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mathroundingnumericalexponential

Avoid round off error in exponential calculation

Due to rounding error cannot get ratio between two numbers: Ratio=exp(x)/(exp(x)+exp(y)) such that x=-1.11e4 and y=-1.12e4. Any mathematical or computational trick to do?

Solution

  • How about some mathematical manipulation in log-space...

    R = exp(x)/[exp(x)+exp(y)]
log(R) = log[exp(x)] - log[exp(x)+exp(y)]
       = log[exp(x)] - log[exp(x)*(1+exp(y)/exp(x))]
       = log[exp(x)] - log[exp(x)*(1+exp(y-x)]
       = log[exp(x)] - log[exp(x)] - log[(1+exp(y-x))]
       = - log[(1+exp(y-x))]

    Now, exp(y-x) should be a reasonable number, so you can calculate that easily. Then convert back to normal space using R = exp(log(R)).

    If that still doesn't work, you can actually taylor expand the last line:

    log[(1+z)] ~ 1 + z^2/2 - z^3/3 ...

    for small z, in this case z = exp(y-x).