1 /* 2 * Single-precision e^x - 1 function. 3 * 4 * Copyright (c) 2022-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "hornerf.h" 9 #include "math_config.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define Shift (0x1.8p23f) 14 #define InvLn2 (0x1.715476p+0f) 15 #define Ln2hi (0x1.62e4p-1f) 16 #define Ln2lo (0x1.7f7d1cp-20f) 17 #define AbsMask (0x7fffffff) 18 #define InfLimit \ 19 (0x1.644716p6) /* Smallest value of x for which expm1(x) overflows. */ 20 #define NegLimit \ 21 (-0x1.9bbabcp+6) /* Largest value of x for which expm1(x) rounds to 1. */ 22 23 #define C(i) __expm1f_poly[i] 24 25 /* Approximation for exp(x) - 1 using polynomial on a reduced interval. 26 The maximum error is 1.51 ULP: 27 expm1f(0x1.8baa96p-2) got 0x1.e2fb9p-2 28 want 0x1.e2fb94p-2. */ 29 float 30 expm1f (float x) 31 { 32 uint32_t ix = asuint (x); 33 uint32_t ax = ix & AbsMask; 34 35 /* Tiny: |x| < 0x1p-23. expm1(x) is closely approximated by x. 36 Inf: x == +Inf => expm1(x) = x. */ 37 if (ax <= 0x34000000 || (ix == 0x7f800000)) 38 return x; 39 40 /* +/-NaN. */ 41 if (ax > 0x7f800000) 42 return __math_invalidf (x); 43 44 if (x >= InfLimit) 45 return __math_oflowf (0); 46 47 if (x <= NegLimit || ix == 0xff800000) 48 return -1; 49 50 /* Reduce argument to smaller range: 51 Let i = round(x / ln2) 52 and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. 53 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 54 where 2^i is exact because i is an integer. */ 55 float j = fmaf (InvLn2, x, Shift) - Shift; 56 int32_t i = j; 57 float f = fmaf (j, -Ln2hi, x); 58 f = fmaf (j, -Ln2lo, f); 59 60 /* Approximate expm1(f) using polynomial. 61 Taylor expansion for expm1(x) has the form: 62 x + ax^2 + bx^3 + cx^4 .... 63 So we calculate the polynomial P(f) = a + bf + cf^2 + ... 64 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ 65 float p = fmaf (f * f, HORNER_4 (f, C), f); 66 /* Assemble the result, using a slight rearrangement to achieve acceptable 67 accuracy. 68 expm1(x) ~= 2^i * (p + 1) - 1 69 Let t = 2^(i - 1). */ 70 float t = ldexpf (0.5f, i); 71 /* expm1(x) ~= 2 * (p * t + (t - 1/2)). */ 72 return 2 * fmaf (p, t, t - 0.5f); 73 } 74 75 PL_SIG (S, F, 1, expm1, -9.9, 9.9) 76 PL_TEST_ULP (expm1f, 1.02) 77 PL_TEST_INTERVAL (expm1f, 0, 0x1p-23, 1000) 78 PL_TEST_INTERVAL (expm1f, -0, -0x1p-23, 1000) 79 PL_TEST_INTERVAL (expm1f, 0x1p-23, 0x1.644716p6, 100000) 80 PL_TEST_INTERVAL (expm1f, -0x1p-23, -0x1.9bbabcp+6, 100000) 81