Fast Inverse Square Root

Fast inverse square root (sometimes referred to as Fast InvSqrt or by the hexadecimal constant 0x5f3759df) is a method of calculating x−½, the reciprocal (or multiplicative inverse) of a square root for a 32-bit floating point number in IEEE 754 floating point format. The algorithm was probably developed at Silicon Graphics in the early 1990s, and an implementation appeared in 1999 in the Quake III Arena source code, but the method did not appear on public forums such as Usenet until 2002 or 2003. At the time, the primary advantage of the algorithm came from avoiding computationally expensive floating point operations in favor of integer operations. Inverse square roots are used to compute angles of incidence and reflection for lighting and shading in computer graphics.

The algorithm accepts a 32-bit floating point number as the input and stores a halved value for later use. Then, treating the bits representing the floating point number as a 32-bit integer, a logical shift right of one bit is performed and the result subtracted from the "magic" value 0x5f3759df. This is the first approximation of the inverse square root of the input. Treating the bits again as floating point it runs one iteration of Newton's method to return a more precise approximation. This computes an approximation of the inverse square root of a floating point number approximately four times faster than floating point division.

The algorithm was originally attributed to John Carmack, but an investigation showed that the code had deeper roots in both the hardware and software side of computer graphics. Adjustments and alterations passed through both Silicon Graphics and 3dfx Interactive, with Gary Tarolli's implementation for the SGI Indigo as the earliest known use. It is not known how the constant was originally derived, though investigation has shed some light on possible methods.