This may be old news to some, but I just discovered today that a MUFU.TANH
instruction was added to the GPU in Turing (sm_75
). While this is exposed in PTX as tanh.approx.f32
, there does not seem to be a matching device function intrinsic __tanhf()
.
The new instruction delivers results accurate to about 16.5 bits. More precisely, the maximum error over the entire input range is 133.96 ulps while the maximum relative error is 1.113e-5. As it turns out, one can construct a software equivalent for older GPU architectures with very closely matching characteristics: maximum error of 108.83 ulps and maximum relative error of 9.345e-6. I combined these into a __tanhf()
device function below:
/*
Copyright (c) 2021, 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.
*/
__forceinline__ __device__ float raw_ex2 (float a)
{
float r;
asm ("ex2.approx.ftz.f32 %0,%1;" : "=f"(r) : "f"(a));
return r;
}
__forceinline__ __device__ float raw_rcp (float a)
{
float r;
asm ("rcp.approx.ftz.f32 %0,%1;" : "=f"(r) : "f"(a));
return r;
}
__forceinline__ __device__ float copysignf_pos (float a, float b)
{
float r;
r = __int_as_float (__float_as_int (a) | (__float_as_int (b) & 0x80000000));
return r;
}
__device__ float __tanhf (float a)
{
#if (__CUDA_ARCH__ >= 750)
// maxulp=133.95290 maxrelerr= 1.11261083e-05
float r;
asm ("tanh.approx.f32 %0,%1; \n\t" : "=f"(r) : "f"(a));
#else // CUDA_ARCH
// maxulp=108.82848 maxrelerr= 9.34501462e-06
const float L2E = 1.442695041f;
float e, r, s, t, d;
s = fabsf (a);
t = -L2E * 2.0f * s;
e = raw_ex2 (t);
d = e + 1.0f;
r = raw_rcp (d);
r = fmaf (e, -r, r);
if (s < 4.997253418e-3f) r = a;
if (!isnan(a)) r = copysignf_pos (r, a);
return r;
#endif // CUDA_ARCH
return r;
}