Square root of 3 billion digits. The fraction (…) for the square root of three can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than (approximately ). The rounded value of 1.732 is correct to within 0.01% of the actual value. Archimedes reported , accurate to (six decimal places) and (four decimal places), respectively. It can be expressed as the continued fraction , expanded on the right. It can also be expressed by generalized continued fractions such as which is evaluated at every second term. The following nested square expressions converge to
Fast inverse square root to compute because this equation is equivalent to where the fraction term is the inverse square root of . At the time, floating-point division was generally expensive compared to multiplication; the fast inverse square root algorithm bypassed the division step, giving it its performance advantage. "Quake III Arena", a first-person shooter video game, used the fast inverse square root algorithm to accelerate graphics computation, but the algorithm has since been implemented in some dedicated hardware vertex shaders using field-programmable gate arrays (FPGA). The following code is the fast inverse square root implementation from "Quake III Arena", stripped of C preprocessor