pl/math/erff_1u5.c

/*
 * Single-precision erf(x) function.
 *
 * Copyright (c) 2020-2023, Arm Limited.
 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
 */
#include "estrinf.h"
#include "hornerf.h"
#include "math_config.h"
#include "pl_sig.h"
#include "pl_test.h"

#define TwoOverSqrtPiMinusOne 0x1.06eba8p-3f
#define A __erff_data.erff_poly_A
#define B __erff_data.erff_poly_B

/* Top 12 bits of a float.  */
static inline uint32_t
top12 (float x)
{
  return asuint (x) >> 20;
}

/* Efficient implementation of erff using either a pure polynomial approximation
   or the exponential of a polynomial. Worst-case error is 1.09ulps at
   0x1.c111acp-1.  */
float
erff (float x)
{
  float r, x2;

  /* Get top word.  */
  uint32_t ix = asuint (x);
  uint32_t sign = ix >> 31;
  uint32_t ia12 = top12 (x) & 0x7ff;

  /* Limit of both intervals is 0.875 for performance reasons but coefficients
     computed on [0.0, 0.921875] and [0.921875, 4.0], which brought accuracy
     from 0.94 to 1.1ulps.  */
  if (ia12 < 0x3f6)
    { /* a = |x| < 0.875.  */

      /* Tiny and subnormal cases.  */
      if (unlikely (ia12 < 0x318))
	{ /* |x| < 2^(-28).  */
	  if (unlikely (ia12 < 0x040))
	    { /* |x| < 2^(-119).  */
	      float y = fmaf (TwoOverSqrtPiMinusOne, x, x);
	      return check_uflowf (y);
	    }
	  return x + TwoOverSqrtPiMinusOne * x;
	}

      x2 = x * x;

      /* Normalized cases (|x| < 0.921875) - Use Horner scheme for x+x*P(x^2).
       */
#define C(i) A[i]
      r = fmaf (HORNER_5 (x2, C), x, x);
#undef C
    }
  else if (ia12 < 0x408)
    { /* |x| < 4.0 - Use a custom Estrin scheme.  */

      float a = fabsf (x);
      /* Use Estrin scheme on high order (small magnitude) coefficients.  */
#define C(i) B[i]
      r = ESTRIN_3_ (a, x * x, C, 3);
#undef C
      /* Then switch to pure Horner scheme.  */
      r = fmaf (r, a, B[2]);
      r = fmaf (r, a, B[1]);
      r = fmaf (r, a, B[0]);
      r = fmaf (r, a, a);
      /* Single precision exponential with ~0.5ulps ensures erff has maximum
	 relative error below 1ulp on [0.921875, 4.0] and below 1.1ulps on
	 [0.875, 4.0].  */
      r = expf (-r);
      /* Explicit copysign (calling copysignf increases latency).  */
      if (sign)
	r = -1.0f + r;
      else
	r = 1.0f - r;
    }
  else
    { /* |x| >= 4.0.  */

      /* Special cases : erff(nan)=nan, erff(+inf)=+1 and erff(-inf)=-1.  */
      if (unlikely (ia12 >= 0x7f8))
	return (1.f - (float) ((ix >> 31) << 1)) + 1.f / x;

      /* Explicit copysign (calling copysignf increases latency).  */
      if (sign)
	r = -1.0f;
      else
	r = 1.0f;
    }
  return r;
}

PL_SIG (S, F, 1, erf, -4.0, 4.0)
PL_TEST_ULP (erff, 0.6)
PL_TEST_INTERVAL (erff, 0, 0xffff0000, 10000)
PL_TEST_INTERVAL (erff, 0x1p-127, 0x1p-26, 40000)
PL_TEST_INTERVAL (erff, -0x1p-127, -0x1p-26, 40000)
PL_TEST_INTERVAL (erff, 0x1p-26, 0x1p3, 40000)
PL_TEST_INTERVAL (erff, -0x1p-26, -0x1p3, 40000)
PL_TEST_INTERVAL (erff, 0, inf, 40000)