1 //===-- Implementation of expk 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 "expk.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 23 do { 22 // hi = i - 11; 23 // e_hi = nearestint(exp(hi) * 2^15) * 2^-15; 24 // print(e_hi, "k,"); 25 // }; 26 static constexpr accum EXP_HI[24] = { 27 0x1p-15k, 0x1p-15k, 0x1p-13k, 0x1.6p-12k, 28 0x1.ep-11k, 0x1.44p-9k, 0x1.bap-8k, 0x1.2cp-6k, 29 0x1.97cp-5k, 0x1.153p-3k, 0x1.78b8p-2k, 0x1p0k, 30 0x1.5bf1p1k, 0x1.d8e68p2k, 0x1.415e6p4k, 0x1.b4c9p5k, 31 0x1.28d388p7k, 0x1.936dc6p8k, 0x1.1228858p10k, 0x1.749ea7cp11k, 32 0x1.fa7157cp12k, 0x1.5829dcf8p14k, 0x1.d3c4489p15k, ACCUM_MAX, 33 }; 34 35 // Generated with Sollya: 36 // > for i from 0 to 15 do { 37 // m = i/16 - 0.0625; 38 // e_m = nearestint(exp(m) * 2^15) * 2^-15; 39 // print(e_m, "k,"); 40 // }; 41 static constexpr accum EXP_MID[16] = { 42 0x1.e0fcp-1k, 0x1p0k, 0x1.1082p0k, 0x1.2216p0k, 43 0x1.34ccp0k, 0x1.48b6p0k, 0x1.5deap0k, 0x1.747ap0k, 44 0x1.8c8p0k, 0x1.a612p0k, 0x1.c14cp0k, 0x1.de46p0k, 45 0x1.fd1ep0k, 0x1.0efap1k, 0x1.2074p1k, 0x1.330ep1k, 46 }; 47 48 } // anonymous namespace 49 50 LLVM_LIBC_FUNCTION(accum, expk, (accum x)) { 51 using FXRep = fixed_point::FXRep<accum>; 52 using StorageType = typename FXRep::StorageType; 53 // Output overflow 54 // > floor(log(2^16) * 2^15) * 2^-15 55 if (LIBC_UNLIKELY(x >= 0x1.62e4p3k)) 56 return FXRep::MAX(); 57 // Lower bound where exp(x) -> 0: 58 // floor(log(2^-16) * 2^15) * 2^-15 59 if (LIBC_UNLIKELY(x <= -0x1.62e44p3k)) 60 return FXRep::ZERO(); 61 62 // Current range of x: 63 // -0x1.62e4p3 <= x <= 0x1.62e3cp3 64 // Range reduction: 65 // x = hi + mid + lo, 66 // where: 67 // hi is an integer 68 // mid * 2^4 is an integer 69 // |lo| <= 2^-5. 70 // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo) 71 // ~ exp(hi) * exp(mid) * (1 + lo + lo^2 / 2) 72 // with relative errors < |lo|^3/2 <= 2^-16. 73 // exp(hi) and exp(mid) are extracted from small lookup tables. 74 75 // Round-to-nearest 1/16, tie-to-(+Int): 76 constexpr accum ONE_THIRTY_SECOND = 0x1.0p-5k; 77 // x_rounded = floor(x + 1/16). 78 accum x_rounded = ((x + ONE_THIRTY_SECOND) >> (FXRep::FRACTION_LEN - 4)) 79 << (FXRep::FRACTION_LEN - 4); 80 accum lo = x - x_rounded; 81 82 // Range of x_rounded: 83 // x_rounded >= floor((-0x1.62e4p3 + 0x1.0p-5) * 2^4) * 2^-4 84 // = -0x1.62p3 = -11.0625 85 // To get the indices, we shift the values so that it start with 0. 86 // Range of indices: 0 <= indices <= 355. 87 StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.62p3k) >> 88 (FXRep::FRACTION_LEN - 4)); 89 // So we have the following relation: 90 // indices = (hi + mid + 177/16) * 16 91 // That implies: 92 // hi + mid = indices/16 - 11.0625 93 // So for lookup tables, we can use the upper 4 bits to get: 94 // exp( floor(indices / 16) - 11 ) 95 // and lower 4 bits for: 96 // exp( (indices - floor(indices)) - 0.0625 ) 97 accum exp_hi = EXP_HI[indices >> 4]; 98 accum exp_mid = EXP_MID[indices & 0xf]; 99 // exp(x) ~ exp(hi) * exp(mid) * (1 + lo); 100 accum l1 = 0x1.0p0k + (lo >> 1); // = 1 + lo / 2 101 accum l2 = 0x1.0p0k + lo * l1; // = 1 + lo * (1 + lo / 2) = 1 + lo + lo^2/2 102 return (exp_hi * (exp_mid * l2)); 103 } 104 105 } // namespace LIBC_NAMESPACE_DECL 106