1*f3087befSAndrew Turner /* 2*f3087befSAndrew Turner * Single-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_f32.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 Shift (0x1.8p23f) 14*f3087befSAndrew Turner #define InvLn2 (0x1.715476p+0f) 15*f3087befSAndrew Turner #define Ln2hi (0x1.62e4p-1f) 16*f3087befSAndrew Turner #define Ln2lo (0x1.7f7d1cp-20f) 17*f3087befSAndrew Turner #define AbsMask (0x7fffffff) 18*f3087befSAndrew Turner #define InfLimit \ 19*f3087befSAndrew Turner (0x1.644716p6) /* Smallest value of x for which expm1(x) overflows. */ 20*f3087befSAndrew Turner #define NegLimit \ 21*f3087befSAndrew Turner (-0x1.9bbabcp+6) /* Largest value of x for which expm1(x) rounds to 1. */ 22*f3087befSAndrew Turner 23*f3087befSAndrew Turner /* Approximation for exp(x) - 1 using polynomial on a reduced interval. 24*f3087befSAndrew Turner The maximum error is 1.51 ULP: 25*f3087befSAndrew Turner expm1f(0x1.8baa96p-2) got 0x1.e2fb9p-2 26*f3087befSAndrew Turner want 0x1.e2fb94p-2. */ 27*f3087befSAndrew Turner float 28*f3087befSAndrew Turner expm1f (float x) 29*f3087befSAndrew Turner { 30*f3087befSAndrew Turner uint32_t ix = asuint (x); 31*f3087befSAndrew Turner uint32_t ax = ix & AbsMask; 32*f3087befSAndrew Turner 33*f3087befSAndrew Turner /* Tiny: |x| < 0x1p-23. expm1(x) is closely approximated by x. 34*f3087befSAndrew Turner Inf: x == +Inf => expm1(x) = x. */ 35*f3087befSAndrew Turner if (ax <= 0x34000000 || (ix == 0x7f800000)) 36*f3087befSAndrew Turner return x; 37*f3087befSAndrew Turner 38*f3087befSAndrew Turner /* +/-NaN. */ 39*f3087befSAndrew Turner if (ax > 0x7f800000) 40*f3087befSAndrew Turner return __math_invalidf (x); 41*f3087befSAndrew Turner 42*f3087befSAndrew Turner if (x >= InfLimit) 43*f3087befSAndrew Turner return __math_oflowf (0); 44*f3087befSAndrew Turner 45*f3087befSAndrew Turner if (x <= NegLimit || ix == 0xff800000) 46*f3087befSAndrew Turner return -1; 47*f3087befSAndrew Turner 48*f3087befSAndrew Turner /* Reduce argument to smaller range: 49*f3087befSAndrew Turner Let i = round(x / ln2) 50*f3087befSAndrew Turner and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 51*f3087befSAndrew Turner exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 52*f3087befSAndrew Turner where 2^i is exact because i is an integer. */ 53*f3087befSAndrew Turner float j = fmaf (InvLn2, x, Shift) - Shift; 54*f3087befSAndrew Turner int32_t i = j; 55*f3087befSAndrew Turner float f = fmaf (j, -Ln2hi, x); 56*f3087befSAndrew Turner f = fmaf (j, -Ln2lo, f); 57*f3087befSAndrew Turner 58*f3087befSAndrew Turner /* Approximate expm1(f) using polynomial. 59*f3087befSAndrew Turner Taylor expansion for expm1(x) has the form: 60*f3087befSAndrew Turner x + ax^2 + bx^3 + cx^4 .... 61*f3087befSAndrew Turner So we calculate the polynomial P(f) = a + bf + cf^2 + ... 62*f3087befSAndrew Turner and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 63*f3087befSAndrew Turner float p = fmaf (f * f, horner_4_f32 (f, __expm1f_poly), f); 64*f3087befSAndrew Turner /* Assemble the result, using a slight rearrangement to achieve acceptable 65*f3087befSAndrew Turner accuracy. 66*f3087befSAndrew Turner expm1(x) ~= 2^i * (p + 1) - 1 67*f3087befSAndrew Turner Let t = 2^(i - 1). */ 68*f3087befSAndrew Turner float t = ldexpf (0.5f, i); 69*f3087befSAndrew Turner /* expm1(x) ~= 2 * (p * t + (t - 1/2)). */ 70*f3087befSAndrew Turner return 2 * fmaf (p, t, t - 0.5f); 71*f3087befSAndrew Turner } 72*f3087befSAndrew Turner 73*f3087befSAndrew Turner TEST_SIG (S, F, 1, expm1, -9.9, 9.9) 74*f3087befSAndrew Turner TEST_ULP (expm1f, 1.02) 75*f3087befSAndrew Turner TEST_SYM_INTERVAL (expm1f, 0, 0x1p-23, 1000) 76*f3087befSAndrew Turner TEST_INTERVAL (expm1f, 0x1p-23, 0x1.644716p6, 100000) 77*f3087befSAndrew Turner TEST_INTERVAL (expm1f, 0x1.644716p6, inf, 1000) 78*f3087befSAndrew Turner TEST_INTERVAL (expm1f, -0x1p-23, -0x1.9bbabcp+6, 100000) 79*f3087befSAndrew Turner TEST_INTERVAL (expm1f, -0x1.9bbabcp+6, -inf, 1000) 80