/* Double-precision e^x function. Copyright (c) 2018 Arm Ltd. All rights reserved. SPDX-License-Identifier: BSD-3-Clause Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the company may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ #include "fdlibm.h" #if !__OBSOLETE_MATH #include #include #include "math_config.h" #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] #define C6 __exp_data.poly[9 - 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 check_oflow (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. */ force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); } y = 0x1p-1022 * y; return check_uflow (y); } /* Top 12 bits of a double (sign and exponent bits). */ static inline uint32_t top12 (double x) { return asuint64 (x) >> 52; } double exp (double x) { uint32_t abstop; uint64_t ki, idx, top, sbits; /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 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. */ #if EXP_POLY_ORDER == 4 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4); #elif EXP_POLY_ORDER == 5 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); #elif EXP_POLY_ORDER == 6 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6); #endif if (unlikely (abstop == 0)) return specialcase (tmp, sbits, ki); scale = asdouble (sbits); /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-739, so there is no spurious underflow here even without fma. */ return scale + scale * tmp; } #endif