1*412f47f9SXin Li /*
2*412f47f9SXin Li * Single-precision e^x - 1 function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2022-2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "poly_scalar_f32.h"
9*412f47f9SXin Li #include "math_config.h"
10*412f47f9SXin Li #include "pl_sig.h"
11*412f47f9SXin Li #include "pl_test.h"
12*412f47f9SXin Li
13*412f47f9SXin Li #define Shift (0x1.8p23f)
14*412f47f9SXin Li #define InvLn2 (0x1.715476p+0f)
15*412f47f9SXin Li #define Ln2hi (0x1.62e4p-1f)
16*412f47f9SXin Li #define Ln2lo (0x1.7f7d1cp-20f)
17*412f47f9SXin Li #define AbsMask (0x7fffffff)
18*412f47f9SXin Li #define InfLimit \
19*412f47f9SXin Li (0x1.644716p6) /* Smallest value of x for which expm1(x) overflows. */
20*412f47f9SXin Li #define NegLimit \
21*412f47f9SXin Li (-0x1.9bbabcp+6) /* Largest value of x for which expm1(x) rounds to 1. */
22*412f47f9SXin Li
23*412f47f9SXin Li /* Approximation for exp(x) - 1 using polynomial on a reduced interval.
24*412f47f9SXin Li The maximum error is 1.51 ULP:
25*412f47f9SXin Li expm1f(0x1.8baa96p-2) got 0x1.e2fb9p-2
26*412f47f9SXin Li want 0x1.e2fb94p-2. */
27*412f47f9SXin Li float
expm1f(float x)28*412f47f9SXin Li expm1f (float x)
29*412f47f9SXin Li {
30*412f47f9SXin Li uint32_t ix = asuint (x);
31*412f47f9SXin Li uint32_t ax = ix & AbsMask;
32*412f47f9SXin Li
33*412f47f9SXin Li /* Tiny: |x| < 0x1p-23. expm1(x) is closely approximated by x.
34*412f47f9SXin Li Inf: x == +Inf => expm1(x) = x. */
35*412f47f9SXin Li if (ax <= 0x34000000 || (ix == 0x7f800000))
36*412f47f9SXin Li return x;
37*412f47f9SXin Li
38*412f47f9SXin Li /* +/-NaN. */
39*412f47f9SXin Li if (ax > 0x7f800000)
40*412f47f9SXin Li return __math_invalidf (x);
41*412f47f9SXin Li
42*412f47f9SXin Li if (x >= InfLimit)
43*412f47f9SXin Li return __math_oflowf (0);
44*412f47f9SXin Li
45*412f47f9SXin Li if (x <= NegLimit || ix == 0xff800000)
46*412f47f9SXin Li return -1;
47*412f47f9SXin Li
48*412f47f9SXin Li /* Reduce argument to smaller range:
49*412f47f9SXin Li Let i = round(x / ln2)
50*412f47f9SXin Li and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
51*412f47f9SXin Li exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
52*412f47f9SXin Li where 2^i is exact because i is an integer. */
53*412f47f9SXin Li float j = fmaf (InvLn2, x, Shift) - Shift;
54*412f47f9SXin Li int32_t i = j;
55*412f47f9SXin Li float f = fmaf (j, -Ln2hi, x);
56*412f47f9SXin Li f = fmaf (j, -Ln2lo, f);
57*412f47f9SXin Li
58*412f47f9SXin Li /* Approximate expm1(f) using polynomial.
59*412f47f9SXin Li Taylor expansion for expm1(x) has the form:
60*412f47f9SXin Li x + ax^2 + bx^3 + cx^4 ....
61*412f47f9SXin Li So we calculate the polynomial P(f) = a + bf + cf^2 + ...
62*412f47f9SXin Li and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
63*412f47f9SXin Li float p = fmaf (f * f, horner_4_f32 (f, __expm1f_poly), f);
64*412f47f9SXin Li /* Assemble the result, using a slight rearrangement to achieve acceptable
65*412f47f9SXin Li accuracy.
66*412f47f9SXin Li expm1(x) ~= 2^i * (p + 1) - 1
67*412f47f9SXin Li Let t = 2^(i - 1). */
68*412f47f9SXin Li float t = ldexpf (0.5f, i);
69*412f47f9SXin Li /* expm1(x) ~= 2 * (p * t + (t - 1/2)). */
70*412f47f9SXin Li return 2 * fmaf (p, t, t - 0.5f);
71*412f47f9SXin Li }
72*412f47f9SXin Li
73*412f47f9SXin Li PL_SIG (S, F, 1, expm1, -9.9, 9.9)
74*412f47f9SXin Li PL_TEST_ULP (expm1f, 1.02)
75*412f47f9SXin Li PL_TEST_SYM_INTERVAL (expm1f, 0, 0x1p-23, 1000)
76*412f47f9SXin Li PL_TEST_INTERVAL (expm1f, 0x1p-23, 0x1.644716p6, 100000)
77*412f47f9SXin Li PL_TEST_INTERVAL (expm1f, 0x1.644716p6, inf, 1000)
78*412f47f9SXin Li PL_TEST_INTERVAL (expm1f, -0x1p-23, -0x1.9bbabcp+6, 100000)
79*412f47f9SXin Li PL_TEST_INTERVAL (expm1f, -0x1.9bbabcp+6, -inf, 1000)
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