Accuracy of sinh and complex cos/sin

[pavpanchekha]

The sin and cos function for complex arguments, and the sinh function for real arguments, are inaccurate when the inputs are very small. This is because Math.exp(x) - Math.exp(-x) returns zero for small x, instead of the more accurate 2x.

This patch replaces sinh by a Taylor expansion when the input is small, which increases accuracy.

[BigFav]

Style note: You have a return statement in the if, so you don't need the else.

if (Math.abs(x) < 1) {
  return x + (x * x * x) / 6 + (x * x * x * x * x) / 120;
}
return (Math.exp(x) - Math.exp(-x)) / 2;

[josdejong]

I think both versions are ok. @BigFav version is more compact, but on the other hand, an explicit else shows very clearly that the second return is only occurring in an else case (there are rules of thumb that say that every if should have an else prevents cases similar to these tricky fall-though cases of switches). Both are fine with me.

[pavpanchekha]

I'll leave it as is for now, but would be happy with whatever you decide.

[BigFav]

Good job catching this case. Most of my comments are style related (with the only exception of the failed test case), so hopefully this will be merged pretty smoothly.

[josdejong]

Thanks @pavpanchekha, nice improvement! Can you please fix this round-off error making the unit tests fail? After that I will merge your PR.

[josdejong]

Thanks a lot @pavpanchekha

[jrus]

This 5th degree taylor expansion of sinh is quite far off when you get close to -1 or 1. For example, for x = 0.99, it’s off by about 1.87e-4. I assume that’s dramatically more error than you want in a math library.

For a graph of the error, here’s Wolfram Alpha: http://www.wolframalpha.com/input/?i=sinh%28x%29+-+%28x+%2B+%281%2F6%29+*+x%5E3+%2B+%281%2F120%29+*+x+%5E+5%29+from+-1+to+1

The GNU Scientific Library uses the degree 17 Taylor expansion for computing sinh in the range [-1, 1]:
https://github.com/ampl/gsl/blob/1a819c2daab24f5f/specfunc/trig.c#L39-L52