### Stable area hyperbolic sine for `Double` and `Float`.

This function was unstable, in particular for negative arguments. https://ghc.haskell.org/trac/ghc/ticket/14927 The reason is that the formula `log (x + sqrt (1 + x*x))` is dominated by the numerical error of the `sqrt` function when x is strongly negative (and thus the summands in the `log` mostly cancel). However, the area hyperbolic sine is an odd function, thus the negative side can as well be calculated by flipping over the positive side, which avoids this instability. Furthermore, for _very_ big arguments, the `x*x` subexpression overflows. However, long before that happens, the square root is anyways completely dominated by that term, so we can neglect the `1 +` and get sqrt (1 + x*x) ≈ sqrt (x*x) = x and therefore asinh x ≈ log (x + x) = log (2*x) = log 2 + log x which does not overflow for any normal-finite positive argument, but perfectly matches the exact formula within the floating-point accuracy.

Please register or sign in to comment