1//===-- Implementation header for exp2m1f ------------------------*- C++-*-===//
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#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_EXP2M1F_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP2M1F_H
11
12#include "exp10f_utils.h"
13#include "src/__support/FPUtil/FEnvImpl.h"
14#include "src/__support/FPUtil/FPBits.h"
15#include "src/__support/FPUtil/PolyEval.h"
16#include "src/__support/FPUtil/except_value_utils.h"
17#include "src/__support/FPUtil/multiply_add.h"
18#include "src/__support/FPUtil/rounding_mode.h"
19#include "src/__support/common.h"
20#include "src/__support/libc_errno.h"
21#include "src/__support/macros/config.h"
22#include "src/__support/macros/optimization.h"
23#include "src/__support/macros/properties/cpu_features.h"
24
25namespace LIBC_NAMESPACE_DECL {
26
27namespace math {
28
29LIBC_INLINE float exp2m1f(float x) {
30#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
31 constexpr size_t N_EXCEPTS_LO = 8;
32
33 constexpr fputil::ExceptValues<float, N_EXCEPTS_LO> EXP2M1F_EXCEPTS_LO = {.values: {
34 // (input, RZ output, RU offset, RD offset, RN offset)
35 // x = 0x1.36dc8ep-36, exp2m1f(x) = 0x1.aef212p-37 (RZ)
36 {.input: 0x2d9b'6e47U, .rnd_towardzero_result: 0x2d57'7909U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 0U},
37 // x = 0x1.224936p-19, exp2m1f(x) = 0x1.926c0ep-20 (RZ)
38 {.input: 0x3611'249bU, .rnd_towardzero_result: 0x35c9'3607U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 1U},
39 // x = 0x1.d16d2p-20, exp2m1f(x) = 0x1.429becp-20 (RZ)
40 {.input: 0x35e8'b690U, .rnd_towardzero_result: 0x35a1'4df6U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 1U},
41 // x = 0x1.17949ep-14, exp2m1f(x) = 0x1.8397p-15 (RZ)
42 {.input: 0x388b'ca4fU, .rnd_towardzero_result: 0x3841'cb80U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 1U},
43 // x = -0x1.9c3e1ep-38, exp2m1f(x) = -0x1.1dbeacp-38 (RZ)
44 {.input: 0xacce'1f0fU, .rnd_towardzero_result: 0xac8e'df56U, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 0U},
45 // x = -0x1.4d89b4p-32, exp2m1f(x) = -0x1.ce61b6p-33 (RZ)
46 {.input: 0xafa6'c4daU, .rnd_towardzero_result: 0xaf67'30dbU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
47 // x = -0x1.a6eac4p-10, exp2m1f(x) = -0x1.24fadap-10 (RZ)
48 {.input: 0xbad3'7562U, .rnd_towardzero_result: 0xba92'7d6dU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
49 // x = -0x1.e7526ep-6, exp2m1f(x) = -0x1.4e53dep-6 (RZ)
50 {.input: 0xbcf3'a937U, .rnd_towardzero_result: 0xbca7'29efU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
51 }};
52
53 constexpr size_t N_EXCEPTS_HI = 3;
54
55 constexpr fputil::ExceptValues<float, N_EXCEPTS_HI> EXP2M1F_EXCEPTS_HI = {.values: {
56 // (input, RZ output, RU offset, RD offset, RN offset)
57 // x = 0x1.16a972p-1, exp2m1f(x) = 0x1.d545b2p-2 (RZ)
58 {.input: 0x3f0b'54b9U, .rnd_towardzero_result: 0x3eea'a2d9U, .rnd_upward_offset: 1U, .rnd_downward_offset: 0U, .rnd_tonearest_offset: 0U},
59 // x = -0x1.9f12acp-5, exp2m1f(x) = -0x1.1ab68cp-5 (RZ)
60 {.input: 0xbd4f'8956U, .rnd_towardzero_result: 0xbd0d'5b46U, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 0U},
61 // x = -0x1.de7b9cp-5, exp2m1f(x) = -0x1.4508f4p-5 (RZ)
62 {.input: 0xbd6f'3dceU, .rnd_towardzero_result: 0xbd22'847aU, .rnd_upward_offset: 0U, .rnd_downward_offset: 1U, .rnd_tonearest_offset: 1U},
63 }};
64#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
65
66 using FPBits = fputil::FPBits<float>;
67 FPBits xbits(x);
68
69 uint32_t x_u = xbits.uintval();
70 uint32_t x_abs = x_u & 0x7fff'ffffU;
71
72 // When |x| >= 128, or x is nan, or |x| <= 2^-5
73 if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) {
74 // |x| <= 2^-5
75 if (x_abs <= 0x3d00'0000U) {
76#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
77 if (auto r = EXP2M1F_EXCEPTS_LO.lookup(x_bits: x_u); LIBC_UNLIKELY(r.has_value()))
78 return r.value();
79#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
80
81 // Minimax polynomial generated by Sollya with:
82 // > display = hexadecimal;
83 // > fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]);
84 constexpr double COEFFS[] = {
85 0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5,
86 0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13};
87 double xd = x;
88 double xsq = xd * xd;
89 double c0 = fputil::multiply_add(x: xd, y: COEFFS[1], z: COEFFS[0]);
90 double c1 = fputil::multiply_add(x: xd, y: COEFFS[3], z: COEFFS[2]);
91 double c2 = fputil::multiply_add(x: xd, y: COEFFS[5], z: COEFFS[4]);
92 double p = fputil::polyeval(x: xsq, a0: c0, a: c1, a: c2);
93 return static_cast<float>(p * xd);
94 }
95
96 // x >= 128, or x is nan
97 if (xbits.is_pos()) {
98 if (xbits.is_finite()) {
99#ifndef LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
100 int rounding = fputil::quick_get_round();
101 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
102 return FPBits::max_normal().get_val();
103#endif
104
105 fputil::set_errno_if_required(ERANGE);
106 fputil::raise_except_if_required(FE_OVERFLOW);
107 }
108
109 // x >= 128 and 2^x - 1 rounds to +inf, or x is +inf or nan
110 return x + FPBits::inf().get_val();
111 }
112 }
113
114 if (LIBC_UNLIKELY(x <= -25.0f)) {
115 // 2^(-inf) - 1 = -1
116 if (xbits.is_inf())
117 return -1.0f;
118 // 2^nan - 1 = nan
119 if (xbits.is_nan())
120 return x;
121
122#ifndef LIBC_MATH_HAS_ASSUME_ROUND_NEAREST_ONLY
123 int rounding = fputil::quick_get_round();
124 if (rounding == FE_UPWARD || rounding == FE_TOWARDZERO)
125 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f
126#endif
127
128 fputil::set_errno_if_required(ERANGE);
129 fputil::raise_except_if_required(FE_UNDERFLOW);
130 return -1.0f;
131 }
132
133#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
134 if (auto r = EXP2M1F_EXCEPTS_HI.lookup(x_bits: x_u); LIBC_UNLIKELY(r.has_value()))
135 return r.value();
136#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
137
138 // For -25 < x < 128, to compute 2^x, we perform the following range
139 // reduction: find hi, mid, lo such that:
140 // x = hi + mid + lo, in which:
141 // hi is an integer,
142 // 0 <= mid * 2^5 < 32 is an integer,
143 // -2^(-6) <= lo <= 2^(-6).
144 // In particular,
145 // hi + mid = round(x * 2^5) * 2^(-5).
146 // Then,
147 // 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
148 // 2^mid is stored in the lookup table of 32 elements.
149 // 2^lo is computed using a degree-4 minimax polynomial generated by Sollya.
150 // We perform 2^hi * 2^mid by simply add hi to the exponent field of 2^mid.
151
152 // kf = (hi + mid) * 2^5 = round(x * 2^5)
153 float kf = 0;
154 int k = 0;
155#ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
156 kf = fputil::nearest_integer(x * 32.0f);
157 k = static_cast<int>(kf);
158#else
159 constexpr float HALF[2] = {0.5f, -0.5f};
160 k = static_cast<int>(fputil::multiply_add(x, y: 32.0f, z: HALF[x < 0.0f]));
161 kf = static_cast<float>(k);
162#endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
163
164 // lo = x - (hi + mid) = x - kf * 2^(-5)
165 double lo = fputil::multiply_add(x: -0x1.0p-5f, y: kf, z: x);
166
167 // hi = floor(kf * 2^(-4))
168 // exp2_hi = shift hi to the exponent field of double precision.
169 int64_t exp2_hi =
170 static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS)
171 << fputil::FPBits<double>::FRACTION_LEN);
172 // mh = 2^hi * 2^mid
173 // mh_bits = bit field of mh
174 int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp2_hi;
175 double mh = fputil::FPBits<double>(static_cast<uint64_t>(mh_bits)).get_val();
176
177 // Degree-4 polynomial approximating (2^x - 1)/x generated by Sollya with:
178 // > display = hexadecimal;
179 // > fpminimax((2^x - 1)/x, 4, [|D...|], [-2^-6, 2^-6]);
180 constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,
181 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,
182 0x1.5d88091198529p-10};
183 double lo_sq = lo * lo;
184 double c1 = fputil::multiply_add(x: lo, y: COEFFS[0], z: 1.0);
185 double c2 = fputil::multiply_add(x: lo, y: COEFFS[2], z: COEFFS[1]);
186 double c3 = fputil::multiply_add(x: lo, y: COEFFS[4], z: COEFFS[3]);
187 double exp2_lo = fputil::polyeval(x: lo_sq, a0: c1, a: c2, a: c3);
188 // 2^x - 1 = 2^(hi + mid + lo) - 1
189 // = 2^(hi + mid) * 2^lo - 1
190 // ~ mh * (1 + lo * P(lo)) - 1
191 // = mh * exp2_lo - 1
192 return static_cast<float>(fputil::multiply_add(x: exp2_lo, y: mh, z: -1.0));
193}
194
195} // namespace math
196
197} // namespace LIBC_NAMESPACE_DECL
198
199#endif // LLVM_LIBC_SRC___SUPPORT_MATH_EXP2M1F_H
200