Why is (1-x)(1+x) preferred to (1-x^2)?

The derivative of ArcTan(X, Sqrt(1-X*X)) with respect to X is 1/Sqrt(1-X*X). This goes to infinity as |X| goes to 1. Therefore, when X is near 1 or -1, any error in evaluation has a huge effect on the result. Thus, it is critical that evaluation minimize error in these cases.

When X is near 1, evaluation of 1-X has no error (in IEEE 754 or any good floating-point system, because the scale of the result is such that its least significant bit is at least as low as the least significant bit in 1 or X, so the exact mathematical result has no bits outside the available significand bits). Since 1-X is exact, consider the effect of the error in 1+X by considering the derivative of ArcTan(X, Sqrt((1-X)*(1+X+e)) with respect to e, where e is the error introduced in the 1+X operation. The derivative is, when X is near 1 and e is small, approximately -1/10. (Taking the derivative with Maple and substituting 1 for x yields -1/(sqrt(4+2e)*(5+2e)). Then substituting 0 for e yields -1/10.) Thus, the error in 1+X is not critical.

Therefore, evaluating the expression as ArcTan(X, Sqrt((1-X)*(1+X)) is a good way to evaluate it.

The situation is symmetric for X near -1. (1+X has no error, and 1-X is not critical.)

Conversely, if we consider the error in X*X, the derivative of ArcTan(X, Sqrt(1-X*X+e)) with respect to e is, when X is near 1, approximately -1/(2sqrt(e)(1+e)), so it is large when e is small. So a small error in evaluating X*X will cause a large error in the result, when X is near 1.

Ask Pascal Cuoq points out, when evaluating a function f(x), we are generally interested in minimizing the relative error in the final result. And, as I have pointed out, the errors that occur during calculation are generally relative errors in intermediate results due to floating-point rounding. I was able to ignore this in the above because I was considering the function when X is near 1, so both the intermediate values under consideration (1+X and X*X) and the final value had magnitudes near 1, so dividing the values by those magnitudes would not change anything significantly.

However, for completeness, I examined the situation more closely. In Maple, I wrote g := arctan(x, sqrt((1-x*x*(1+e0))*(1+e1))*(1+e2)), thus allowing for relative errors e0, e1, and e2 in the calculations of x*x, 1-x*x, and the sqrt, respectively, and I wrote h:= arctan(x, sqrt((1-x)*(1+x)*(1+e0))*(1+e2)) for the alternative. Note that e0 in this case combines the three errors in 1-x, 1+x, and the multiplication of them; the full error term could be (1+ea)*(1+eb)*(1+ec), but this is effectively 1+e0 with a larger possible range for e0.

Then I examined the derivatives of these functions with respect to (one at a time) e0, e1, and e2 divided by abs(f(x)), where f was the ideal function, arctan(x, sqrt(1-x*x)). E.g., in Maple, I examined diff(g, e0) / abs(f(x)). I did not perform a full analytic evaluation of these; I examined the values for some values of x near 0 and near 1 and for values of e0, e1, and e2 at one of their limits, -2-54.

For x near 0, the values were all of magnitude about 1 or less. That is, any relative error in calculation resulted in a similar relative error in the result, or less.

For x near 1, the values with the derivatives of e1 and e2 were tiny, about 10-8 or less. However, the values with the derivatives of e0 were hugely different for the two methods. For the 1-x*x method, the value was about 2•107 (using x = 1-2-53). For the (1-x)*(1+x) method, the value was about 5•10-9.

In summary, the two methods do not differ much near x = 0, but the (1-x)*(1+x) method is significantly better near x = 1.

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