A very compact implementation of the inverse error function erfinvf()

Accelerated Computing CUDA CUDA Programming and Performance
1

In a specific context, I was looking for the most compact implementation of erfinvf() possible while maintaining the accuracy and performance characteristics of CUDA’s built-in erfinvf(). “Compact” is defined as a small register foot print along with a small code foot print. While I succeeded in terms of compactness, I got exceedingly close but slightly missed on the accuracy and performance side. I am posting this as work-in-progress in case this is turns out to be useful to someone else.

I used CUDA 8 for my work with Quadro P2000 (sm_61) hardware. The throughput of CUDA’s built-in erfinvf() implementation on this platform is 77.595 billion function evaluations per second, while my code reaches 75.67 billion function evaluations per second, or 97.5% of the built-in math function. In terms of accuracy, CUDA achieves an error bound of 2.443247 ulp, while my implementation achieves the slightly larger error bound of 2.446816 ulp, which probably makes no difference in practical terms. In the programming contexts I examined, register pressure appeared to be reduced by four registers, with some reduction in the number of static instructions (maximum reduction by about 1/3 in code that consist essentially of many erfinvf() calls and not much more).

The general structure of the algorithm follows roughly the approach pioneered by Mike Giles, but tries to simplify it to the maximum extent possible by reducing it to a fast SFU-supplied log2() operation plus two straightforward polynomial approximations, one covering the common cases and the second for edge cases – arguments near unity.

M. Giles, “Approximating the erfinv function.” In GPU Computing Gems Jade Edition, pp. 109-116. 2011.

/*
  Copyright (c) 2017-2018, 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 the inverse of the error function. Maximum error = 2.446816 ulp */
__device__ float my_erfinvf (float a)
{
    float p, r, t;
    t = fmaf (a, -a, 1.0f);
    asm ("lg2.approx.ftz.f32 %0,%0;" : "+f"(t));
    if (t < -8.8359375f) { // max ulp err = 2.446816
        p =              1.50166858e-11f;  //  0x1.082d24p-36
        p = fmaf (p, t,  2.10561724e-09f); //  0x1.2164d2p-29
        p = fmaf (p, t,  1.27291500e-07f); //  0x1.115b3ep-23
        p = fmaf (p, t,  4.29223974e-06f); //  0x1.200c1ep-18
        p = fmaf (p, t,  8.60039654e-05f); //  0x1.68ba0ep-14
        p = fmaf (p, t,  9.49774636e-04f); //  0x1.f1f498p-11
        p = fmaf (p, t,  1.88959786e-03f); //  0x1.ef58c4p-10
        p = fmaf (p, t, -1.85239315e-01f); // -0x1.7b5ec0p-3
        p = fmaf (p, t,  8.36782515e-01f); //  0x1.ac6ec2p-1
        r = fmaf (p, a, 0.0f);
    } else { // max ulp err = 2.443247
        p =              2.00998329e-10f;  //  0x1.ba0000p-33
        p = fmaf (p, t,  7.63788321e-09f); //  0x1.066f88p-27
        p = fmaf (p, t,  9.44120231e-08f); //  0x1.957f1ep-24
        p = fmaf (p, t,  1.27279494e-08f); //  0x1.b5543ap-27
        p = fmaf (p, t, -8.98369126e-06f); // -0x1.2d7152p-17
        p = fmaf (p, t, -3.40910883e-05f); // -0x1.1dfa0ep-15
        p = fmaf (p, t,  7.70850747e-04f); //  0x1.9425d6p-11
        p = fmaf (p, t,  5.54407062e-03f); //  0x1.6b5612p-8
        p = fmaf (p, t, -1.60820886e-01f); // -0x1.495c76p-3
        p = fmaf (p, t,  8.86226892e-01f); //  0x1.c5bf88p-1
        r = a * p;
    }        
    return r;
}
1 Like