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Diffstat (limited to 'pl/math/expm1_2u5.c')
-rw-r--r-- | pl/math/expm1_2u5.c | 86 |
1 files changed, 86 insertions, 0 deletions
diff --git a/pl/math/expm1_2u5.c b/pl/math/expm1_2u5.c new file mode 100644 index 0000000..a3faff7 --- /dev/null +++ b/pl/math/expm1_2u5.c @@ -0,0 +1,86 @@ +/* + * Double-precision e^x - 1 function. + * + * Copyright (c) 2022-2023, Arm Limited. + * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception + */ + +#include "estrin.h" +#include "math_config.h" +#include "pl_sig.h" +#include "pl_test.h" + +#define InvLn2 0x1.71547652b82fep0 +#define Ln2hi 0x1.62e42fefa39efp-1 +#define Ln2lo 0x1.abc9e3b39803fp-56 +#define Shift 0x1.8p52 +#define TinyBound \ + 0x3cc0000000000000 /* 0x1p-51, below which expm1(x) is within 2 ULP of x. */ +#define BigBound 0x1.63108c75a1937p+9 /* Above which expm1(x) overflows. */ +#define NegBound -0x1.740bf7c0d927dp+9 /* Below which expm1(x) rounds to 1. */ +#define AbsMask 0x7fffffffffffffff + +#define C(i) __expm1_poly[i] + +/* Approximation for exp(x) - 1 using polynomial on a reduced interval. + The maximum error observed error is 2.17 ULP: + expm1(0x1.63f90a866748dp-2) got 0x1.a9af56603878ap-2 + want 0x1.a9af566038788p-2. */ +double +expm1 (double x) +{ + uint64_t ix = asuint64 (x); + uint64_t ax = ix & AbsMask; + + /* Tiny, +Infinity. */ + if (ax <= TinyBound || ix == 0x7ff0000000000000) + return x; + + /* +/-NaN. */ + if (ax > 0x7ff0000000000000) + return __math_invalid (x); + + /* Result is too large to be represented as a double. */ + if (x >= 0x1.63108c75a1937p+9) + return __math_oflow (0); + + /* Result rounds to -1 in double precision. */ + if (x <= NegBound) + return -1; + + /* Reduce argument to smaller range: + Let i = round(x / ln2) + and f = x - i * ln2, then f is in [-ln2/2, ln2/2]. + exp(x) - 1 = 2^i * (expm1(f) + 1) - 1 + where 2^i is exact because i is an integer. */ + double j = fma (InvLn2, x, Shift) - Shift; + int64_t i = j; + double f = fma (j, -Ln2hi, x); + f = fma (j, -Ln2lo, f); + + /* Approximate expm1(f) using polynomial. + Taylor expansion for expm1(x) has the form: + x + ax^2 + bx^3 + cx^4 .... + So we calculate the polynomial P(f) = a + bf + cf^2 + ... + and assemble the approximation expm1(f) ~= f + f^2 * P(f). */ + double f2 = f * f; + double f4 = f2 * f2; + double p = fma (f2, ESTRIN_10 (f, f2, f4, f4 * f4, C), f); + + /* Assemble the result, using a slight rearrangement to achieve acceptable + accuracy. + expm1(x) ~= 2^i * (p + 1) - 1 + Let t = 2^(i - 1). */ + double t = ldexp (0.5, i); + /* expm1(x) ~= 2 * (p * t + (t - 1/2)). */ + return 2 * fma (p, t, t - 0.5); +} + +PL_SIG (S, D, 1, expm1, -9.9, 9.9) +PL_TEST_ULP (expm1, 1.68) +PL_TEST_INTERVAL (expm1, 0, 0x1p-51, 1000) +PL_TEST_INTERVAL (expm1, -0, -0x1p-51, 1000) +PL_TEST_INTERVAL (expm1, 0x1p-51, 0x1.63108c75a1937p+9, 100000) +PL_TEST_INTERVAL (expm1, -0x1p-51, -0x1.740bf7c0d927dp+9, 100000) +PL_TEST_INTERVAL (expm1, 0x1.63108c75a1937p+9, inf, 100) +PL_TEST_INTERVAL (expm1, -0x1.740bf7c0d927dp+9, -inf, 100) |