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 <errno.h> 25 26 #include "explogxf.h" 27 28 namespace LIBC_NAMESPACE_DECL { 29 namespace generic { 30 31 LIBC_INLINE float exp2f(float x) { 32 constexpr uint32_t EXVAL1 = 0x3b42'9d37U; 33 constexpr uint32_t EXVAL2 = 0xbcf3'a937U; 34 constexpr uint32_t EXVAL_MASK = EXVAL1 & EXVAL2; 35 36 using FPBits = typename fputil::FPBits<float>; 37 FPBits xbits(x); 38 39 uint32_t x_u = xbits.uintval(); 40 uint32_t x_abs = x_u & 0x7fff'ffffU; 41 42 // When |x| >= 128, or x is nan, or |x| <= 2^-5 43 if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) { 44 // |x| <= 2^-5 45 if (x_abs <= 0x3d00'0000) { 46 // |x| < 2^-25 47 if (LIBC_UNLIKELY(x_abs <= 0x3280'0000U)) { 48 return 1.0f + x; 49 } 50 51 // Check exceptional values. 52 if (LIBC_UNLIKELY((x_u & EXVAL_MASK) == EXVAL_MASK)) { 53 if (LIBC_UNLIKELY(x_u == EXVAL1)) { // x = 0x1.853a6ep-9f 54 return fputil::round_result_slightly_down(0x1.00870ap+0f); 55 } else if (LIBC_UNLIKELY(x_u == EXVAL2)) { // x = -0x1.e7526ep-6f 56 return fputil::round_result_slightly_down(0x1.f58d62p-1f); 57 } 58 } 59 60 // Minimax polynomial generated by Sollya with: 61 // > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]); 62 constexpr double COEFFS[] = { 63 0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5, 64 0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13}; 65 double xd = static_cast<double>(x); 66 double xsq = xd * xd; 67 double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]); 68 double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]); 69 double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]); 70 double p = fputil::polyeval(xsq, c0, c1, c2); 71 double r = fputil::multiply_add(p, xd, 1.0); 72 return static_cast<float>(r); 73 } 74 75 // x >= 128 76 if (xbits.is_pos()) { 77 // x is finite 78 if (x_u < 0x7f80'0000U) { 79 int rounding = fputil::quick_get_round(); 80 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 81 return FPBits::max_normal().get_val(); 82 83 fputil::set_errno_if_required(ERANGE); 84 fputil::raise_except_if_required(FE_OVERFLOW); 85 } 86 // x is +inf or nan 87 return x + FPBits::inf().get_val(); 88 } 89 // x <= -150 90 if (x_u >= 0xc316'0000U) { 91 // exp(-Inf) = 0 92 if (xbits.is_inf()) 93 return 0.0f; 94 // exp(nan) = nan 95 if (xbits.is_nan()) 96 return x; 97 if (fputil::fenv_is_round_up()) 98 return FPBits::min_subnormal().get_val(); 99 if (x != 0.0f) { 100 fputil::set_errno_if_required(ERANGE); 101 fputil::raise_except_if_required(FE_UNDERFLOW); 102 } 103 return 0.0f; 104 } 105 } 106 107 // For -150 < x < 128, to compute 2^x, we perform the following range 108 // reduction: find hi, mid, lo such that: 109 // x = hi + mid + lo, in which 110 // hi is an integer, 111 // 0 <= mid * 2^5 < 32 is an integer 112 // -2^(-6) <= lo <= 2^-6. 113 // In particular, 114 // hi + mid = round(x * 2^5) * 2^(-5). 115 // Then, 116 // 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo. 117 // 2^mid is stored in the lookup table of 32 elements. 118 // 2^lo is computed using a degree-5 minimax polynomial 119 // generated by Sollya. 120 // We perform 2^hi * 2^mid by simply add hi to the exponent field 121 // of 2^mid. 122 123 // kf = (hi + mid) * 2^5 = round(x * 2^5) 124 float kf; 125 int k; 126 #ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT 127 kf = fputil::nearest_integer(x * 32.0f); 128 k = static_cast<int>(kf); 129 #else 130 constexpr float HALF[2] = {0.5f, -0.5f}; 131 k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f])); 132 kf = static_cast<float>(k); 133 #endif // LIBC_TARGET_CPU_HAS_NEAREST_INT 134 135 // dx = lo = x - (hi + mid) = x - kf * 2^(-5) 136 double dx = fputil::multiply_add(-0x1.0p-5f, kf, x); 137 138 // hi = floor(kf * 2^(-4)) 139 // exp_hi = shift hi to the exponent field of double precision. 140 int64_t exp_hi = 141 static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS) 142 << fputil::FPBits<double>::FRACTION_LEN); 143 // mh = 2^hi * 2^mid 144 // mh_bits = bit field of mh 145 int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp_hi; 146 double mh = fputil::FPBits<double>(uint64_t(mh_bits)).get_val(); 147 148 // Degree-5 polynomial approximating (2^x - 1)/x generating by Sollya with: 149 // > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-1/32. 1/32]); 150 constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3, 151 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7, 152 0x1.5d88091198529p-10}; 153 double dx_sq = dx * dx; 154 double c1 = fputil::multiply_add(dx, COEFFS[0], 1.0); 155 double c2 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]); 156 double c3 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]); 157 double p = fputil::multiply_add(dx_sq, c3, c2); 158 // 2^x = 2^(hi + mid + lo) 159 // = 2^(hi + mid) * 2^lo 160 // ~ mh * (1 + lo * P(lo)) 161 // = mh + (mh*lo) * P(lo) 162 return static_cast<float>(fputil::multiply_add(p, dx_sq * mh, c1 * mh)); 163 } 164 165 } // namespace generic 166 } // namespace LIBC_NAMESPACE_DECL 167 168 #endif // LLVM_LIBC_SRC_MATH_GENERIC_EXP2F_IMPL_H 169