1//===-- Implementation of exp2m1f 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 "src/math/exp2m1f.h"
10#include "src/__support/FPUtil/FEnvImpl.h"
11#include "src/__support/FPUtil/FPBits.h"
12#include "src/__support/FPUtil/PolyEval.h"
13#include "src/__support/FPUtil/except_value_utils.h"
14#include "src/__support/FPUtil/multiply_add.h"
15#include "src/__support/FPUtil/rounding_mode.h"
16#include "src/__support/common.h"
17#include "src/__support/macros/optimization.h"
18#include "src/__support/macros/properties/cpu_features.h"
19#include "src/errno/libc_errno.h"
20
21#include "explogxf.h"
22
23namespace LIBC_NAMESPACE {
24
25static constexpr size_t N_EXCEPTS_LO = 8;
26
27static constexpr fputil::ExceptValues<float, N_EXCEPTS_LO> EXP2M1F_EXCEPTS_LO =
28 {.values: {
29 // (input, RZ output, RU offset, RD offset, RN offset)
30 // x = 0x1.36dc8ep-36, exp2m1f(x) = 0x1.aef212p-37 (RZ)
31 {.input: 0x2d9b'6e47U, .rnd_towardzero_result: 0x2d57'7909U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 0U},
32 // x = 0x1.224936p-19, exp2m1f(x) = 0x1.926c0ep-20 (RZ)
33 {.input: 0x3611'249bU, .rnd_towardzero_result: 0x35c9'3607U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 1U},
34 // x = 0x1.d16d2p-20, exp2m1f(x) = 0x1.429becp-20 (RZ)
35 {.input: 0x35e8'b690U, .rnd_towardzero_result: 0x35a1'4df6U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 1U},
36 // x = 0x1.17949ep-14, exp2m1f(x) = 0x1.8397p-15 (RZ)
37 {.input: 0x388b'ca4fU, .rnd_towardzero_result: 0x3841'cb80U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 1U},
38 // x = -0x1.9c3e1ep-38, exp2m1f(x) = -0x1.1dbeacp-38 (RZ)
39 {.input: 0xacce'1f0fU, .rnd_towardzero_result: 0xac8e'df56U, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 0U},
40 // x = -0x1.4d89b4p-32, exp2m1f(x) = -0x1.ce61b6p-33 (RZ)
41 {.input: 0xafa6'c4daU, .rnd_towardzero_result: 0xaf67'30dbU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
42 // x = -0x1.a6eac4p-10, exp2m1f(x) = -0x1.24fadap-10 (RZ)
43 {.input: 0xbad3'7562U, .rnd_towardzero_result: 0xba92'7d6dU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
44 // x = -0x1.e7526ep-6, exp2m1f(x) = -0x1.4e53dep-6 (RZ)
45 {.input: 0xbcf3'a937U, .rnd_towardzero_result: 0xbca7'29efU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
46 }};
47
48static constexpr size_t N_EXCEPTS_HI = 3;
49
50static constexpr fputil::ExceptValues<float, N_EXCEPTS_HI> EXP2M1F_EXCEPTS_HI =
51 {.values: {
52 // (input, RZ output, RU offset, RD offset, RN offset)
53 // x = 0x1.16a972p-1, exp2m1f(x) = 0x1.d545b2p-2 (RZ)
54 {.input: 0x3f0b'54b9U, .rnd_towardzero_result: 0x3eea'a2d9U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 0U},
55 // x = -0x1.9f12acp-5, exp2m1f(x) = -0x1.1ab68cp-5 (RZ)
56 {.input: 0xbd4f'8956U, .rnd_towardzero_result: 0xbd0d'5b46U, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 0U},
57 // x = -0x1.de7b9cp-5, exp2m1f(x) = -0x1.4508f4p-5 (RZ)
58 {.input: 0xbd6f'3dceU, .rnd_towardzero_result: 0xbd22'847aU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
59 }};
60
61LLVM_LIBC_FUNCTION(float, exp2m1f, (float x)) {
62 using FPBits = fputil::FPBits<float>;
63 FPBits xbits(x);
64
65 uint32_t x_u = xbits.uintval();
66 uint32_t x_abs = x_u & 0x7fff'ffffU;
67
68 // When |x| >= 128, or x is nan, or |x| <= 2^-5
69 if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) {
70 // |x| <= 2^-5
71 if (x_abs <= 0x3d00'0000U) {
72 if (auto r = EXP2M1F_EXCEPTS_LO.lookup(x_bits: x_u); LIBC_UNLIKELY(r.has_value()))
73 return r.value();
74
75 // Minimax polynomial generated by Sollya with:
76 // > display = hexadecimal;
77 // > fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]);
78 constexpr double COEFFS[] = {
79 0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5,
80 0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13};
81 double xd = x;
82 double xsq = xd * xd;
83 double c0 = fputil::multiply_add(x: xd, y: COEFFS[1], z: COEFFS[0]);
84 double c1 = fputil::multiply_add(x: xd, y: COEFFS[3], z: COEFFS[2]);
85 double c2 = fputil::multiply_add(x: xd, y: COEFFS[5], z: COEFFS[4]);
86 double p = fputil::polyeval(x: xsq, a0: c0, a: c1, a: c2);
87 return static_cast<float>(p * xd);
88 }
89
90 // x >= 128, or x is nan
91 if (xbits.is_pos()) {
92 if (xbits.is_finite()) {
93 int rounding = fputil::quick_get_round();
94 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
95 return FPBits::max_normal().get_val();
96
97 fputil::set_errno_if_required(ERANGE);
98 fputil::raise_except_if_required(FE_OVERFLOW);
99 }
100
101 // x >= 128 and 2^x - 1 rounds to +inf, or x is +inf or nan
102 return x + FPBits::inf().get_val();
103 }
104 }
105
106 if (LIBC_UNLIKELY(x <= -25.0f)) {
107 // 2^(-inf) - 1 = -1
108 if (xbits.is_inf())
109 return -1.0f;
110 // 2^nan - 1 = nan
111 if (xbits.is_nan())
112 return x;
113
114 int rounding = fputil::quick_get_round();
115 if (rounding == FE_UPWARD || rounding == FE_TOWARDZERO)
116 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f
117
118 fputil::set_errno_if_required(ERANGE);
119 fputil::raise_except_if_required(FE_UNDERFLOW);
120 return -1.0f;
121 }
122
123 if (auto r = EXP2M1F_EXCEPTS_HI.lookup(x_bits: x_u); LIBC_UNLIKELY(r.has_value()))
124 return r.value();
125
126 // For -25 < x < 128, to compute 2^x, we perform the following range
127 // reduction: find hi, mid, lo such that:
128 // x = hi + mid + lo, in which:
129 // hi is an integer,
130 // 0 <= mid * 2^5 < 32 is an integer,
131 // -2^(-6) <= lo <= 2^(-6).
132 // In particular,
133 // hi + mid = round(x * 2^5) * 2^(-5).
134 // Then,
135 // 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
136 // 2^mid is stored in the lookup table of 32 elements.
137 // 2^lo is computed using a degree-4 minimax polynomial generated by Sollya.
138 // We perform 2^hi * 2^mid by simply add hi to the exponent field of 2^mid.
139
140 // kf = (hi + mid) * 2^5 = round(x * 2^5)
141 float kf;
142 int k;
143#ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
144 kf = fputil::nearest_integer(x: x * 32.0f);
145 k = static_cast<int>(kf);
146#else
147 constexpr float HALF[2] = {0.5f, -0.5f};
148 k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f]));
149 kf = static_cast<float>(k);
150#endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
151
152 // lo = x - (hi + mid) = x - kf * 2^(-5)
153 double lo = fputil::multiply_add(x: -0x1.0p-5f, y: kf, z: x);
154
155 // hi = floor(kf * 2^(-4))
156 // exp2_hi = shift hi to the exponent field of double precision.
157 int64_t exp2_hi =
158 static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS)
159 << fputil::FPBits<double>::FRACTION_LEN);
160 // mh = 2^hi * 2^mid
161 // mh_bits = bit field of mh
162 int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp2_hi;
163 double mh = fputil::FPBits<double>(static_cast<uint64_t>(mh_bits)).get_val();
164
165 // Degree-4 polynomial approximating (2^x - 1)/x generated by Sollya with:
166 // > display = hexadecimal;
167 // > fpminimax((2^x - 1)/x, 4, [|D...|], [-2^-6, 2^-6]);
168 constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,
169 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,
170 0x1.5d88091198529p-10};
171 double lo_sq = lo * lo;
172 double c1 = fputil::multiply_add(x: lo, y: COEFFS[0], z: 1.0);
173 double c2 = fputil::multiply_add(x: lo, y: COEFFS[2], z: COEFFS[1]);
174 double c3 = fputil::multiply_add(x: lo, y: COEFFS[4], z: COEFFS[3]);
175 double exp2_lo = fputil::polyeval(x: lo_sq, a0: c1, a: c2, a: c3);
176 // 2^x - 1 = 2^(hi + mid + lo) - 1
177 // = 2^(hi + mid) * 2^lo - 1
178 // ~ mh * (1 + lo * P(lo)) - 1
179 // = mh * exp2_lo - 1
180 return static_cast<float>(fputil::multiply_add(x: exp2_lo, y: mh, z: -1.0));
181}
182
183} // namespace LIBC_NAMESPACE
184

source code of libc/src/math/generic/exp2m1f.cpp