Power of number close to 1

Question

I'm guessing there is some standard trick that I wasn't able to find: Anyway I want to compute a large power of a number very close to 1(think 1-p where p<1e-17) in a numerically stable fashion. 1-p is truncated to 1 on my system.

Using the taylor expansion of the logarithm I obtain the following bounds

Is there anything smarter I can do?

Answer 1

You may calculate log(1+x) more accurately for |x| <= 1 by using the log1p function.

An example:

> p <- 1e-17
> log(1-p)
[1] 0
> log1p(-p)
[1] -1e-17

And another one:

> print((1+1e-17)^100, digits=22)
[1] 1
> print(exp(100*log1p(-1e-17)), digits=22)
[1] 0.9999999999999990007993

Here, however, we're limited with the accuracy of double type-based FP arithmetic (see What Every Computer Scientist Should Know About Floating-Point Arithmetic).

Another way is to use e.g. the Rmpfr (a.k.a. Multiple Precision Floating-Point Reliable) package:

> options(digits=22)
> library(Rmpfr)
> .N <- function(.) mpfr(., precBits = 200) # see the package's vignette
> (1-.N(1e-20))^100
1 'mpfr' number of precision  200   bits 
[1] 0.99999999999999999900000000000000005534172854579042829381053529

The package uses the gsl and mpfr library to implement arbitrary precision FP operations (at the cost of slower computation speed, of course).