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/libc_errno.h"
18#include "src/__support/macros/config.h"
19#include "src/__support/macros/optimization.h"
20#include "src/__support/macros/properties/cpu_features.h"
21
22#include "explogxf.h"
23
24namespace LIBC_NAMESPACE_DECL {
25
26#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
27static constexpr size_t N_EXCEPTS_LO = 8;
28
29static constexpr fputil::ExceptValues<float, N_EXCEPTS_LO> EXP2M1F_EXCEPTS_LO =
30 {{
31 // (input, RZ output, RU offset, RD offset, RN offset)
32 // x = 0x1.36dc8ep-36, exp2m1f(x) = 0x1.aef212p-37 (RZ)
33 {0x2d9b'6e47U, 0x2d57'7909U, 1U, 0U, 0U},
34 // x = 0x1.224936p-19, exp2m1f(x) = 0x1.926c0ep-20 (RZ)
35 {0x3611'249bU, 0x35c9'3607U, 1U, 0U, 1U},
36 // x = 0x1.d16d2p-20, exp2m1f(x) = 0x1.429becp-20 (RZ)
37 {0x35e8'b690U, 0x35a1'4df6U, 1U, 0U, 1U},
38 // x = 0x1.17949ep-14, exp2m1f(x) = 0x1.8397p-15 (RZ)
39 {0x388b'ca4fU, 0x3841'cb80U, 1U, 0U, 1U},
40 // x = -0x1.9c3e1ep-38, exp2m1f(x) = -0x1.1dbeacp-38 (RZ)
41 {0xacce'1f0fU, 0xac8e'df56U, 0U, 1U, 0U},
42 // x = -0x1.4d89b4p-32, exp2m1f(x) = -0x1.ce61b6p-33 (RZ)
43 {0xafa6'c4daU, 0xaf67'30dbU, 0U, 1U, 1U},
44 // x = -0x1.a6eac4p-10, exp2m1f(x) = -0x1.24fadap-10 (RZ)
45 {0xbad3'7562U, 0xba92'7d6dU, 0U, 1U, 1U},
46 // x = -0x1.e7526ep-6, exp2m1f(x) = -0x1.4e53dep-6 (RZ)
47 {0xbcf3'a937U, 0xbca7'29efU, 0U, 1U, 1U},
48 }};
49
50static constexpr size_t N_EXCEPTS_HI = 3;
51
52static constexpr fputil::ExceptValues<float, N_EXCEPTS_HI> EXP2M1F_EXCEPTS_HI =
53 {{
54 // (input, RZ output, RU offset, RD offset, RN offset)
55 // x = 0x1.16a972p-1, exp2m1f(x) = 0x1.d545b2p-2 (RZ)
56 {0x3f0b'54b9U, 0x3eea'a2d9U, 1U, 0U, 0U},
57 // x = -0x1.9f12acp-5, exp2m1f(x) = -0x1.1ab68cp-5 (RZ)
58 {0xbd4f'8956U, 0xbd0d'5b46U, 0U, 1U, 0U},
59 // x = -0x1.de7b9cp-5, exp2m1f(x) = -0x1.4508f4p-5 (RZ)
60 {0xbd6f'3dceU, 0xbd22'847aU, 0U, 1U, 1U},
61 }};
62#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
63
64LLVM_LIBC_FUNCTION(float, exp2m1f, (float x)) {
65 using FPBits = fputil::FPBits<float>;
66 FPBits xbits(x);
67
68 uint32_t x_u = xbits.uintval();
69 uint32_t x_abs = x_u & 0x7fff'ffffU;
70
71 // When |x| >= 128, or x is nan, or |x| <= 2^-5
72 if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) {
73 // |x| <= 2^-5
74 if (x_abs <= 0x3d00'0000U) {
75#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
76 if (auto r = EXP2M1F_EXCEPTS_LO.lookup(x_u); LIBC_UNLIKELY(r.has_value()))
77 return r.value();
78#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
79
80 // Minimax polynomial generated by Sollya with:
81 // > display = hexadecimal;
82 // > fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]);
83 constexpr double COEFFS[] = {
84 0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5,
85 0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13};
86 double xd = x;
87 double xsq = xd * xd;
88 double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]);
89 double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]);
90 double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]);
91 double p = fputil::polyeval(xsq, c0, c1, c2);
92 return static_cast<float>(p * xd);
93 }
94
95 // x >= 128, or x is nan
96 if (xbits.is_pos()) {
97 if (xbits.is_finite()) {
98 int rounding = fputil::quick_get_round();
99 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
100 return FPBits::max_normal().get_val();
101
102 fputil::set_errno_if_required(ERANGE);
103 fputil::raise_except_if_required(FE_OVERFLOW);
104 }
105
106 // x >= 128 and 2^x - 1 rounds to +inf, or x is +inf or nan
107 return x + FPBits::inf().get_val();
108 }
109 }
110
111 if (LIBC_UNLIKELY(x <= -25.0f)) {
112 // 2^(-inf) - 1 = -1
113 if (xbits.is_inf())
114 return -1.0f;
115 // 2^nan - 1 = nan
116 if (xbits.is_nan())
117 return x;
118
119 int rounding = fputil::quick_get_round();
120 if (rounding == FE_UPWARD || rounding == FE_TOWARDZERO)
121 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f
122
123 fputil::set_errno_if_required(ERANGE);
124 fputil::raise_except_if_required(FE_UNDERFLOW);
125 return -1.0f;
126 }
127
128#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
129 if (auto r = EXP2M1F_EXCEPTS_HI.lookup(x_u); LIBC_UNLIKELY(r.has_value()))
130 return r.value();
131#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
132
133 // For -25 < x < 128, to compute 2^x, we perform the following range
134 // reduction: find hi, mid, lo such that:
135 // x = hi + mid + lo, in which:
136 // hi is an integer,
137 // 0 <= mid * 2^5 < 32 is an integer,
138 // -2^(-6) <= lo <= 2^(-6).
139 // In particular,
140 // hi + mid = round(x * 2^5) * 2^(-5).
141 // Then,
142 // 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
143 // 2^mid is stored in the lookup table of 32 elements.
144 // 2^lo is computed using a degree-4 minimax polynomial generated by Sollya.
145 // We perform 2^hi * 2^mid by simply add hi to the exponent field of 2^mid.
146
147 // kf = (hi + mid) * 2^5 = round(x * 2^5)
148 float kf;
149 int k;
150#ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
151 kf = fputil::nearest_integer(x * 32.0f);
152 k = static_cast<int>(kf);
153#else
154 constexpr float HALF[2] = {0.5f, -0.5f};
155 k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f]));
156 kf = static_cast<float>(k);
157#endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
158
159 // lo = x - (hi + mid) = x - kf * 2^(-5)
160 double lo = fputil::multiply_add(-0x1.0p-5f, kf, x);
161
162 // hi = floor(kf * 2^(-4))
163 // exp2_hi = shift hi to the exponent field of double precision.
164 int64_t exp2_hi =
165 static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS)
166 << fputil::FPBits<double>::FRACTION_LEN);
167 // mh = 2^hi * 2^mid
168 // mh_bits = bit field of mh
169 int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp2_hi;
170 double mh = fputil::FPBits<double>(static_cast<uint64_t>(mh_bits)).get_val();
171
172 // Degree-4 polynomial approximating (2^x - 1)/x generated by Sollya with:
173 // > display = hexadecimal;
174 // > fpminimax((2^x - 1)/x, 4, [|D...|], [-2^-6, 2^-6]);
175 constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,
176 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,
177 0x1.5d88091198529p-10};
178 double lo_sq = lo * lo;
179 double c1 = fputil::multiply_add(lo, COEFFS[0], 1.0);
180 double c2 = fputil::multiply_add(lo, COEFFS[2], COEFFS[1]);
181 double c3 = fputil::multiply_add(lo, COEFFS[4], COEFFS[3]);
182 double exp2_lo = fputil::polyeval(lo_sq, c1, c2, c3);
183 // 2^x - 1 = 2^(hi + mid + lo) - 1
184 // = 2^(hi + mid) * 2^lo - 1
185 // ~ mh * (1 + lo * P(lo)) - 1
186 // = mh * exp2_lo - 1
187 return static_cast<float>(fputil::multiply_add(exp2_lo, mh, -1.0));
188}
189
190} // namespace LIBC_NAMESPACE_DECL
191

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