Re: Finding the inverf

Ray Koopman <koopman@xxxxxx> wrote:
On Dec 2, 9:43 pm, morg...@xxxxxxxxx wrote:
Could someone tell me how to calculate inverf (x) if I am given
the value of erf (x).

Here's a relatively simple approximation for inverf[x]:

With[{t = -2*Log[1-|x|]}, Sign[x]*Sqrt[(t -
Log[1 + t + (.01167845*t+.1066561)*t^2/
((.02118035*t+.3710243)*t+1)])/2]]

The maximum absolute error is about 3.6*10^-5
for all x in (-1,1) in 64-bit IEEE format.

Thanks for that approximation. I have some comments and what some might
consider to be an improvement.

First, I believe your statement about the error to be correct. (N.B. I've
forgotten much about floating-point arithmetic, so anything I say related
to that here should be "taken with a grain of salt".) The reason that your
statement is correct is that numbers only very slightly less than 1 are not
representable in 64-bit IEEE format. But forgetting floating-point and just
considering the interval (-1, 1) of real numbers, the maximum error of your
approximation is about 1.2*10^-4, occurring when |x| is about 1 - 10^-79 .

Sometimes we prefer to consider relative, rather than absolute, error.
That's what I would normally choose to do for erf. Your approximation has
relative error exceeding 8*10^-4 at x = 0.

With the constraint that relative error = 0 at x = 0 and then
minimizing |relative error| on the real interval (-1, 1), for an
approximation of erf(x) in the form

With[{t = -2*Log[1-|x|]},
Sign[x]*Sqrt[(t - Log[1 + t + (a*t + b)*t^2/((c*t + d)*t+1)])/2]]

we find that b = (4 - pi)/8, implied by the constraint, and
a = 0.01857224, c = 0.03453839 and d = 0.44739441 .

This gives |relative error| < 3.7*10^-5 . (If we were to drop the
constraint, then worst relative error could be reduced a bit further.
However, if for nothing but aesthetics, I prefer to keep the constraint.)

BTW, with my coefficients above, if we were to approximate the inverse
of the complementary error function, inverfc(x), on (0, 2), my bound
on |relative error| would remain unchanged, of course. But with your
original coefficients, if we were to approximate inverfc(x) on (0, 2)
in 64-bit IEEE format, your error bound would increase (since
floating-point allows representation of numbers much closer to 0
than to 1).

Best regards,
David W. Cantrell
.

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