Multiplying by the inverse is not the same as the division

In school, we learn that the division is the same as multiplying the inverse (or reciprocal), that is x / y = x (1/y). In software, this does not work as-is when working with integers since 1/y is not an integer when y is an integer (except for y = 1 or y = -1). However, computers have floating-point numbers which are meant to emulate real numbers. It is reasonable to expect that x / y = x (1/y) will work out with floating-point numbers. It does not.

For example, dividing by 3.1416 and multiplying by 1/3.1416 do not result in the same numbers, when doing the computation on a computer.

To prove it, let me pull out my node (JavaScript) console…

> x = 651370000000
> x / 3.1416 * 3.1416
651370000000
> invd = 1 / 3.1416
> x * invd * 3.1416
651370000000.0001        

Maybe only JavaScript is broken? Let us try in Python…

>>> x = 651370000000
>>> x / 3.1416 * 3.1416
651370000000.0
>>> invd = 1 / 3.1416
>>> x * invd * 3.1416
651370000000.0001        

Always keep in mind that floating-point numbers are different from real numbers… for example, half of all floating-point numbers are in the interval [-1,1].