计算平方根时，IEEE-754正确的结果是什么？

Question

I have an Newton-Raphson Square Root Algorithm I am using which computes the single-precision square root of an input value. However using a testbench I input I found that certain input values don't converge to an answer which is closest to the actual square root. When I say actual square root, I mean the result you would get with more precision than 32-bit IEEE-754. As a result, I was wondering what is considered the correct value to be obtained when performing the square root in IEEE-754. Some people on this forum have told me that the closest value is not necessarily the most correct, that is why I am asking.

When computing the square root of the single precision IEEE-754 32-bit value 0x3f7fffff, what is considered the correct result and why?

Furthermore, what is considered the correct result when compute the square root of 0x7F7FFFFF?

Answer 1

0x3f7fffff is 1.0 - u , where u = 2**-24 . The Taylor Series for sqrt(1 + x) is:

sqrt(1 + x) = 1 + x/2 - x^2/8 + O(x^3)

If we plug -u in for x , we get:

sqrt(1 - u) = 1 - u/2 - u^2/8 - O(u^3)

The value 1 - u/2 is the exact halfway point between the two closest representable floating-point numbers, 1-u and 1 ; since the next term in the Taylor series is negative, the value of sqrt(1 - u) is just a tiny bit smaller, and so the result rounds down to 1 - u .

0x7f7fffff is just 2**128*(1-u) , so the mathematically exact square root is 2**64*(1 - u/2 - u^2/8 - ...) , which rounds down to 2**64 * (1-u) , as described above.