/*
* Double-precision erf(x) function.
*
* Copyright (c) 2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "math_config.h"
#include "pl_sig.h"
#include "pl_test.h"
#define TwoOverSqrtPiMinusOne 0x1.06eba8214db69p-3
#define Shift 0x1p45
/* Polynomial coefficients. */
#define OneThird 0x1.5555555555555p-2
#define TwoThird 0x1.5555555555555p-1
#define TwoOverFifteen 0x1.1111111111111p-3
#define TwoOverFive 0x1.999999999999ap-2
#define Tenth 0x1.999999999999ap-4
#define TwoOverNine 0x1.c71c71c71c71cp-3
#define TwoOverFortyFive 0x1.6c16c16c16c17p-5
#define Sixth 0x1.555555555555p-3
/* Fast erf approximation based on series expansion near x rounded to
nearest multiple of 1/128.
Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
erf(x) ~ erf(r)
+ scale * d * [
+ 1
- r d
+ 1/3 (2 r^2 - 1) d^2
- 1/6 (r (2 r^2 - 3)) d^3
+ 1/30 (4 r^4 - 12 r^2 + 3) d^4
- 1/90 (4 r^4 - 20 r^2 + 15) d^5
]
Maximum measure error: 2.29 ULP
erf(-0x1.00003c924e5d1p-8) got -0x1.20dd59132ebadp-8
want -0x1.20dd59132ebafp-8. */
double
erf (double x)
{
/* Get absolute value and sign. */
uint64_t ix = asuint64 (x);
uint64_t ia = ix & 0x7fffffffffffffff;
uint64_t sign = ix & ~0x7fffffffffffffff;
/* |x| < 0x1p-508. Triggers exceptions. */
if (unlikely (ia < 0x2030000000000000))
return fma (TwoOverSqrtPiMinusOne, x, x);
if (ia < 0x4017f80000000000) /* |x| < 6 - 1 / 128 = 5.9921875. */
{
/* Set r to multiple of 1/128 nearest to |x|. */
double a = asdouble (ia);
double z = a + Shift;
uint64_t i = asuint64 (z) - asuint64 (Shift);
double r = z - Shift;
/* Lookup erf(r) and scale(r) in table.
Set erf(r) to 0 and scale to 2/sqrt(pi) for |x| <= 0x1.cp-9. */
double erfr = __erf_data.tab[i].erf;
double scale = __erf_data.tab[i].scale;
/* erf(x) ~ erf(r) + scale * d * poly (d, r). */
double d = a - r;
double r2 = r * r;
double d2 = d * d;
/* poly (d, r) = 1 + p1(r) * d + p2(r) * d^2 + ... + p5(r) * d^5. */
double p1 = -r;
double p2 = fma (TwoThird, r2, -OneThird);
double p3 = -r * fma (OneThird, r2, -0.5);
double p4 = fma (fma (TwoOverFifteen, r2, -TwoOverFive), r2, Tenth);
double p5
= -r * fma (fma (TwoOverFortyFive, r2, -TwoOverNine), r2, Sixth);
double p34 = fma (p4, d, p3);
double p12 = fma (p2, d, p1);
double y = fma (p5, d2, p34);
y = fma (y, d2, p12);
y = fma (fma (y, d2, d), scale, erfr);
return asdouble (asuint64 (y) | sign);
}
/* Special cases : erf(nan)=nan, erf(+inf)=+1 and erf(-inf)=-1. */
if (unlikely (ia >= 0x7ff0000000000000))
return (1.0 - (double) (sign >> 62)) + 1.0 / x;
/* Boring domain (|x| >= 6.0). */
return asdouble (sign | asuint64 (1.0));
}
PL_SIG (S, D, 1, erf, -6.0, 6.0)
PL_TEST_ULP (erf, 1.79)
PL_TEST_SYM_INTERVAL (erf, 0, 5.9921875, 40000)
PL_TEST_SYM_INTERVAL (erf, 5.9921875, inf, 40000)
PL_TEST_SYM_INTERVAL (erf, 0, inf, 40000)
