81 lines
2.5 KiB
C
81 lines
2.5 KiB
C
/*
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* Single-precision e^x - 1 function.
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*
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* Copyright (c) 2022-2023, Arm Limited.
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* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
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*/
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#include "hornerf.h"
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#include "math_config.h"
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#include "pl_sig.h"
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#include "pl_test.h"
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#define Shift (0x1.8p23f)
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#define InvLn2 (0x1.715476p+0f)
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#define Ln2hi (0x1.62e4p-1f)
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#define Ln2lo (0x1.7f7d1cp-20f)
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#define AbsMask (0x7fffffff)
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#define InfLimit \
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(0x1.644716p6) /* Smallest value of x for which expm1(x) overflows. */
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#define NegLimit \
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(-0x1.9bbabcp+6) /* Largest value of x for which expm1(x) rounds to 1. */
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#define C(i) __expm1f_poly[i]
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/* Approximation for exp(x) - 1 using polynomial on a reduced interval.
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The maximum error is 1.51 ULP:
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expm1f(0x1.8baa96p-2) got 0x1.e2fb9p-2
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want 0x1.e2fb94p-2. */
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float
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expm1f (float x)
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{
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uint32_t ix = asuint (x);
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uint32_t ax = ix & AbsMask;
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/* Tiny: |x| < 0x1p-23. expm1(x) is closely approximated by x.
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Inf: x == +Inf => expm1(x) = x. */
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if (ax <= 0x34000000 || (ix == 0x7f800000))
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return x;
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/* +/-NaN. */
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if (ax > 0x7f800000)
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return __math_invalidf (x);
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if (x >= InfLimit)
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return __math_oflowf (0);
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if (x <= NegLimit || ix == 0xff800000)
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return -1;
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/* Reduce argument to smaller range:
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Let i = round(x / ln2)
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and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
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exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
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where 2^i is exact because i is an integer. */
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float j = fmaf (InvLn2, x, Shift) - Shift;
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int32_t i = j;
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float f = fmaf (j, -Ln2hi, x);
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f = fmaf (j, -Ln2lo, f);
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/* Approximate expm1(f) using polynomial.
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Taylor expansion for expm1(x) has the form:
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x + ax^2 + bx^3 + cx^4 ....
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So we calculate the polynomial P(f) = a + bf + cf^2 + ...
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and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
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float p = fmaf (f * f, HORNER_4 (f, C), f);
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/* Assemble the result, using a slight rearrangement to achieve acceptable
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accuracy.
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expm1(x) ~= 2^i * (p + 1) - 1
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Let t = 2^(i - 1). */
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float t = ldexpf (0.5f, i);
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/* expm1(x) ~= 2 * (p * t + (t - 1/2)). */
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return 2 * fmaf (p, t, t - 0.5f);
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}
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PL_SIG (S, F, 1, expm1, -9.9, 9.9)
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PL_TEST_ULP (expm1f, 1.02)
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PL_TEST_INTERVAL (expm1f, 0, 0x1p-23, 1000)
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PL_TEST_INTERVAL (expm1f, -0, -0x1p-23, 1000)
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PL_TEST_INTERVAL (expm1f, 0x1p-23, 0x1.644716p6, 100000)
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PL_TEST_INTERVAL (expm1f, -0x1p-23, -0x1.9bbabcp+6, 100000)
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