The document discusses Newton's method of approximation for calculating square roots, using a specific example with n = 2. It details the iterative calculations and their convergence to the square root of 2, alongside a reference to a notable performance change in modern CPUs compared to earlier models. Additionally, it includes links to relevant GitHub and Netlib resources for further exploration.

1. PWL SF 2017

5. 19th August 2005

7. https://github.com/id-Software/Quake-III-Arena/blob/master/
code/game/q_math.c#L552

12. Newton’s Method of Approximation

13. n = 2
1 / sqrt(n) = 0.70710678119
g = 1

14. n = 2
1 / sqrt(n) = 0.70710678119
g = 1
1st ng = g*(1.5 - (n/2)*g*g) 0.5 70.71067811831%

15. n = 2
1 / sqrt(n) = 0.70710678119
g = 1
1st ng = g*(1.5 - (n/2)*g*g) 0.5 70.71067811831%
2nd ng = g*(1.5 - (n/2)*g*g) 0.625 88.38834764789%

16. n = 2
1 / sqrt(n) = 0.70710678119
g = 1
1st ng = g*(1.5 - (n/2)*g*g) 0.5 70.71067811831%
2nd ng = g*(1.5 - (n/2)*g*g) 0.625 88.38834764789%
3rd ng = g*(1.5 - (n/2)*g*g) 0.693359375 98.05582317187%
4th ng = g*(1.5 - (n/2)*g*g) 0.7067084685 99.94367007917%
5th ng = g*(1.5 - (n/2)*g*g) 0.7071064447 99.99995241313%

18. 0x5f3759df

21. http://www.netlib.org/fdlibm/e_sqrt.c

23. May 2017

26. modern CPU - Intel Xeon E5-2667v3
the hack is actually ~x2 slower in 2017
used to be ~x4 times faster back in ~2003

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