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Subtract extremely small number from one in C++

I need to subtract extremely small double number x from 1 i.e. to calculate 1-x in C++ for 0<x<1e-16. Because of machine precision restrictions for small enoughf x I will always get 1-x=1. Simple solution is to switch from double to some more precise format like long. But because of some restrictions I can't switch to more precise number formats.

What is the most efficient way to get accurate value of 1-x where x is an extremely small double if I can't use more precise formats and I need to store the result of the subtraction as a double? In practice I would like to avoid percentage errors greater then 1% (between double representation of 1-x and its actual value).

P.S. I am using Rcpp to calculate the quantiles of standard normal distribution via qnorm function. This function is symmetric around 0.5 being much more accurate for values close to 0. Therefore instead of qnorm(1-(1e-30)) I would like to calculate -qnorm(1e-30) but to derive 1e-30 from 1-(1e-30) I need to deal with a precision problem. The restriction on double is due to the fact that as I know it is not safe to use more precise numeric formats in Rcpp. Note that my inputs to qnorm could be sought of exogeneous so I can't to derive 1-x from x durning some preliminary calculations.

Solution

  • Simple solution is to switch from double to some more precise format like long [presumably, double]

    In that case you have no solution. long double is an alias for double on all modern machines. I stand corrected, gcc and icc still support it, only cl has dropped support for it for a long time.

    So you have two solutions, and they're not mutually exclusive:

    1. Use an arbitrary precision library instead of the built-in types. They're orders of magnitude slower, but if that's the best your algorithm can work with then that's that.

    2. Use a better algorithm, or at least rearrange your equation variables, to not have this need in the first place. Use distribution and cancellation rules to avoid the problem entirely. Without a more in depth description of your problem we can't help you, but I can tell you with certainty that double is more than enough to allow us to model airplane AI and flight parameters anywhere in the world.