Introduction

This article introduces the fast inv-sqrt algorithm, which is firstly used in Quake III Arena.

And btw, this article is simply translated from the blog I wrote on CSDN: https://blog.csdn.net/noahzuo/article/details/51555161

The Source Code

So here is the source code:

float Q_rsqrt( float number ) 
{ 
    long i; 
    float x2, y; 
    const float threehalfs = 1.5F ;
    x2 = number * 0.5F ; 
    y = number ; 
    i = * ( long * ) &y; // evil floating point bit level hacking 
    i = 0x5f3759df - ( i >> 1 ); // what the fuck? 
    y = * ( float * ) &i; 
    y = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration 
//  y = y * ( threehalfs - ( x2 * y * y ) ); // 2nd iteration, this can be removed
    return y ;
}

And this function takes a 32-bit floating-point number $x$, and output the reciprocal of the square root $\frac{1}{\sqrt{x}}$of this number.

32-bit Floating-Point Number

Since this algorithm was designed with the IEEE 754-1985 32-bit floating-point specification in mind, it is necessary for us to get to know the structure of a floating-point number.

Float_example

You can refer to Single-precision floating-point format for more information and I’ll skip the explanation since this is not the point of this article.

In short, for a positive floating-point number $x$, we have

$$ x = 2^{e_x}(1 + m_{x}) $$

which $e_x$ is the 8-bit exponent and $m_x$ is the fraction part.

Moreover, to convert a floating-point number to a integer, we have

$$ I_{x} = E_{X}L + M_{x} = L(e_{x} + B + m_{x}) $$

which L is $2^{23}$ and B is $127$.

Algorithm

Newton’s Method

We can find successively better approximations to the roots of a real-valued function using Newton's method.

Suppose a single-variable function $ f $ defined for a real variable $ x $, if the function satisfied sufficient assumptions and the initial guess is close, then

$$x_{1} = x_{0} - \frac{f(x_{0})}{f^{‘}(x_{n})}$$

is a better approximation of the root than $x_{0}$.

The process is repeated as

$$x_{n + 1} = x_{n} - \frac{f(x_{n})}{f^{‘}(x_{n})}$$

until a sufficiently precise value is reached.

The process iteration is like:

Illustration of Newton's method

Black Magic

The magic is i = 0x5f3759df - ( i >> 1 ), but how does it work?

For function

$$y = \frac{1}{\sqrt{x}}$$

, firstly we have:

$$f(y) = \frac{1}{y^{2}} - x = 0$$

then we convert it to Newton's method style:

$$y_{n+1} = \frac{y_{n}(3 - xy^{2}_{n})}{2}$$

So for $ x = 2^{e_{x}}(1 + m_{x}) $, we have:
$$ log_{2}{x} = e_{x} + \log{2}{1 + m_{x}}$$

and since $m_{x} \in [0, 1)$, we have:

$$ log_{x}{1 + m_{x}} \approx m_{x} + \sigma $$

where $\sigma \approx 0.0430357$ yields the optimal approximation.

Thus there is the approximation
$$ log_{2}{x} \approx e_{x} + m_{x} + \sigma $$

Finally we have:
$$I_{x} = E_{x}L + M_{x} = L(e_{x} + m_{x} + \sigma + B - \sigma) \
\approx Llog_{2}{(x)} + L(B - \sigma)$$.

So:
$$I_{y} \approx \frac{3}{2}L(B -sigma) - \frac{1}{2}I_{x}$$,

which is, in the form of code:

1	i = 0x5f3759df - ( i >> 1 );

$$\frac{3}{2}L(B - \sigma) = 0x5f3759df$$