1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Double-precision e^x - 1 function. 3*f3087befSAndrew Turner * 4*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited. 5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6*f3087befSAndrew Turner */ 7*f3087befSAndrew Turner 8*f3087befSAndrew Turner #include "poly_scalar_f64.h" 9*f3087befSAndrew Turner #include "math_config.h" 10*f3087befSAndrew Turner #include "test_sig.h" 11*f3087befSAndrew Turner #include "test_defs.h" 12*f3087befSAndrew Turner 13*f3087befSAndrew Turner #define InvLn2 0x1.71547652b82fep0 14*f3087befSAndrew Turner #define Ln2hi 0x1.62e42fefa39efp-1 15*f3087befSAndrew Turner #define Ln2lo 0x1.abc9e3b39803fp-56 16*f3087befSAndrew Turner #define Shift 0x1.8p52 17*f3087befSAndrew Turner /* 0x1p-51, below which expm1(x) is within 2 ULP of x. */ 18*f3087befSAndrew Turner #define TinyBound 0x3cc0000000000000 19*f3087befSAndrew Turner /* Above which expm1(x) overflows. */ 20*f3087befSAndrew Turner #define BigBound 0x1.63108c75a1937p+9 21*f3087befSAndrew Turner /* Below which expm1(x) rounds to 1. */ 22*f3087befSAndrew Turner #define NegBound -0x1.740bf7c0d927dp+9 23*f3087befSAndrew Turner #define AbsMask 0x7fffffffffffffff 24*f3087befSAndrew Turner 25*f3087befSAndrew Turner /* Approximation for exp(x) - 1 using polynomial on a reduced interval. 26*f3087befSAndrew Turner The maximum error observed error is 2.17 ULP: 27*f3087befSAndrew Turner expm1(0x1.63f90a866748dp-2) got 0x1.a9af56603878ap-2 28*f3087befSAndrew Turner want 0x1.a9af566038788p-2. */ 29*f3087befSAndrew Turner double 30*f3087befSAndrew Turner expm1 (double x) 31*f3087befSAndrew Turner { 32*f3087befSAndrew Turner uint64_t ix = asuint64 (x); 33*f3087befSAndrew Turner uint64_t ax = ix & AbsMask; 34*f3087befSAndrew Turner 35*f3087befSAndrew Turner /* Tiny, +Infinity. */ 36*f3087befSAndrew Turner if (ax <= TinyBound || ix == 0x7ff0000000000000) 37*f3087befSAndrew Turner return x; 38*f3087befSAndrew Turner 39*f3087befSAndrew Turner /* +/-NaN. */ 40*f3087befSAndrew Turner if (ax > 0x7ff0000000000000) 41*f3087befSAndrew Turner return __math_invalid (x); 42*f3087befSAndrew Turner 43*f3087befSAndrew Turner /* Result is too large to be represented as a double. */ 44*f3087befSAndrew Turner if (x >= 0x1.63108c75a1937p+9) 45*f3087befSAndrew Turner return __math_oflow (0); 46*f3087befSAndrew Turner 47*f3087befSAndrew Turner /* Result rounds to -1 in double precision. */ 48*f3087befSAndrew Turner if (x <= NegBound) 49*f3087befSAndrew Turner return -1; 50*f3087befSAndrew Turner 51*f3087befSAndrew Turner /* Reduce argument to smaller range: 52*f3087befSAndrew Turner Let i = round(x / ln2) 53*f3087befSAndrew Turner and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 54*f3087befSAndrew Turner exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 55*f3087befSAndrew Turner where 2^i is exact because i is an integer. */ 56*f3087befSAndrew Turner double j = fma (InvLn2, x, Shift) - Shift; 57*f3087befSAndrew Turner int64_t i = j; 58*f3087befSAndrew Turner double f = fma (j, -Ln2hi, x); 59*f3087befSAndrew Turner f = fma (j, -Ln2lo, f); 60*f3087befSAndrew Turner 61*f3087befSAndrew Turner /* Approximate expm1(f) using polynomial. 62*f3087befSAndrew Turner Taylor expansion for expm1(x) has the form: 63*f3087befSAndrew Turner x + ax^2 + bx^3 + cx^4 .... 64*f3087befSAndrew Turner So we calculate the polynomial P(f) = a + bf + cf^2 + ... 65*f3087befSAndrew Turner and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 66*f3087befSAndrew Turner double f2 = f * f; 67*f3087befSAndrew Turner double f4 = f2 * f2; 68*f3087befSAndrew Turner double p = fma (f2, estrin_10_f64 (f, f2, f4, f4 * f4, __expm1_poly), f); 69*f3087befSAndrew Turner 70*f3087befSAndrew Turner /* Assemble the result, using a slight rearrangement to achieve acceptable 71*f3087befSAndrew Turner accuracy. 72*f3087befSAndrew Turner expm1(x) ~= 2^i * (p + 1) - 1 73*f3087befSAndrew Turner Let t = 2^(i - 1). */ 74*f3087befSAndrew Turner double t = ldexp (0.5, i); 75*f3087befSAndrew Turner /* expm1(x) ~= 2 * (p * t + (t - 1/2)). */ 76*f3087befSAndrew Turner return 2 * fma (p, t, t - 0.5); 77*f3087befSAndrew Turner } 78*f3087befSAndrew Turner 79*f3087befSAndrew Turner TEST_SIG (S, D, 1, expm1, -9.9, 9.9) 80*f3087befSAndrew Turner TEST_ULP (expm1, 1.68) 81*f3087befSAndrew Turner TEST_SYM_INTERVAL (expm1, 0, 0x1p-51, 1000) 82*f3087befSAndrew Turner TEST_INTERVAL (expm1, 0x1p-51, 0x1.63108c75a1937p+9, 100000) 83*f3087befSAndrew Turner TEST_INTERVAL (expm1, -0x1p-51, -0x1.740bf7c0d927dp+9, 100000) 84*f3087befSAndrew Turner TEST_INTERVAL (expm1, 0x1.63108c75a1937p+9, inf, 100) 85*f3087befSAndrew Turner TEST_INTERVAL (expm1, -0x1.740bf7c0d927dp+9, -inf, 100) 86