A little trick which can be used in programming, to reduce the CPU load, if the value of a Hypotenuse is being Compared.

A scenario which often happens in computing is, that there exists a quantity, call that (a), which will result accurately by squaring the quantities (x), (y) and (z) first, and then computing the square root of the sum. It could then also be said, that the following explicit function has been defined:

F(x, y, z) := sqrt(x^2 + y^2 + z^2)

Further, the idea exists in Computing, that when all one wants to compute, is (x2) for example, it takes fewer CPU cycles actually to compute (x*x), than it takes, to compute a real power function.

But, the object of the exercise could actually be, not to derive (a) from (x), (y) and (z), but rather, to compare two instances of F(x, y, z).

The biggest issue as such, with actually computing F(x, y, z), is, that to compute the square root, is even slower, than it was to compute (x2), (y2) and (z2). Therefore, if one has the luxury of knowing what (a) is in advance, what one can do, for real-number comparisons, is just to square (a), and then, not to compute the square root, which should exist within the function F(). Therefore, when two known quantities are simply being compared, the following way to do it, will run slightly faster:

a^2 < (x^2 + y^2 + z^2)

In Modern Computing, what is often done is, that actual CPU usage is ignored, to make the task of writing complex code easier, and, the situation may not always be recognizable, that two values are going to be compared, which would both have been computed as the square root of one other value. And so, to avoid having to stare at some code cross-eyed, the practice can be just as valid, to compute two instances of F(x, y, z), but, to compute them with the square root function in each case, and somewhere later in the code execution, just to compare the two resulting values.