Best Square Root Method - Algorithm - Function (Precision VS Speed)

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Introduction

I enjoy Game Programming with Directx and I noticed that the most called method throughout most of my games is the standard sqrt method in the Math.h and this made me search for faster functions than the standard sqrt. And after some searching, I found lots of functions that were much much faster but it's always a compromise between speed and precision. The main purpose of this article is to help people choose the best square-root method that suits their program.

Background

In this article, I compare 14 different methods for computing the square root with the standard sqrt function as a reference, and for each method I show its precision and speed compared to the sqrt method.

What this Article is Not About

1. Explaining how each method works
2. New ways to compute the square root

Using the Code

The code is simple, it basically contains:

1. main.cpp

Calls all the methods and for each one of them, it computes the speed and precision relative to the sqrt function.

2. SquareRootmethods.h

This Header contains the implementation of the functions, and the reference of where I got them from.

First I calculate the Speed and Precision of the sqrt method which will be my reference.

For computing the Speed, I measure the time it takes to call the sqrt function (M-1) times and I assign this value to RefSpeed which will be my reference.

And for computing the Precision, I add the current result to the previous result in RefTotalPrecision every time I call the method. <code>RefTotalPrecision will be my reference.

For measuring runtime duration (Speed) of the methods, I use the CDuration class found in this link, and I use dur as an instance of that class.

for(int j=0;j<AVG;j++)	
{	
  dur.Start();
		
	for(int i=1;i<M;i++)
	   RefTotalPrecision+=sqrt((float) i);
	
  dur.Stop();
	
  Temp+=dur.GetDuration();	
}
	
RefTotalPrecision/=AVG;
Temp/=AVG;

RefSpeed=(float)(Temp)/CLOCKS_PER_SEC;      

And for the other methods I do the same calculations, but in the end, I reference them to the sqrt.

	for(int j=0;j<AVG;j++)	
	{	
	  dur.Start();
		
		for(int i=1;i<M;i++)
		    TotalPrecision+=sqrt1((float) i);
	
          dur.Stop();
	
	  Temp+=dur.GetDuration();	
	}
	
	TotalPrecision/=AVG;
	Temp/=AVG;
		
	Speed=(float)(Temp)/CLOCKS_PER_SEC;

cout<<"Precision = "
<<(double)(1-abs((TotalPrecision-RefTotalPrecision)/(RefTotalPrecision)))*100<<endl;         

NOTES:

1. I assume that the error in Precision whether larger or smaller than the reference is equal, that's why I use "abs".
2. The Speed is referenced as the actual percentage, while the Precision is referenced as a decrease percentage.

You can modify the value of M as you like, I initially assign it with 10000.

You can modify AVG as well, the higher it is, the more accurate the results.

#define M 10000   
#define AVG 10   

Points of Interest

Precision wise, the sqrt standard method is the best. But the other functions can be much faster even 5 times faster. I would personally choose Method N# 14 as it has high precision and high speed, but I'll leave it for you to choose. :)

I took 5 samples and averaged them and here is the output:

According to the analysis the above Methods Performance Ranks (Speed x Precision) is:

NOTE: The performance of these methods depends highly on your processor and may change from one computer to another.

The METHODS

Sqrt1

Reference: http://ilab.usc.edu/wiki/index.php/Fast_Square_Root

Algorithm: Babylonian Method + some manipulations on IEEE 32 bit floating point representation

float sqrt1(const float x)  
{
  union
  {
    int i;
    float x;
  } u;
  u.x = x;
  u.i = (1<<29) + (u.i >> 1) - (1<<22); 
  
  // Two Babylonian Steps (simplified from:)
  // u.x = 0.5f * (u.x + x/u.x);
  // u.x = 0.5f * (u.x + x/u.x);
  u.x =       u.x + x/u.x;
  u.x = 0.25f*u.x + x/u.x;

  return u.x;
}  

Sqrt2

Reference: http://ilab.usc.edu/wiki/index.php/Fast_Square_Root

Algorithm: The Magic Number (Quake 3)

#define SQRT_MAGIC_F 0x5f3759df 
 float  sqrt2(const float x)
{
  const float xhalf = 0.5f*x;
 
  union // get bits for floating value
  {
    float x;
    int i;
  } u;
  u.x = x;
  u.i = SQRT_MAGIC_F - (u.i >> 1);  // gives initial guess y0
  return x*u.x*(1.5f - xhalf*u.x*u.x);// Newton step, repeating increases accuracy 
}   

Sqrt3

Reference: http://ilab.usc.edu/wiki/index.php/Fast_Square_Root

Algorithm: Log base 2 approximation and Newton's Method

float sqrt3(const float x)  
{
  union
  {
    int i;
    float x;
  } u;

  u.x = x;
  u.i = (1<<29) + (u.i >> 1) - (1<<22); 
  return u.x;
} 

Sqrt4

Reference: I got it a long time a go from a forum and I forgot, please contact me if you know its reference.

Algorithm: Bakhsali Approximation

float sqrt4(const float m)
{
   int i=0; 
   while( (i*i) <= m )
          i++;
    i--; 
   float d = m - i*i; 
 float p=d/(2*i); 
 float a=i+p; 
   return a-(p*p)/(2*a);
}  

Sqrt5

Reference: http://www.dreamincode.net/code/snippet244.htm

Algorithm: Babylonian Method

   float sqrt5(const float m)
{
   float i=0;
   float x1,x2;
   while( (i*i) <= m )
          i+=0.1f;
   x1=i;
   for(int j=0;j<10;j++)
   {
       x2=m;
      x2/=x1;
      x2+=x1;
      x2/=2;
      x1=x2;
   }
   return x2;
}   

Sqrt6

Reference: http://www.azillionmonkeys.com/qed/sqroot.html#calcmeth

Algorithm: Dependant on IEEE representation and only works for 32 bits

double sqrt6 (double y) 
{
    double x, z, tempf;
    unsigned long *tfptr = ((unsigned long *)&tempf) + 1;
    tempf = y;
   *tfptr = (0xbfcdd90a - *tfptr)>>1; 
 x =  tempf;
 z =  y*0.5;                       
 x = (1.5*x) - (x*x)*(x*z);    //The more you make replicates of this statement 
                               //the higher the accuracy, here only 2 replicates are used  
  x = (1.5*x) - (x*x)*(x*z);       
  return x*y; 
  }  

Sqrt7

Reference: http://bits.stephan-brumme.com/squareRoot.html

Algorithm: Dependant on IEEE representation and only works for 32 bits

float sqrt7(float x)
 {
   unsigned int i = *(unsigned int*) &x; 
   // adjust bias
   i  += 127 << 23;
   // approximation of square root
   i >>= 1; 
   return *(float*) &i;
 }   

Sqrt8

Reference: http://forums.techarena.in/software-development/1290144.htm

Algorithm: Babylonian Method

double sqrt9( const double fg)
{ 
 double n = fg / 2.0;
 double lstX = 0.0; 
 while(n != lstX)  
 { 
 lstX = n;
 n = (n + fg/n) / 2.0; 
 }
 return n;
 }  

Sqrt9

Reference: http://www.functionx.com/cpp/examples/squareroot.htm

Algorithm: Babylonian Method

 double Abs(double Nbr)
{ 
 if( Nbr >= 0 ) 
  return Nbr; 
 else
  return -Nbr;
}

double sqrt10(double Nbr)
{
 double Number = Nbr / 2; 
 const double Tolerance = 1.0e-7; 
 do
 {
  Number = (Number + Nbr / Number) / 2;
 }while( Abs(Number * Number - Nbr) > Tolerance);
	
 return Number;
}   

Sqrt10

Reference: http://www.cs.uni.edu/~jacobson/C++/newton.html

Algorithm: Newton's Approximation Method

double sqrt11(const double number)e
{
const double ACCURACY=0.001;
double lower, upper, guess;

 if (number < 1)
 {
  lower = number;
  upper = 1;
 }
 else
 {
  lower = 1;
  upper = number;
 }

 while ((upper-lower) > ACCURACY)
 {
  guess = (lower + upper)/2;
  if(guess*guess > number)
   upper =guess;
  else
   lower = guess; 
 }
 return (lower + upper)/2;

}  

Sqrt11

Reference: http://www.drdobbs.com/184409869;jsessionid=AIDFL0EBECDYLQE1GHOSKH4ATMY32JVN

Algorithm: Newton's Approximation Method

 double sqrt12( unsigned long N )
{
    double n, p, low, high;
    if( 2 > N )
        return( N );
    low  = 0;
    high = N;
    while( high > low + 1 )
    {
        n = (high + low) / 2;
        p = n * n;
        if( N < p )
            high = n;
        else if( N > p )
            low = n;
        else
            break;
    }
    return( N == p ? n : low );
}  

Sqrt12

Reference: http://cjjscript.q8ieng.com/?p=32

Algorithm: Babylonian Method

double sqrt13( int n )
{
	// double a = (eventually the main method will plug values into a)
	double a = (double) n;
	double x = 1;
 
	// For loop to get the square root value of the entered number.
	for( int i = 0; i < n; i++)
	{
		x = 0.5 * ( x+a / x );
	}
 
	return x;
}    

Sqrt13

Reference: N/A

Algorithm: Assembly fsqrt

double sqrt13(double n)
{
  __asm{
     fld n
     fsqrt
   }
}

Sqrt14

Reference: N/A

Algorithm: Assembly fsqrt 2

double inline __declspec (naked) __fastcall sqrt14(double n)
{
	_asm fld qword ptr [esp+4]
	_asm fsqrt
	_asm ret 8
} 

History

1.3 (15 September 2010)

• Added Method N#14 (which is the best method till now)
• Added modified source code

1.2 (24 June 2010)

• Added Method N#13
• Added the Methods Performance Rank
• Added modified source code

1.1 (3 April 2010)

• Added Precision Timer instead of clock because it's more precise
• Added the average feature

1.0 (31 March 2010)

• Initial release

I hope that this article would at least slightly help those who are interested in this issue.