Consider the expression fx squareroot 1 x2 squareroot 1

Consider the expression f(x) = squareroot 1 - x^2 - squareroot 1 + x^2. Rearrange f(x) in a manner so that it will be computed accurately for small x. Give a good approximate value for f(10^-7). At about what value of x will your algorithm produce f(x) = 0 due to underflow in single precision IEEE arithmetic?

Solution

the good approximte value for the function where x = 10^-7 is zero.

if we substitute x = 0, we get 1-1 =0;

at what value of x, we consider x² as zero the given function will be zero for that value of x

of we consider x= 0.1 , x^2 becomes 0.01 where we cant consider the value as zero

based on the approximations to 3, 5 ,7 decimal places we consider value as zero

approx. 7 decimal digits

based on IEEE single precision the value of X should be 10^-4

level	width	range at full precision	precision*
single precision	32 bits	±1.18×1038 to ±3.4×1038	approx. 7 decimal digits