Fast hyperbolic tangent approximation in Javascript

Question

I'm doing some digital signal processing calculations in javascript, and I found that calculating the hyperbolic tangent (tanh) is a bit too expensive. This is how I currently approximate tanh:

function tanh (arg) {
    // sinh(number)/cosh(number)
    return (Math.exp(arg) - Math.exp(-arg)) / (Math.exp(arg) + Math.exp(-arg));
}

Anyone knows a faster way to calculate it?

Answer 1

For an accurate answer using fewer Math.exp()s, you can use the relationship between tanh and the logistic function. Tanh(x) is exactly 2 * logistic(2 * x) - 1, and expanding out the logistic function, you get:

  function one_exp_tanh(x){
      return 2.0 / (1.0 + exp(-2.0 * x)) - 1.0;
  }

I don't know whether that is faster in javascript though.

Answer 2

From here.

function rational_tanh(x)
{
    if( x < -3 )
        return -1;
    else if( x > 3 )
        return 1;
    else
        return x * ( 27 + x * x ) / ( 27 + 9 * x * x );
}

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.

The Padé approximation is magnitudes better than the Taylor expansion. The clamping may also be an issue (depending on your range).

Answer 3

You could do this and cut your performance time in half:

function tanh(arg) {
    var pos = Math.exp(arg);
    var neg = Math.exp(-arg);
    return (pos - neg) / (pos + neg);
}