$(25$pts$)$ Consider the function

$f(x)=\sqrt[\phantom{2}]{1-x^2}-\sqrt[\phantom{2}]{1+x^2}$

We want to compute it accurately in the IEEE single precision standard $(,t,e_{\text{m}\text{i}\text{n}},e_{\text{m}\text{a}\text{x}})=$

$(2,24,-126,128).$

A$)$ Rearrange $f(x)$ in a manner so that it will be computed accurately for small $x.$

$f(x)=\frac{-2x^2}{\sqrt[\phantom{2}]{1-x^2}+\sqrt[\phantom{2}]{1+x^2}}$

B$)$ Give a good approximate value for $f({10}^{-7})$ computed by your algorithm. Briefly justify your

answer.

$f({10}^{-7})=\frac{-2({10}^{-7}{)}^2}{\sqrt[\phantom{2}]{1-({10}^{-7}{)}^2}+\sqrt[\phantom{2}]{1+({10}^{-7}{)}^2}}=\frac{-2{10}^{-14}}{\sqrt[\phantom{2}]{1-{10}^{-14}}+\sqrt[\phantom{2}]{1+{10}^{-14}}}$~~$-{10}^{-14}$~~$0$

The single precision standard can hold $24$ bits or less, so the relative error ${10}^{-14}$ becomes zero.

C$)$ Argue why your algorithm can produce a better result than computing with the original

expression.