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105 lines
2.6 KiB
105 lines
2.6 KiB
/*
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* Single-precision erf(x) function.
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*
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* Copyright (c) 2020, Arm Limited.
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* SPDX-License-Identifier: MIT
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*/
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#include <stdint.h>
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#include <math.h>
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#include "math_config.h"
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#define TwoOverSqrtPiMinusOne 0x1.06eba8p-3f
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#define A __erff_data.erff_poly_A
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#define B __erff_data.erff_poly_B
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/* Top 12 bits of a float. */
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static inline uint32_t
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top12 (float x)
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{
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return asuint (x) >> 20;
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}
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/* Efficient implementation of erff
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using either a pure polynomial approximation or
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the exponential of a polynomial.
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Worst-case error is 1.09ulps at 0x1.c111acp-1. */
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float
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erff (float x)
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{
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float r, x2, u;
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/* Get top word. */
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uint32_t ix = asuint (x);
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uint32_t sign = ix >> 31;
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uint32_t ia12 = top12 (x) & 0x7ff;
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/* Limit of both intervals is 0.875 for performance reasons but coefficients
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computed on [0.0, 0.921875] and [0.921875, 4.0], which brought accuracy
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from 0.94 to 1.1ulps. */
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if (ia12 < 0x3f6)
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{ /* a = |x| < 0.875. */
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/* Tiny and subnormal cases. */
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if (unlikely (ia12 < 0x318))
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{ /* |x| < 2^(-28). */
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if (unlikely (ia12 < 0x040))
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{ /* |x| < 2^(-119). */
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float y = fmaf (TwoOverSqrtPiMinusOne, x, x);
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return check_uflowf (y);
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}
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return x + TwoOverSqrtPiMinusOne * x;
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}
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x2 = x * x;
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/* Normalized cases (|x| < 0.921875). Use Horner scheme for x+x*P(x^2). */
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r = A[5];
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r = fmaf (r, x2, A[4]);
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r = fmaf (r, x2, A[3]);
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r = fmaf (r, x2, A[2]);
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r = fmaf (r, x2, A[1]);
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r = fmaf (r, x2, A[0]);
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r = fmaf (r, x, x);
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}
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else if (ia12 < 0x408)
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{ /* |x| < 4.0 - Use a custom Estrin scheme. */
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float a = fabsf (x);
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/* Start with Estrin scheme on high order (small magnitude) coefficients. */
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r = fmaf (B[6], a, B[5]);
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u = fmaf (B[4], a, B[3]);
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x2 = x * x;
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r = fmaf (r, x2, u);
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/* Then switch to pure Horner scheme. */
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r = fmaf (r, a, B[2]);
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r = fmaf (r, a, B[1]);
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r = fmaf (r, a, B[0]);
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r = fmaf (r, a, a);
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/* Single precision exponential with ~0.5ulps,
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ensures erff has max. rel. error
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< 1ulp on [0.921875, 4.0],
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< 1.1ulps on [0.875, 4.0]. */
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r = expf (-r);
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/* Explicit copysign (calling copysignf increases latency). */
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if (sign)
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r = -1.0f + r;
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else
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r = 1.0f - r;
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}
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else
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{ /* |x| >= 4.0. */
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/* Special cases : erff(nan)=nan, erff(+inf)=+1 and erff(-inf)=-1. */
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if (unlikely (ia12 >= 0x7f8))
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return (1.f - (float) ((ix >> 31) << 1)) + 1.f / x;
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/* Explicit copysign (calling copysignf increases latency). */
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if (sign)
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r = -1.0f;
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else
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r = 1.0f;
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}
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return r;
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}
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