
44if (LIBC_UNLIKELY(x >= 0x1.64p2hk))
45return FXRep::MAX();
46// Lower bound where exp(x) -> 0:
47// floor(log(2^-8) * 2^7) * 2^-7
48if (LIBC_UNLIKELY(x <= -0x1.63p2hk))
49return FXRep::ZERO();
50
51// Current range of x:
52// -0x1.628p2 <= x <= 0x1.638p2
53// Range reduction:
54// x = hi + mid + lo,
55// where:
56// hi is an integer
57// mid * 2^3 is an integer
58// |lo| <= 2^-4.
59// Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo)
60// ~ exp(hi) * exp(mid) * (1 + lo)
61// with relative errors < |lo|^2 <= 2^-8.
62// exp(hi) and exp(mid) are extracted from small lookup tables.
63
64// Round-to-nearest 1/8, tie-to-(+Int):
65constexpr short accum ONE_SIXTEENTH = 0x1.0p-4hk;
66// x_rounded = floor(x + 1/16).
67short accum x_rounded = ((x + ONE_SIXTEENTH) >> (FXRep::FRACTION_LEN - 3))
68<< (FXRep::FRACTION_LEN - 3);
69short accum lo = x - x_rounded;
70
71// Range of x_rounded:
72// x_rounded >= floor((-0x1.628p2 + 0x1.0p-4) * 2^3) * 2^-3
73// = -0x1.6p2 = -5.5
74// To get the indices, we shift the values so that it start with 0.
75// Range of indices: 0 <= indices <= 89
76StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.6p2hk) >>
77(FXRep::FRACTION_LEN - 3));
78// So we have the following relation:
79// indices = (hi + mid + 44/8) * 8
80// That implies:
81// hi + mid = indices/8 - 5.5
82// So for lookup tables, we can use the upper 4 bits to get:
83// exp( floor(indices / 8) - 5 )
84// and lower 3 bits for:
85// exp( (indices - floor(indices)) - 0.5 )
86short accum exp_hi = EXP_HI[indices >> 3];
87short accum exp_mid = EXP_MID[indices & 0x7];
88// exp(x) ~ exp(hi) * exp(mid) * (1 + lo);
89return (exp_hi * (exp_mid * (0x1.0p0hk + lo)));
90}
91
92} // namespace LIBC_NAMESPACE
93