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/*
* Single-precision e^x - 1 function.
*
* Copyright (c) 2022, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "hornerf.h"
#include "math_config.h"
#include "pl_sig.h"
#define Shift (0x1.8p23f)
#define InvLn2 (0x1.715476p+0f)
#define Ln2hi (0x1.62e4p-1f)
#define Ln2lo (0x1.7f7d1cp-20f)
#define AbsMask (0x7fffffff)
#define InfLimit \
(0x1.644716p6) /* Smallest value of x for which expm1(x) overflows. */
#define NegLimit \
(-0x1.9bbabcp+6) /* Largest value of x for which expm1(x) rounds to 1. */
#define C(i) __expm1f_poly[i]
/* Approximation for exp(x) - 1 using polynomial on a reduced interval.
The maximum error is 1.51 ULP:
expm1f(0x1.8baa96p-2) got 0x1.e2fb9p-2
want 0x1.e2fb94p-2. */
float
expm1f (float x)
{
uint32_t ix = asuint (x);
uint32_t ax = ix & AbsMask;
/* Tiny: |x| < 0x1p-23. expm1(x) is closely approximated by x.
Inf: x == +Inf => expm1(x) = x. */
if (ax <= 0x34000000 || (ix == 0x7f800000))
return x;
/* +/-NaN. */
if (ax > 0x7f800000)
return __math_invalidf (x);
if (x >= InfLimit)
return __math_oflowf (0);
if (x <= NegLimit || ix == 0xff800000)
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. */
float j = fmaf (InvLn2, x, Shift) - Shift;
int32_t i = j;
float f = fmaf (j, -Ln2hi, x);
f = fmaf (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). */
float p = fmaf (f * f, HORNER_4 (f, 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). */
float t = ldexpf (0.5f, i);
/* expm1(x) ~= 2 * (p * t + (t - 1/2)). */
return 2 * fmaf (p, t, t - 0.5f);
}
PL_SIG (S, F, 1, expm1, -9.9, 9.9)
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