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/*
* Double-precision e^x - 1 function.
*
* Copyright (c) 2022, 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)
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