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 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