As3 code tricks, optimizations, and small discoveries

Math.sin tricks

Jul 30, 2013 ~ 2 Comments ~ Written by benjamin guihaire

Math.sin is slow, we have many options to optimize it, a good post from Jackson dunstan “Even Faster Trig Through Inlining” show lots of methods (http://jacksondunstan.com/articles/1213)
But there is a major problem with all the proposed methods : it only works for a certain range of angle, and if you want to use those functions with any values,
you need to first compute the modulo angle , (angle % (2*PI)) which is pretty slow, so today I am going to explore better and faster solutions !

Taylor serie

so lets start simple, Taylor series:

sin x = x - x^3/3! + x^5/5! - x^7/7!...

we can refactor that code:
sin x= x ( 1- x^2/3! + x^4/5! - x^6/7!...)

we can further do the re-factoring by computing x^2:
x2 = x*x
sin x= x ( 1 + x2* (-1/3! + x2 * (1/5! - x2* (-1/7! ...)

what about the modulo?
the period of the sin function is 2PI.

for our functions, after the modulo, we want values from [-PI;PI], which makes the modulo computation to take even more time:

//modulo 2PI
angle %= 6.283185307179586476925286766559;
//-PI PI range
if(angle>PI) angle-=2*PI else if(angle<-PI) angle +=2*PI;

this can be optimized by first offsetting the values by PI, so we only need to test for values < -PI:

//always wrap input angle to -PI..PI
angle += 3.14159265;
angle %= 6.283185307179586476925286766559;
angle -= 3.14159265;
if (angle < -3.14159265)
     angle += 6.28318531;

and the full code:

		static public function sinTaylor7(rad:Number): Number
		{
			rad += 3.14159265;
			rad %= 6.283185307179586476925286766559;
			rad -= 3.14159265;
			
			//always wrap input angle to -PI..PI
			if (rad < -3.14159265)
				rad += 6.28318531;
			
			const p2:Number = rad * rad;			
			return rad + p2 * rad * ( -0.16666666666 + p2 * ( 0.008333333333333333 - p2 * 0.0001984126984126984));			
		}

hummm, can we do better ?
If we divide the angle by 2PI, the period becomes 1:
angle = angle / (2PI)
or:
angle01 = angle * 0.15915494309

and to compute the modulo, we just need to get the fractional part:
angle01 = angl01 - int(angle01);

so we have:

angle*= 0.15915494309189533576888376337251;	
angle-= (angle>>0);
if(angle>0.5)
   angle-=1.0;
else if(angle<-0.5)
  angle+=1.0;
//then we might want to go back to normal range:
//angle *= 2*PI;

now, back to the taylor serie, we need to re-multiply by 2PI, but we can re-integrate directly the 2PI value in the taylor serie.
and the result is:
x2 = x*x*((2PI)^2)
sin x= x*2PI ( 1 + x2* (-1/3! + x2 * (1/5! - x2* (-1/7! ...)

or better, further develop and refactor, pre compute 1/3! , 1/5! ... 

	static public function sinTaylor7_2(rad:Number): Number
		{
			rad*= 0.15915494309189533576888376337251;	
			rad-= int(rad);
			if(rad>0.5)
				rad-=1.0;
			else if(rad<-0.5)
				rad+=1.0;
			
			const p2:Number = rad * rad;			
			return rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 - p2 * 76.7058597530)));
		}

I also wrote a post before on how to optimize the int conversion by using alchemy opcode tricks:

		static public function sinTaylor7_Normalized(rad:Number): Number
		{
			rad*= 0.15915494309189533576888376337251;	
			fastmem.fastSetDouble(rad+6755399441055744.0,0)
			rad-= fastmem.fastGetI32(0);
			if(rad>0.5)
				rad-=1.0;
			else if(rad<-0.5)
				rad+=1.0;			
			const p2:Number = rad * rad;			
			return rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 - p2 * 76.7058597530)));
		}

If we want to further develop taylor series to degree 9 for bigger precision, we get:

		static public function sinTaylor9(rad:Number): Number
		{
			rad*= 0.15915494309189533576888376337251;	
			fastmem.fastSetDouble(rad+6755399441055744.0,0)
			rad-= fastmem.fastGetI32(0);
			if(rad>0.5)
				rad-=1.0;
			else if(rad<-0.5)
				rad+=1.0;			
			
			const p2:Number = rad * rad;
			return rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 + p2 * ( -76.7058597530 + p2* 42.058693944897653144986811111526))));
		}

Sin LUT

What about Lookup Table solutions ?
The quickest way to compute a modulo is for power of 2 values:
modval = val & (pow2mod-1);
So if we build a table of lets say 2048 values, the index in our lut is:
angleLUT = int(angle * (2048/2PI))
and with the modulo
angleLUT = angleLUT & 2047;

so lets first initialize our LUT table:

for(var luti:int=0;luti<2048;luti++)
			{
				lut[luti&2047] = Math.sin(luti*0.00306796157577128245943617517898); //0.003067 = 2PI/2048
			}

and to get the sin, we just need to do:

sinangle = lut[(a*325.94932345220164765467394738691)&2047]; //325.949 is 2048/2PI

We can also store our lookup table in alchemy byteArray as float or as double, for our test, lets do both:

for(var luti:int=0;luti<2048;luti++)
			{
				lut[luti&2047] = Math.sin(luti*0.00306796157577128245943617517898);
				fastmem.fastSetFloat(lut[luti&2047],((luti&2047)<<2)+1024);
				fastmem.fastSetDouble(lut[luti&2047],((luti&2047)<<3)+10000);
				//trace("sin:",luti,luti*360/6.2831853071795864769,lut[luti&2047]);
			}

and we can once again use our int conversion trick:

for double:

fastmem.fastSetDouble((a*325.94932345220164765467394738691)+6755399441055744.0,0) //325.949 = 2048/2PI
sinangle = fastmem.fastGetDouble(((fastmem.fastGetI32(0)&2047)<<3)+10000); //doubles are 8bytes , so we shift by <<3, and re-add the hardcoded offset of 10000 of the LUT

for float:

fastmem.fastSetDouble((a*325.94932345220164765467394738691)+6755399441055744.0,0) //325.949 = 2048/2PI
sinangle = fastmem.fastGetFloat(((fastmem.fastGetI32(0)&2047)<<2)+1024); //floats are 4 bytes, so we shift by <<2, and re-add the hardcoded offset of 1024 of the LUT

other solutions?

this was the recommended solution in Jackson post, here with the modulo

		static public function sin2(theta:Number):Number
		{
			theta += 3.14159265;
			theta %= 6.283185307179586476925286766559;
			theta -= 3.14159265;
			
			//always wrap input angle to -PI..PI
			if (theta < -3.14159265)
				theta += 6.28318531;
			
			if (theta < 0)
				theta *= 1.27323954  + .405284735 * theta;				
			else
				theta *= 1.27323954 - 0.405284735 * theta;
			
			if (theta < 0)
				theta *= -0.225 * theta + 0.775;
			else
				theta *=  0.225 * theta + 0.775;
			
			return theta;
		}

and here an alternate solution using the [0-1] range to work with any values:

		static public function sin3(angle:Number):Number
		{
			var ax:Number = 0.15915494309189533576888376337251*angle; //angle*1/(2PI)
			var x:Number = ax - (ax>>0);
			var s:Number;
			if(ax<0)
				x += 1.0;
			if(x >= .5)
			{
				x = 2.0*x - 1.5;
				s = -3.6419789056581784278305460054775;//4.0/(1.048*1.048);
			}
			else
			{
				x = 2.0*x - 0.5;
				s = 3.6419789056581784278305460054775;//4.0/(1.048*1.048);
			}
			x*=x;
			return s*(x - .25)*(x - 1.098304);//1.048*1.048
}

Profiling

moment of truth!!!

sin

and the winner is ... LUT !

the full code:

		private function sinTest():void
		{
			//show curve:
			visibleitems.addElement(curve);
			
			initSinLut();
			
			resetCurve();
			drawCurve("Math.sin",Math.sin,-18,18,0.15);
			
			drawCurve("sinTaylor7N",sinTaylor7_Normalized,-18,18,0.15);
			//drawCurve("sinTaylor7",sinTaylor7,-18,18,0.15);
			drawCurve("sinTaylor9",sinTaylor9,-18,18,0.15);
			drawCurve("sin2",sin2,-18,18,0.15);
			drawCurve("sin3",sin3,-18,18,0.15);
			drawCurve("sinLutAlchemyDouble",sinLutAlchemyDouble,-18,18,0.15);			
			drawCurve("sinLUT",sinLut,-18,18,0.15);
			
			var i:int;
			var angle:Number;
			var angleMin:Number = -140;
			var angleMax:Number =  140;
			var angleInc:Number =  0.1;
			var a:Number = 0;
			var rad:Number;
			var p2:Number;
			
			const REPS:int = 3000;	
			
			
			var time1:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					a += Math.sin(angle);
				}
			}
			var time2:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					//a = TestPerf.sinTaylor7_Normalized(angle);
					rad= angle * 0.15915494309189533576888376337251;	
					fastmem.fastSetDouble(rad+6755399441055744.0,0)
					rad-= fastmem.fastGetI32(0);
					if(rad>0.5)
						rad-=1.0;
					else if(rad<-0.5)
						rad+=1.0;			
					p2 = rad * rad;			
					a += rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 - p2 * 76.7058597530)));
				}
			}
			var time3:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					//a = TestPerf.sinTaylor9(angle);
					rad= angle * 0.15915494309189533576888376337251;	
					fastmem.fastSetDouble(rad+6755399441055744.0,0)
					rad-= fastmem.fastGetI32(0);
					if(rad>0.5)
						rad-=1.0;
					else if(rad<-0.5)
						rad+=1.0;			
					
					p2 = rad * rad;
					a += rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 + p2 * ( -76.7058597530 + p2* 42.058693944897653144986811111526))));
					
				}
			}
			var time4:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					a += TestPerf.sin2(angle);
				}
			}
			var time5:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					a += TestPerf.sin3(angle);
				}
			}
			var time6:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					//a = TestPerf.sinTaylor7(angle);
					rad = angle + 3.14159265;
					rad %= 6.283185307179586476925286766559;
					rad -= 3.14159265;
					
					//always wrap input angle to -PI..PI
					if (rad < -3.14159265)
						rad += 6.28318531;
					
					p2  = rad * rad;			
					a += rad + p2 * rad * ( -0.16666666666 + p2 * ( 0.008333333333333333 - p2 * 0.0001984126984126984));			
					
				}
			}
			var time7:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					//a = TestPerf.sinTaylor7_2(angle);
					rad= angle * 0.15915494309189533576888376337251;	
					rad-= (rad>>0);
					if(rad>0.5)
						rad-=1.0;
					else if(rad<-0.5)
						rad+=1.0;
					
					p2 = rad * rad;			
					a += rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 - p2 * 76.7058597530)));

				}
			}
			var time8:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					a += lut[(angle*325.94932345220164765467394738691)&2047]
				}
			}
			var time9:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					a += fastmem.fastGetFloat((((angle*325.94932345220164765467394738691)&2047)<<2)+1024);
				}
			}
			var time10:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					a += fastmem.fastGetDouble((((angle*325.94932345220164765467394738691)&2047)<<3)+10000);
				}
			}
			var time11:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					fastmem.fastSetDouble((angle*325.94932345220164765467394738691)+6755399441055744.0,0)
					a += fastmem.fastGetDouble(((fastmem.fastGetI32(0)&2047)<<3)+10000);
				}
			}
			var time12:uint = getTimer();
			for(i=0;i<REPS;i++)
			{						
				for(angle=angleMin;angle<angleMax;angle+=angleInc)
				{
					fastmem.fastSetDouble((angle*325.94932345220164765467394738691)+6755399441055744.0,0)
					a += fastmem.fastGetFloat(((fastmem.fastGetI32(0)&2047)<<2)+1024);
				}
			}
			var time13:uint = getTimer();
								
			startTest();
			setTest( "Math.sin"       , "sin", time2-time1  );
			setTest( "sinTaylor7_Normalized"       , "sin", time3-time2  );
			setTest( "sinTaylor7"       , "sin", time7-time6  );
			setTest( "sinTaylor7_2"       , "sin", time8-time7  );
			setTest( "sinTaylor9"       , "sin", time4-time3  );
			setTest( "sin2"       , "sin", time5-time4  );
			setTest( "sin3"       , "sin", time6-time5  );
			setTest( "sinLUTVector"       , "sin", time9-time8  );
			setTest( "sinLUTAlchemyFloat"       , "sin", time10-time9  );
			setTest( "sinLUTAlchemyFloat2"       , "sin", time13-time12  );
			setTest( "sinLUTAlchemyDouble"       , "sin", time11-time10  );
			setTest( "sinLUTAlchemyDouble2"      , "sin", time12-time11  );
			endTest();		

		}

		static public function sin2(theta:Number):Number
		{
			theta += 3.14159265;
			theta %= 6.283185307179586476925286766559;
			theta -= 3.14159265;
			
			//always wrap input angle to -PI..PI
			if (theta < -3.14159265)
				theta += 6.28318531;
			
			if (theta < 0)
				theta *= 1.27323954  + .405284735 * theta;				
			else
				theta *= 1.27323954 - 0.405284735 * theta;
			
			if (theta < 0)
				theta *= -0.225 * theta + 0.775;
			else
				theta *=  0.225 * theta + 0.775;
			
			return theta;
		}
		
		static public function sinTaylor7(rad:Number): Number
		{
			rad += 3.14159265;
			rad %= 6.283185307179586476925286766559;
			rad -= 3.14159265;
			
			//always wrap input angle to -PI..PI
			if (rad < -3.14159265)
				rad += 6.28318531;
			
			const p2:Number = rad * rad;			
			return rad + p2 * rad * ( -0.16666666666 + p2 * ( 0.008333333333333333 - p2 * 0.0001984126984126984));			
		}
		static public function sinTaylor7_2(rad:Number): Number
		{
			rad*= 0.15915494309189533576888376337251;	
			rad-= (rad>>0);
			if(rad>0.5)
				rad-=1.0;
			else if(rad<-0.5)
				rad+=1.0;
			
			const p2:Number = rad * rad;			
			return rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 - p2 * 76.7058597530)));
		}
			
		//fastmem.fastSetDouble(val+6755399441055744.0,0);
		//fastmem.fastGetI32(0);
		
		static public function sinTaylor7_Normalized(rad:Number): Number
		{
			rad*= 0.15915494309189533576888376337251;	
			fastmem.fastSetDouble(rad+6755399441055744.0,0)
			rad-= fastmem.fastGetI32(0);
			if(rad>0.5)
				rad-=1.0;
			else if(rad<-0.5)
				rad+=1.0;			
			const p2:Number = rad * rad;			
			return rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 - p2 * 76.7058597530)));
		}
		
		
		static public function sinTaylor9(rad:Number): Number
		{
			rad*= 0.15915494309189533576888376337251;	
			fastmem.fastSetDouble(rad+6755399441055744.0,0)
			rad-= fastmem.fastGetI32(0);
			if(rad>0.5)
				rad-=1.0;
			else if(rad<-0.5)
				rad+=1.0;			
			
			const p2:Number = rad * rad;
			return rad* (6.2831853071795864769 + p2 *  ( -41.341702240399760 + p2 * ( 81.60524927 + p2 * ( -76.7058597530 + p2* 42.058693944897653144986811111526))));
		}
		
		
		
		/** optimized sin cos function  */
		static public function sin3(angle:Number):Number
		{
			var ax:Number = 0.15915494309189533576888376337251*angle; //angle*1/(2PI)
			var x:Number = ax - (ax>>0);
			var s:Number;
			if(ax<0)
				x += 1.0;
			if(x >= .5)
			{
				x = 2.0*x - 1.5;
				s = -3.6419789056581784278305460054775;//4.0/(1.048*1.048);
			}
			else
			{
				x = 2.0*x - 0.5;
				s = 3.6419789056581784278305460054775;//4.0/(1.048*1.048);
			}
			x*=x;
			return s*(x - .25)*(x - 1.098304);//1.048*1.048
			/*			ax += 0.25;
			x = ax - int(ax);
			if(ax < 0.0)
			x += 1.0;
			if(x >= .5)
			{
			x = 2.0*x - 1.5;
			c = -B_FACT;
			}
			else
			{
			x = 2.0*x - 0.5;
			c = B_FACT;
			}
			x*=x;
			c *= (x - .25)*(x - A_FACT);
			*/			
		}

What do you get?

Posted in Math - Tagged , ,
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2 Comments
  1. Jackson Dunstan's Gravatar Jackson Dunstan
    July 30, 2013 at 5:12 pm | Permalink

    LUT Vector is fastest for me by a little on Windows 7 64-bit, 2.67 GHz Intel Xeon X550, Flash Player 11.8. It’s nice to not have to use any assembly or “alchemy” opcodes. :)

    Reply
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