expm1f.h

Go to the documentation of this file.

// Copyright © 2023 Apple Inc.
 
#pragma once
 
#include <metal_math>
 
// Original license copied below:
//  Copyright (c) 2015-2023 Norbert Juffa
//  All rights reserved.
//
//  Redistribution and use in source and binary forms, with or without
//  modification, are permitted provided that the following conditions
//  are met:
//
//  1. Redistributions of source code must retain the above copyright
//     notice, this list of conditions and the following disclaimer.
//
//  2. Redistributions in binary form must reproduce the above copyright
//     notice, this list of conditions and the following disclaimer in the
//     documentation and/or other materials provided with the distribution.
//
//  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
//  "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
//  LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
//  A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
//  HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
//  SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
//  LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
//  DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
//  THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
//  (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
//  OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 
/* Compute exponential base e minus 1. Maximum ulp error = 0.997458
 
   i = rint(a/log(2)), f = a-i*log(2). Then expm1(a) = 2**i * (expm1(f)+1) - 1.
   Compute r = expm1(f). Then expm1(a)= 2 * (0.5 * 2**i * r + 0.5 * 2**i - 0.5).
   With t = 0.5*2**i, expm1(a) = 2*(r * t + t-0.5). However, for best accuracy,
   when i == 1, expm1(a)= 2*(r + 0.5), and when i == 0, expm1(a) = r.
 
   NOTE: Scale factor b is only applied if i < 0 or i > 1 (should be power of 2)
*/
float expm1f_scaled_unchecked(float a, float b) {
  float f, j, r, s, t, u, v, x, y;
  int i;
 
  // exp(a) = 2**i * exp(f); i = rintf (a / log(2))
  j = fma(1.442695f, a, 12582912.f); // 0x1.715476p0, 0x1.8p23
  j = j - 12582912.0f; // 0x1.8p23
  i = (int)j;
  f = fma(j, -6.93145752e-1f, a);
 
  // approximate r = exp(f)-1 on interval [-log(2)/2, +log(2)/2]
  s = f * f;
  if (a == 0.0f)
    s = a; // ensure -0 is passed through
  // err = 0.997458  ulp1 = 11081805
  r = 1.97350979e-4f; // 0x1.9de000p-13
  r = fma(r, f, 1.39309070e-3f); // 0x1.6d30bcp-10
  r = fma(r, f, 8.33343994e-3f); // 0x1.1111f6p-7
  r = fma(r, f, 4.16668020e-2f); // 0x1.55559ep-5
  r = fma(r, f, 1.66666716e-1f); // 0x1.55555cp-3
  r = fma(r, f, 4.99999970e-1f); // 0x1.fffffep-2
  u = (j == 1) ? (f + 0.5f) : f;
  v = fma(r, s, u);
  s = 0.5f * b;
  t = ldexp(s, i);
  y = t - s;
  x = (t - y) - s; // double-float canonicalization of difference
  r = fma(v, t, x) + y;
  r = r + r;
  if (j == 0)
    r = v;
  if (j == 1)
    r = v + v;
  return r;
}
 
/* Compute exponential base e minus 1. max ulp err = 0.99746 */
float expm1f(float a) {
  float r;
 
  r = expm1f_scaled_unchecked(a, 1.0f);
  /* handle severe overflow and underflow */
  if (abs(a - 1.0f) > 88.0f) {
    r = pow(2, a);
    r = fma(r, r, -1.0f);
  }
  return r;
}