Rational tanh approximation

  • Author or source: cschueler
  • Type: saturation
  • Created: 2006-11-15 17:29:12
notes
This is a rational function to approximate a tanh-like soft clipper. It is based on the
pade-approximation of the tanh function with tweaked coefficients.

The function is in the range x=-3..3 and outputs the range y=-1..1. Beyond this range the
output must be clamped to -1..1.

The first to derivatives of the function vanish at -3 and 3, so the transition to the hard
clipped region is C2-continuous.
code
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float rational_tanh(x)
{
    if( x < -3 )
        return -1;
    else if( x > 3 )
        return 1;
    else
        return x * ( 27 + x * x ) / ( 27 + 9 * x * x );
}

Comments

Works fine. If you want only a little overdrive, you don't even need the clipping, just the last line for faster processing.

float rational_tanh_noclip(x)
{
  return x * ( 27 + x * x ) / ( 27 + 9 * x * x );
}

The maximum error of this function in the -4.5 .. 4.5 range is about 2.6%.

By the way this is the fastest tanh() approximation in the archive so far.
  • Date: 2006-12-08 21:21:02
  • By: cschueler
Yep, I thought so.
That's why I thought it would be worth sharing.

Especially fast when using SSE you can do a 4-way parallel implementation, with MIN/MAX and the RCP instruction.
  • Date: 2007-01-26 12:13:50
  • By: mdsp
nice one

BTW if you google about "pade-approximation" you'll find a nice page with many solutions for common functions.

there's exp, log, sin, cos, tan, gaussian...

Yep, but the RCP increases the noise floor somewhat, giving a quantized sound, so I'd refrain from using it for high quality audio.