/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│ │vi: set et ft=c ts=8 tw=8 fenc=utf-8 :vi│ ╚──────────────────────────────────────────────────────────────────────────────╝ │ │ │ Optimized Routines │ │ Copyright (c) 1999-2022, Arm Limited. │ │ │ │ Permission is hereby granted, free of charge, to any person obtaining │ │ a copy of this software and associated documentation files (the │ │ "Software"), to deal in the Software without restriction, including │ │ without limitation the rights to use, copy, modify, merge, publish, │ │ distribute, sublicense, and/or sell copies of the Software, and to │ │ permit persons to whom the Software is furnished to do so, subject to │ │ the following conditions: │ │ │ │ The above copyright notice and this permission notice shall be │ │ included in all copies or substantial portions of the Software. │ │ │ │ THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, │ │ EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF │ │ MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. │ │ IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY │ │ CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, │ │ TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE │ │ SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. │ │ │ ╚─────────────────────────────────────────────────────────────────────────────*/ #include "libc/intrin/likely.h" #include "libc/math.h" #include "libc/tinymath/exp_data.internal.h" #include "libc/tinymath/internal.h" asm(".ident\t\"\\n\\n\ Double-precision math functions (MIT License)\\n\ Copyright 2018 ARM Limited\""); asm(".include \"libc/disclaimer.inc\""); // clang-format off /* * Double-precision e^x function. * * Copyright (c) 2018, Arm Limited. * SPDX-License-Identifier: MIT */ #define N (1 << EXP_TABLE_BITS) #define InvLn2N __exp_data.invln2N #define NegLn2hiN __exp_data.negln2hiN #define NegLn2loN __exp_data.negln2loN #define Shift __exp_data.shift #define T __exp_data.tab #define C2 __exp_data.poly[5 - EXP_POLY_ORDER] #define C3 __exp_data.poly[6 - EXP_POLY_ORDER] #define C4 __exp_data.poly[7 - EXP_POLY_ORDER] #define C5 __exp_data.poly[8 - EXP_POLY_ORDER] /* Handle cases that may overflow or underflow when computing the result that is scale*(1+TMP) without intermediate rounding. The bit representation of scale is in SBITS, however it has a computed exponent that may have overflown into the sign bit so that needs to be adjusted before using it as a double. (int32_t)KI is the k used in the argument reduction and exponent adjustment of scale, positive k here means the result may overflow and negative k means the result may underflow. */ static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki) { double_t scale, y; if ((ki & 0x80000000) == 0) { /* k > 0, the exponent of scale might have overflowed by <= 460. */ sbits -= 1009ull << 52; scale = asdouble(sbits); y = 0x1p1009 * (scale + scale * tmp); return eval_as_double(y); } /* k < 0, need special care in the subnormal range. */ sbits += 1022ull << 52; scale = asdouble(sbits); y = scale + scale * tmp; if (y < 1.0) { /* Round y to the right precision before scaling it into the subnormal range to avoid double rounding that can cause 0.5+E/2 ulp error where E is the worst-case ulp error outside the subnormal range. So this is only useful if the goal is better than 1 ulp worst-case error. */ double_t hi, lo; lo = scale - y + scale * tmp; hi = 1.0 + y; lo = 1.0 - hi + y + lo; y = eval_as_double(hi + lo) - 1.0; /* Avoid -0.0 with downward rounding. */ if (WANT_ROUNDING && y == 0.0) y = 0.0; /* The underflow exception needs to be signaled explicitly. */ fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022); } y = 0x1p-1022 * y; return eval_as_double(y); } /* Top 12 bits of a double (sign and exponent bits). */ static inline uint32_t top12(double x) { return asuint64(x) >> 52; } /** * Returns 𝑒^x. */ double exp(double x) { uint32_t abstop; uint64_t ki, idx, top, sbits; double_t kd, z, r, r2, scale, tail, tmp; abstop = top12(x) & 0x7ff; if (UNLIKELY(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) { if (abstop - top12(0x1p-54) >= 0x80000000) /* Avoid spurious underflow for tiny x. */ /* Note: 0 is common input. */ return WANT_ROUNDING ? 1.0 + x : 1.0; if (abstop >= top12(1024.0)) { if (asuint64(x) == asuint64(-INFINITY)) return 0.0; if (abstop >= top12(INFINITY)) return 1.0 + x; if (asuint64(x) >> 63) return __math_uflow(0); else return __math_oflow(0); } /* Large x is special cased below. */ abstop = 0; } /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ z = InvLn2N * x; #if TOINT_INTRINSICS kd = roundtoint(z); ki = converttoint(z); #elif EXP_USE_TOINT_NARROW /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ kd = eval_as_double(z + Shift); ki = asuint64(kd) >> 16; kd = (double_t)(int32_t)ki; #else /* z - kd is in [-1, 1] in non-nearest rounding modes. */ kd = eval_as_double(z + Shift); ki = asuint64(kd); kd -= Shift; #endif r = x + kd * NegLn2hiN + kd * NegLn2loN; /* 2^(k/N) ~= scale * (1 + tail). */ idx = 2 * (ki % N); top = ki << (52 - EXP_TABLE_BITS); tail = asdouble(T[idx]); /* This is only a valid scale when -1023*N < k < 1024*N. */ sbits = T[idx + 1] + top; /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ /* Evaluation is optimized assuming superscalar pipelined execution. */ r2 = r * r; /* Without fma the worst case error is 0.25/N ulp larger. */ /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); if (UNLIKELY(abstop == 0)) return specialcase(tmp, sbits, ki); scale = asdouble(sbits); /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there is no spurious underflow here even without fma. */ return eval_as_double(scale + scale * tmp); }