1 //===-- Implementation of exphk 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 "exphk.h" 10 #include "src/__support/CPP/bit.h" 11 #include "src/__support/common.h" 12 #include "src/__support/fixed_point/fx_bits.h" 13 #include "src/__support/macros/config.h" 14 15 namespace LIBC_NAMESPACE_DECL { 16 17 namespace { 18 19 // Look up tables for exp(hi) and exp(mid). 20 // Generated with Sollya: 21 // > for i from 0 to 89 do { 22 // hi = floor(i/8) - 5; 23 // m = i/8 - floor(i/8) - 0.5; 24 // e_hi = nearestint(exp(hi) * 2^7) * 2^-7; 25 // e_mid = nearestint(exp(m) * 2^7) * 2^-7; 26 // print(hi, e_hi, m, e_mid); 27 // }; 28 // Notice that when i = 88 and 89, e_hi will overflow short accum range. 29 static constexpr short accum EXP_HI[12] = { 30 0x1.0p-7hk, 0x1.0p-6hk, 0x1.8p-5hk, 0x1.1p-3hk, 0x1.78p-2hk, 0x1.0p0hk, 31 0x1.5cp1hk, 0x1.d9p2hk, 0x1.416p4hk, 0x1.b4dp5hk, 0x1.28d4p7hk, SACCUM_MAX, 32 }; 33 34 static constexpr short accum EXP_MID[8] = { 35 0x1.38p-1hk, 0x1.6p-1hk, 0x1.9p-1hk, 0x1.c4p-1hk, 36 0x1.0p0hk, 0x1.22p0hk, 0x1.48p0hk, 0x1.74p0hk, 37 }; 38 39 } // anonymous namespace 40 41 LLVM_LIBC_FUNCTION(short accum, exphk, (short accum x)) { 42 using FXRep = fixed_point::FXRep<short accum>; 43 using StorageType = typename FXRep::StorageType; 44 // Output overflow 45 if (LIBC_UNLIKELY(x >= 0x1.64p2hk)) 46 return FXRep::MAX(); 47 // Lower bound where exp(x) -> 0: 48 // floor(log(2^-8) * 2^7) * 2^-7 49 if (LIBC_UNLIKELY(x <= -0x1.63p2hk)) 50 return FXRep::ZERO(); 51 52 // Current range of x: 53 // -0x1.628p2 <= x <= 0x1.638p2 54 // Range reduction: 55 // x = hi + mid + lo, 56 // where: 57 // hi is an integer 58 // mid * 2^3 is an integer 59 // |lo| <= 2^-4. 60 // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo) 61 // ~ exp(hi) * exp(mid) * (1 + lo) 62 // with relative errors < |lo|^2 <= 2^-8. 63 // exp(hi) and exp(mid) are extracted from small lookup tables. 64 65 // Round-to-nearest 1/8, tie-to-(+Int): 66 constexpr short accum ONE_SIXTEENTH = 0x1.0p-4hk; 67 // x_rounded = floor(x + 1/16). 68 short accum x_rounded = ((x + ONE_SIXTEENTH) >> (FXRep::FRACTION_LEN - 3)) 69 << (FXRep::FRACTION_LEN - 3); 70 short accum lo = x - x_rounded; 71 72 // Range of x_rounded: 73 // x_rounded >= floor((-0x1.628p2 + 0x1.0p-4) * 2^3) * 2^-3 74 // = -0x1.6p2 = -5.5 75 // To get the indices, we shift the values so that it start with 0. 76 // Range of indices: 0 <= indices <= 89 77 StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.6p2hk) >> 78 (FXRep::FRACTION_LEN - 3)); 79 // So we have the following relation: 80 // indices = (hi + mid + 44/8) * 8 81 // That implies: 82 // hi + mid = indices/8 - 5.5 83 // So for lookup tables, we can use the upper 4 bits to get: 84 // exp( floor(indices / 8) - 5 ) 85 // and lower 3 bits for: 86 // exp( (indices - floor(indices)) - 0.5 ) 87 short accum exp_hi = EXP_HI[indices >> 3]; 88 short accum exp_mid = EXP_MID[indices & 0x7]; 89 // exp(x) ~ exp(hi) * exp(mid) * (1 + lo); 90 return (exp_hi * (exp_mid * (0x1.0p0hk + lo))); 91 } 92 93 } // namespace LIBC_NAMESPACE_DECL 94