Square Root and Inverse Square Root Approximations

by Iain Paterson Stephens

It is often necessary to be able to calculate the square root or inverse square root of a non-negative floating-point number to a high degree of accuracy. This may be used, for example, when calculating the magnitude of the output from an FFT or for Feedforward Automatic Gain Control. A simple, effective and fast algorithm for this is based on the Newton-Raphson iteration method. For example, the inverse square root 'x' of the floating point number 'y' can be calculated as follows:

       if x = 1/sqrt(y)

Step 1: Give Give an initial approximation for x_n the inverse square root.

Step 2: Use the initial approximation,x_n, in the Newton-Raphson iteration, as follows:-

       x_n+1 = (0.5x_n)(3-y(x_n)²)

Step 3: Repeat the Newton-Raphson iteration, this time replacing the input value x_n, with the calculated 'second' approximation x_n+1.

Step 4: Repeat step 3 a number of times according to the required accuracy, for example only two iterations are required for an accuracy down to +/- 1 LSB for an IEEE 32-bit floating point number.

And what about calculating x = srqt(y)?

To calculate the square root, just use,

       x = y.(1/sqrt(y))

that is, try using the inverse square root approximation first and then multiply the result by the original number.