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-exception
6//
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 namespace
49
50LLVM_LIBC_FUNCTION(accum, expk, (accum x)) {
51 using FXRep = fixed_point::FXRep<accum>;
52 using StorageType = typename FXRep::StorageType;
53 // Output overflow
54 // > floor(log(2^16) * 2^15) * 2^-15
55 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^-15
59 if (LIBC_UNLIKELY(x <= -0x1.62e44p3k))
60 return FXRep::ZERO();
61
62 // Current range of x:
63 // -0x1.62e4p3 <= x <= 0x1.62e3cp3
64 // Range reduction:
65 // x = hi + mid + lo,
66 // where:
67 // hi is an integer
68 // mid * 2^4 is an integer
69 // |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^-4
84 // = -0x1.62p3 = -11.0625
85 // 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) * 16
91 // That implies:
92 // hi + mid = indices/16 - 11.0625
93 // 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 / 2
101 accum l2 = 0x1.0p0k + lo * l1; // = 1 + lo * (1 + lo / 2) = 1 + lo + lo^2/2
102 return (exp_hi * (exp_mid * l2));
103}
104
105} // namespace LIBC_NAMESPACE_DECL
106

source code of libc/src/stdfix/expk.cpp