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 14 namespace LIBC_NAMESPACE { 15 16 namespace { 17 18 // Look up tables for exp(hi) and exp(mid). 19 // Generated with Sollya: 20 // > for i from 0 to 23 do { 21 // hi = i - 11; 22 // e_hi = nearestint(exp(hi) * 2^15) * 2^-15; 23 // print(e_hi, "k,"); 24 // }; 25 static constexpr accum EXP_HI[24] = { 26 0x1p-15k, 0x1p-15k, 0x1p-13k, 0x1.6p-12k, 27 0x1.ep-11k, 0x1.44p-9k, 0x1.bap-8k, 0x1.2cp-6k, 28 0x1.97cp-5k, 0x1.153p-3k, 0x1.78b8p-2k, 0x1p0k, 29 0x1.5bf1p1k, 0x1.d8e68p2k, 0x1.415e6p4k, 0x1.b4c9p5k, 30 0x1.28d388p7k, 0x1.936dc6p8k, 0x1.1228858p10k, 0x1.749ea7cp11k, 31 0x1.fa7157cp12k, 0x1.5829dcf8p14k, 0x1.d3c4489p15k, ACCUM_MAX, 32 }; 33 34 // Generated with Sollya: 35 // > for i from 0 to 15 do { 36 // m = i/16 - 0.0625; 37 // e_m = nearestint(exp(m) * 2^15) * 2^-15; 38 // print(e_m, "k,"); 39 // }; 40 static constexpr accum EXP_MID[16] = { 41 0x1.e0fcp-1k, 0x1p0k, 0x1.1082p0k, 0x1.2216p0k, 42 0x1.34ccp0k, 0x1.48b6p0k, 0x1.5deap0k, 0x1.747ap0k, 43 0x1.8c8p0k, 0x1.a612p0k, 0x1.c14cp0k, 0x1.de46p0k, 44 0x1.fd1ep0k, 0x1.0efap1k, 0x1.2074p1k, 0x1.330ep1k, 45 }; 46 47 } // anonymous namespace 48 49 LLVM_LIBC_FUNCTION(accum, expk, (accum x)) { 50 using FXRep = fixed_point::FXRep<accum>; 51 using StorageType = typename FXRep::StorageType; 52 // Output overflow 53 // > floor(log(2^16) * 2^15) * 2^-15 54 if (LIBC_UNLIKELY(x >= 0x1.62e4p3k)) 55 return FXRep::MAX(); 56 // Lower bound where exp(x) -> 0: 57 // floor(log(2^-16) * 2^15) * 2^-15 58 if (LIBC_UNLIKELY(x <= -0x1.62e44p3k)) 59 return FXRep::ZERO(); 60 61 // Current range of x: 62 // -0x1.62e4p3 <= x <= 0x1.62e3cp3 63 // Range reduction: 64 // x = hi + mid + lo, 65 // where: 66 // hi is an integer 67 // mid * 2^4 is an integer 68 // |lo| <= 2^-5. 69 // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo) 70 // ~ exp(hi) * exp(mid) * (1 + lo + lo^2 / 2) 71 // with relative errors < |lo|^3/2 <= 2^-16. 72 // exp(hi) and exp(mid) are extracted from small lookup tables. 73 74 // Round-to-nearest 1/16, tie-to-(+Int): 75 constexpr accum ONE_THIRTY_SECOND = 0x1.0p-5k; 76 // x_rounded = floor(x + 1/16). 77 accum x_rounded = ((x + ONE_THIRTY_SECOND) >> (FXRep::FRACTION_LEN - 4)) 78 << (FXRep::FRACTION_LEN - 4); 79 accum lo = x - x_rounded; 80 81 // Range of x_rounded: 82 // x_rounded >= floor((-0x1.62e4p3 + 0x1.0p-5) * 2^4) * 2^-4 83 // = -0x1.62p3 = -11.0625 84 // To get the indices, we shift the values so that it start with 0. 85 // Range of indices: 0 <= indices <= 355. 86 StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.62p3k) >> 87 (FXRep::FRACTION_LEN - 4)); 88 // So we have the following relation: 89 // indices = (hi + mid + 177/16) * 16 90 // That implies: 91 // hi + mid = indices/16 - 11.0625 92 // So for lookup tables, we can use the upper 4 bits to get: 93 // exp( floor(indices / 16) - 11 ) 94 // and lower 4 bits for: 95 // exp( (indices - floor(indices)) - 0.0625 ) 96 accum exp_hi = EXP_HI[indices >> 4]; 97 accum exp_mid = EXP_MID[indices & 0xf]; 98 // exp(x) ~ exp(hi) * exp(mid) * (1 + lo); 99 accum l1 = 0x1.0p0k + (lo >> 1); // = 1 + lo / 2 100 accum l2 = 0x1.0p0k + lo * l1; // = 1 + lo * (1 + lo / 2) = 1 + lo + lo^2/2 101 return (exp_hi * (exp_mid * l2)); 102 } 103 104 } // namespace LIBC_NAMESPACE 105