(Please read the following as a not-very-precise set of intuitions; I haven't been able to work out all of the details.) One thing that I forgot to say about sinh/asinh representation is the potential symmetry in "positional" representation. This symmetry is more easily seen in the "binary" version of sinh: sinh2(x)=(2^x-2^-x)/2 sinh2(k)=(2^k-2^-k)/2, k a positive integer =a long string of 1's in binary positional notation. and cosh2(x)=(2^k+2^-k)/2 So the output of cosh2(x) should -- in general -- be a *palindrome* in a "suitable" positional notation. Knuth discusses "balanced" positional notations, in which the digits 0-9 are replaced by -5..+5; these representations are often utilized in hardware multipliers to reduce the amount of circuitry required. Are there number representations which make the palidromic nature of cosh2(x) obvious? Once we have such a representation, then we can consider sinh2(x), which will have a similar type of anti-palidromic representation. We should be able to decompose *any* number into a sum(?) of elementary anti-palindromes.