y = asinh(x) = log(x + sqrt(1+x^2)) the "obvious" formula. We have several choices: * Calculate using the obvious log formula * Calculate using Newton's method on sinh(y) * More complicated methods One more complicated method is to factor the above expression as: y = asinh(x) = f(g(x)), where f(x) = log(x) and g(x) = x+sqrt(1+x^2) Note that if y=g(x), then x=(y-1/y)/2. So we could first invert g(x) using Newton's method, and then apply the log function. The iteration to invert y=g(x) is x -> 2*(y+(x-y)/(1+x^2)) with an initial guess of 2*y. This iteration has quadratic convergence, and 6 iterations provides fpprec=128 precision. It remains to be seen if this is faster and/or more precise than the obvious formula. The other possibility is to get a very good approximation, and finish it off with a very small (1,2,3 ?) iterations of Newton to invert y=sinh(x). The iteration is x -> x-tanh(x)+y/cosh(x), with quadratic convergence: x0+eps -> x0+eps^2*tanh(x0)/2