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Fast trigonometric approximations

 
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Ttelmah



Joined: 11 Mar 2010
Posts: 19531

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Fast trigonometric approximations
PostPosted: Fri Jun 09, 2017 2:50 pm     Reply with quote

Some time ago, I posted a much faster atan2 in the general forum.
Decided to post it here together with another fast set of code for sin and cosine:

The accuracy is better than half a degree, and it is _fast_. Takes about 70uSec on a 40Mhz PIC, while atan2 takes at least ten times longer. It is also smaller.... It only gives positive angles, from 0 to 360 as it's result. This works in degrees but is easy to convert to radians if required.

Code:

//Processor defines here....
#include <math.h>

float angle(float X, float Y)
{
   //routine to give a fast solution for angle, from X/Y co-ordinates - result in degrees
   float AX,AY,ival,oval,aival;
   int8 quad;
   AX=fabs(X);
   AY=fabs(Y);
   //Now the approximation used works for tan from -1 to 1, so we have to keep the
   //values inside this range and adjust the input/output.
   //Four 'quadrants' are decoded -1 to 1, (315 to 45 degrees), then 45 to 135,
   //135 to 225, 225 to 315
   If (X >= 0) //Right hand half of the circle
   {
      If (AY > X)
      {
         If (Y < 0)
         {
             quad = 4;
             ival = -X / Y;
         }
         Else
         {
             quad = 2;
             ival = X / -Y;
         }
      }
      Else
      {
         If (AY > X)
         {
             quad = 4;
             ival = -Y / X;
         }
         else
         {
             quad = 1;
             ival = Y / X;
         }
      }
   }
   else
   {
      //Now the others
      If (Y > AX)
      {
         quad = 2;
         ival = X / -Y;
      }
      Else
      {
         If (AY > AX)
         {         
             quad = 4;
             ival = -X / Y;
         }
         Else
         {
             quad = 3;
             ival = -Y / -X;
         }
      }
   }
   //A lot of lines of code, but small and quick really.....
   //Now the solution
   //Now approximation for atan from -1 to +1, giving an answer in degrees.

   aival = fAbs(ival);
   oval = 45 * ival - ival * (aival - 1) * (14.02 + 3.79 * aival);
   
   //Now solve back to the final result
   If (quad != 1)
   {
      If (quad == 2)
          oval = oval + 90;
      Else
      {
          If (quad == 3)
              oval = oval + 180;
          Else
              oval = oval + 270;
      }
   }
   if (oval<0)
      oval+=360;
   return oval;
}   

//Demo program using pairs of numbers from the array to test
void main()
{
   const signed int16 source[] = {0,300,600,1000,0,-300,-600,-1000};
   int8 ctr,ctr2=0;
   signed int16 X,Y;

   while (TRUE)
   {
      for (ctr=0;ctr<8;ctr++)
      {
         //Now loop through the array, using pairs from two counters as X/Y
         X=source[ctr];
         Y=source[ctr2];
         printf("X %ld, Y %ld, angle %5.1f\n\r", X,Y,angle(X,Y));
      }
      if (ctr2<7)
         ctr2++;
      else
         ctr2=0;
   }
}


Now I have added this code for sin/cosine. This is in radians for +PI to -PI input range.
Code:

//A fast float approximation to sin over the range +/-PI
#define PI2 (PI*PI)
#define INV4  (4/PI)
#define INV4SQ  (-4/(PI2))
#define P  0.225

float fast_sin(float x)
{
    float y;
    y = (INV4 * x) + (INV4SQ * x * fabs(x));
    return P * (y * fabs(y) - y) + y;   
}

float fast_cos(float x)
{  //This is done by shifting the quadrants
   x += PIDIV2;
   if(x > PI)   // Original x > pi/2
   {
      x -= PI2;   // Wrap: cos(x) = cos(x - 2 pi)
   }

   return fast_sin(x);
}

void main(void)
{
  //Test the fast sin algorithm for angles from 0 to PI in steps of PI/128
  float an, res, sres,;

  for (an=0.0;an<(PI);an+=(PI/128))
  {
     //res1=fast_sin1(an);
     res=fast_sin(an);
     sres=sin(an);
     printf ("AN=%5.3f sinfast=%5.3f sin=%5.3f\n",an,res,sres);
  }
  while(TRUE)
     delay_cycles(1);
}


I don't show a test for the cos, but this works exactly the same. Slightly slower.
tinley



Joined: 09 May 2006
Posts: 67

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Great code thanks
PostPosted: Tue Jan 07, 2020 7:03 am     Reply with quote

Thanks Ttelmah. This code is exactly what I was looking for. You mention that the quadrants may not be correct. They don't match CCS atan2 function and I think you were right to suspect. I substituted the following:

result = (atan2( input_value_X , input_value_Y ) * 57.2958 ); // 180/pi = 57.2958

With:

result = angle(input_value_X, input_value_Y);

And I had to swap the X and Y to get it to work.

But it does work very well and quickly! Thank you!
tinley



Joined: 09 May 2006
Posts: 67

View user's profile Send private message Visit poster's website

Fast COS
PostPosted: Tue Jan 07, 2020 9:07 am     Reply with quote

As it happens, I can also use your fast COS in the same project. But this did have a couple of bugs which are fixed in the following:

Code:

//A fast float approximation to sin over the range +/-PI
#define PI2 (PI*PI)
#define INV4  (4/PI)
#define INV4SQ  (-4/(PI2))
#define P  0.225
#define PIDIV2 (PI/2)
#define PIX2 (PI*2)

float fast_sin(float x)
{
    float y;
    y = (INV4 * x) + (INV4SQ * x * fabs(x));
    return P * (y * fabs(y) - y) + y;   
}

float fast_cos(float x)
{  //This is done by shifting the quadrants
   x += PIDIV2;
   if(x > PI)   // Original x > pi/2
   {
      x -= PIX2;   // Wrap: cos(x) = cos(x - 2 pi)
   }

   return fast_sin(x);
}


But still save me heaps of time trying to save time! Thank you!
Ttelmah



Joined: 11 Mar 2010
Posts: 19531

View user's profile Send private message

PostPosted: Mon Jan 27, 2020 1:16 pm     Reply with quote

Well done.
Glad you got it sorted. Very Happy
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