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# volkansalma/atan2_approximation.c

Created June 22, 2012 11:35
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optimized atan2 approximation
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 #include #include #include float atan2_approximation1(float y, float x); float atan2_approximation2(float y, float x); int main() { float x = 1; float y = 0; for( y = 0; y < 2*M_PI; y+= 0.1 ) { for(x = 0; x < 2*M_PI; x+= 0.1) { printf("atan2 for %f,%f: %f \n", y, x, atan2(y, x)); printf("approx1 for %f,%f: %f \n", y, x, atan2_approximation1(y, x)); printf("approx2 for %f,%f: %f \n \n", y, x, atan2_approximation2(y, x)); getch(); } } return 0; } float atan2_approximation1(float y, float x) { //http://pubs.opengroup.org/onlinepubs/009695399/functions/atan2.html //Volkan SALMA const float ONEQTR_PI = M_PI / 4.0; const float THRQTR_PI = 3.0 * M_PI / 4.0; float r, angle; float abs_y = fabs(y) + 1e-10f; // kludge to prevent 0/0 condition if ( x < 0.0f ) { r = (x + abs_y) / (abs_y - x); angle = THRQTR_PI; } else { r = (x - abs_y) / (x + abs_y); angle = ONEQTR_PI; } angle += (0.1963f * r * r - 0.9817f) * r; if ( y < 0.0f ) return( -angle ); // negate if in quad III or IV else return( angle ); } #define PI_FLOAT 3.14159265f #define PIBY2_FLOAT 1.5707963f // |error| < 0.005 float atan2_approximation2( float y, float x ) { if ( x == 0.0f ) { if ( y > 0.0f ) return PIBY2_FLOAT; if ( y == 0.0f ) return 0.0f; return -PIBY2_FLOAT; } float atan; float z = y/x; if ( fabs( z ) < 1.0f ) { atan = z/(1.0f + 0.28f*z*z); if ( x < 0.0f ) { if ( y < 0.0f ) return atan - PI_FLOAT; return atan + PI_FLOAT; } } else { atan = PIBY2_FLOAT - z/(z*z + 0.28f); if ( y < 0.0f ) return atan - PI_FLOAT; } return atan; }

### karussell commented May 10, 2014

Can I use and publish this under Apache 2 license with proper attribution? It would be for graphhopper

### mpaperno commented Feb 2, 2018

In my (somewhat limited but concise) testing on both a very fast Gen 7 Xeon and an STM32F4 (w/FPU) ARM micro show `atan2_approximation1` to be faster (and much more accurate) than the 2nd version.

On STM32F4 `a1` is ~2.3 times faster than stdlib (with gcc-arm 4.7.4)., while `a2` is ~2.1 times faster than stdlib.

On the Xeon under Windows with MSVC `a1` shows maybe a very slight improvement in speed over std (within margin of error, really), while with MinGW-w64 on same system the std version is a whopping 12 times slower!

Thanks for this snip, very handy!

### mooman219 commented Sep 6, 2020 • edited

For anyone looking at this in the future, this function was first referenced on an Apple mailing list in 2005 that's not in their current archives (but it is on the wayback machine!). It was a derivative of a similar function form 1999 which can be found here. That function was in the `public domain` if anyone is concerned about licensing.

### pswiatki commented Aug 16, 2021 • edited

``````if ( x == 0.0f )  // line 62
if ( y == 0.0f ) return 0.0f;  // line 65
``````

Do these lines ever get executed (in cases where 0.0f is not explicitly used as the argument, but rather is a result of some calculation that evaluates "sufficiently close" to 0.0f)?

### Pflugshaupt commented Aug 26, 2021

approximation1 can be made branchless using std::copysign and std::fabs - which boil down to simple bitwise logic.

``````float atan2_approx(float y, float x) {
float abs_y = std::fabs(y) + 1e-10f;      // kludge to prevent 0/0 condition
float r = (x - std::copysign(abs_y, x)) / (abs_y + std::fabs(x));
float angle = M_PI/2.f - std::copysign(M_PI/4.f, x);

angle += (0.1963f * r * r - 0.9817f) * r;
return std::copysign(angle, y);
}
``````

### pswiatki commented Sep 7, 2021 • edited

I was pointing out the fact they used a compare operation to 0.0 float. That will be true very seldom (to put it mildly) and - according to my knowledge - shall be avoided. Let me think about your code snippet above (interesting kludge of 1e-10f ;)

### Pflugshaupt commented Sep 7, 2021 • edited

the kludge line comes from approximation1, you were pointing out the comparisons in approximation2. Sorry, I didn't mean to answer your question, but wanted to comment on the original code because it inspired me to get to a branchless version.
The beauty of a branchless atan2 is that it can easily be performed using SIMD instructions.

### pswiatki commented Sep 7, 2021

very nice, indeed.