1 //===-- Single-precision 2^x 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 #ifndef LLVM_LIBC_SRC_MATH_GENERIC_EXP2F_IMPL_H
10 #define LLVM_LIBC_SRC_MATH_GENERIC_EXP2F_IMPL_H
11
12 #include "src/__support/FPUtil/FEnvImpl.h"
13 #include "src/__support/FPUtil/FPBits.h"
14 #include "src/__support/FPUtil/PolyEval.h"
15 #include "src/__support/FPUtil/except_value_utils.h"
16 #include "src/__support/FPUtil/multiply_add.h"
17 #include "src/__support/FPUtil/nearest_integer.h"
18 #include "src/__support/FPUtil/rounding_mode.h"
19 #include "src/__support/common.h"
20 #include "src/__support/macros/config.h"
21 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
22 #include "src/__support/macros/properties/cpu_features.h"
23
24 #include "explogxf.h"
25
26 namespace LIBC_NAMESPACE_DECL {
27 namespace generic {
28
exp2f(float x)29 LIBC_INLINE float exp2f(float x) {
30 constexpr uint32_t EXVAL1 = 0x3b42'9d37U;
31 constexpr uint32_t EXVAL2 = 0xbcf3'a937U;
32 constexpr uint32_t EXVAL_MASK = EXVAL1 & EXVAL2;
33
34 using FPBits = typename fputil::FPBits<float>;
35 FPBits xbits(x);
36
37 uint32_t x_u = xbits.uintval();
38 uint32_t x_abs = x_u & 0x7fff'ffffU;
39
40 // When |x| >= 128, or x is nan, or |x| <= 2^-5
41 if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) {
42 // |x| <= 2^-5
43 if (x_abs <= 0x3d00'0000) {
44 // |x| < 2^-25
45 if (LIBC_UNLIKELY(x_abs <= 0x3280'0000U)) {
46 return 1.0f + x;
47 }
48
49 // Check exceptional values.
50 if (LIBC_UNLIKELY((x_u & EXVAL_MASK) == EXVAL_MASK)) {
51 if (LIBC_UNLIKELY(x_u == EXVAL1)) { // x = 0x1.853a6ep-9f
52 return fputil::round_result_slightly_down(0x1.00870ap+0f);
53 } else if (LIBC_UNLIKELY(x_u == EXVAL2)) { // x = -0x1.e7526ep-6f
54 return fputil::round_result_slightly_down(0x1.f58d62p-1f);
55 }
56 }
57
58 // Minimax polynomial generated by Sollya with:
59 // > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]);
60 constexpr double COEFFS[] = {
61 0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5,
62 0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13};
63 double xd = static_cast<double>(x);
64 double xsq = xd * xd;
65 double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]);
66 double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]);
67 double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]);
68 double p = fputil::polyeval(xsq, c0, c1, c2);
69 double r = fputil::multiply_add(p, xd, 1.0);
70 return static_cast<float>(r);
71 }
72
73 // x >= 128
74 if (xbits.is_pos()) {
75 // x is finite
76 if (x_u < 0x7f80'0000U) {
77 int rounding = fputil::quick_get_round();
78 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
79 return FPBits::max_normal().get_val();
80
81 fputil::set_errno_if_required(ERANGE);
82 fputil::raise_except_if_required(FE_OVERFLOW);
83 }
84 // x is +inf or nan
85 return x + FPBits::inf().get_val();
86 }
87 // x <= -150
88 if (x_u >= 0xc316'0000U) {
89 // exp(-Inf) = 0
90 if (xbits.is_inf())
91 return 0.0f;
92 // exp(nan) = nan
93 if (xbits.is_nan())
94 return x;
95 if (fputil::fenv_is_round_up())
96 return FPBits::min_subnormal().get_val();
97 if (x != 0.0f) {
98 fputil::set_errno_if_required(ERANGE);
99 fputil::raise_except_if_required(FE_UNDERFLOW);
100 }
101 return 0.0f;
102 }
103 }
104
105 // For -150 < x < 128, to compute 2^x, we perform the following range
106 // reduction: find hi, mid, lo such that:
107 // x = hi + mid + lo, in which
108 // hi is an integer,
109 // 0 <= mid * 2^5 < 32 is an integer
110 // -2^(-6) <= lo <= 2^-6.
111 // In particular,
112 // hi + mid = round(x * 2^5) * 2^(-5).
113 // Then,
114 // 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo.
115 // 2^mid is stored in the lookup table of 32 elements.
116 // 2^lo is computed using a degree-5 minimax polynomial
117 // generated by Sollya.
118 // We perform 2^hi * 2^mid by simply add hi to the exponent field
119 // of 2^mid.
120
121 // kf = (hi + mid) * 2^5 = round(x * 2^5)
122 float kf;
123 int k;
124 #ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
125 kf = fputil::nearest_integer(x * 32.0f);
126 k = static_cast<int>(kf);
127 #else
128 constexpr float HALF[2] = {0.5f, -0.5f};
129 k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f]));
130 kf = static_cast<float>(k);
131 #endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
132
133 // dx = lo = x - (hi + mid) = x - kf * 2^(-5)
134 double dx = fputil::multiply_add(-0x1.0p-5f, kf, x);
135
136 // hi = floor(kf * 2^(-4))
137 // exp_hi = shift hi to the exponent field of double precision.
138 int64_t exp_hi =
139 static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS)
140 << fputil::FPBits<double>::FRACTION_LEN);
141 // mh = 2^hi * 2^mid
142 // mh_bits = bit field of mh
143 int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp_hi;
144 double mh = fputil::FPBits<double>(uint64_t(mh_bits)).get_val();
145
146 // Degree-5 polynomial approximating (2^x - 1)/x generating by Sollya with:
147 // > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-1/32. 1/32]);
148 constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3,
149 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7,
150 0x1.5d88091198529p-10};
151 double dx_sq = dx * dx;
152 double c1 = fputil::multiply_add(dx, COEFFS[0], 1.0);
153 double c2 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]);
154 double c3 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]);
155 double p = fputil::multiply_add(dx_sq, c3, c2);
156 // 2^x = 2^(hi + mid + lo)
157 // = 2^(hi + mid) * 2^lo
158 // ~ mh * (1 + lo * P(lo))
159 // = mh + (mh*lo) * P(lo)
160 return static_cast<float>(fputil::multiply_add(p, dx_sq * mh, c1 * mh));
161 }
162
163 } // namespace generic
164 } // namespace LIBC_NAMESPACE_DECL
165
166 #endif // LLVM_LIBC_SRC_MATH_GENERIC_EXP2F_IMPL_H
167