llvm-project
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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
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 23 do {
21// hi = i - 11;
22// e_hi = nearestint(exp(hi) * 2^15) * 2^-15;
23// print(e_hi, "k,");
24// };
25static constexpr accum EXP_HI[24] = {
260x1p-15k, 0x1p-15k, 0x1p-13k, 0x1.6p-12k,
270x1.ep-11k, 0x1.44p-9k, 0x1.bap-8k, 0x1.2cp-6k,
280x1.97cp-5k, 0x1.153p-3k, 0x1.78b8p-2k, 0x1p0k,
290x1.5bf1p1k, 0x1.d8e68p2k, 0x1.415e6p4k, 0x1.b4c9p5k,
300x1.28d388p7k, 0x1.936dc6p8k, 0x1.1228858p10k, 0x1.749ea7cp11k,
310x1.fa7157cp12k, 0x1.5829dcf8p14k, 0x1.d3c4489p15k, ACCUM_MAX,
32};
33
34// Generated with Sollya:
35// > for i from 0 to 15 do {
36// m = i/16 - 0.0625;
37// e_m = nearestint(exp(m) * 2^15) * 2^-15;
38// print(e_m, "k,");
39// };
40static constexpr accum EXP_MID[16] = {
410x1.e0fcp-1k, 0x1p0k, 0x1.1082p0k, 0x1.2216p0k,
420x1.34ccp0k, 0x1.48b6p0k, 0x1.5deap0k, 0x1.747ap0k,
430x1.8c8p0k, 0x1.a612p0k, 0x1.c14cp0k, 0x1.de46p0k,
440x1.fd1ep0k, 0x1.0efap1k, 0x1.2074p1k, 0x1.330ep1k,
45};
46
47} // anonymous namespace
48
49LLVM_LIBC_FUNCTION(accum, expk, (accum x)) {
50using FXRep = fixed_point::FXRep<accum>;
51using StorageType = typename FXRep::StorageType;
52// Output overflow
53// > floor(log(2^16) * 2^15) * 2^-15
54if (LIBC_UNLIKELY(x >= 0x1.62e4p3k))
55return FXRep::MAX();
56// Lower bound where exp(x) -> 0:
57// floor(log(2^-16) * 2^15) * 2^-15
58if (LIBC_UNLIKELY(x <= -0x1.62e44p3k))
59return FXRep::ZERO();
60
61// Current range of x:
62// -0x1.62e4p3 <= x <= 0x1.62e3cp3
63// Range reduction:
64// x = hi + mid + lo,
65// where:
66// hi is an integer
67// mid * 2^4 is an integer
68// |lo| <= 2^-5.
69// Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo)
70// ~ exp(hi) * exp(mid) * (1 + lo + lo^2 / 2)
71// with relative errors < |lo|^3/2 <= 2^-16.
72// exp(hi) and exp(mid) are extracted from small lookup tables.
73
74// Round-to-nearest 1/16, tie-to-(+Int):
75constexpr accum ONE_THIRTY_SECOND = 0x1.0p-5k;
76// x_rounded = floor(x + 1/16).
77accum x_rounded = ((x + ONE_THIRTY_SECOND) >> (FXRep::FRACTION_LEN - 4))
78<< (FXRep::FRACTION_LEN - 4);
79accum lo = x - x_rounded;
80
81// Range of x_rounded:
82// x_rounded >= floor((-0x1.62e4p3 + 0x1.0p-5) * 2^4) * 2^-4
83// = -0x1.62p3 = -11.0625
84// To get the indices, we shift the values so that it start with 0.
85// Range of indices: 0 <= indices <= 355.
86StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.62p3k) >>
87(FXRep::FRACTION_LEN - 4));
88// So we have the following relation:
89// indices = (hi + mid + 177/16) * 16
90// That implies:
91// hi + mid = indices/16 - 11.0625
92// So for lookup tables, we can use the upper 4 bits to get:
93// exp( floor(indices / 16) - 11 )
94// and lower 4 bits for:
95// exp( (indices - floor(indices)) - 0.0625 )
96accum exp_hi = EXP_HI[indices >> 4];
97accum exp_mid = EXP_MID[indices & 0xf];
98// exp(x) ~ exp(hi) * exp(mid) * (1 + lo);
99accum l1 = 0x1.0p0k + (lo >> 1); // = 1 + lo / 2
100accum l2 = 0x1.0p0k + lo * l1; // = 1 + lo * (1 + lo / 2) = 1 + lo + lo^2/2
101return (exp_hi * (exp_mid * l2));
102}
103
104} // namespace LIBC_NAMESPACE
105