Calculate atan2 without std functions or C99

Question

I am calculating angles from a 3-axis accelerometer, but my compiler doesn\'t have a atan or atan2 function. It has a reserved memory slot, but it calls a function i can\'t

Answer 1

The following code uses a rational approximation to get the arctangent normalized to the [0 1) interval (you can multiply the result by Pi/2 to get the real arctangent)

normalized_atan(x) ~ (b x + x^2) / (1 + 2 b x + x^2)

where b = 0.596227

The maximum error is 0.1620º

#include 
#include 

// Approximates atan(x) normalized to the [-1,1] range
// with a maximum error of 0.1620 degrees.

float normalized_atan( float x )
{
    static const uint32_t sign_mask = 0x80000000;
    static const float b = 0.596227f;

    // Extract the sign bit
    uint32_t ux_s  = sign_mask & (uint32_t &)x;

    // Calculate the arctangent in the first quadrant
    float bx_a = ::fabs( b * x );
    float num = bx_a + x * x;
    float atan_1q = num / ( 1.f + bx_a + num );

    // Restore the sign bit
    uint32_t atan_2q = ux_s | (uint32_t &)atan_1q;
    return (float &)atan_2q;
}

// Approximates atan2(y, x) normalized to the [0,4) range
// with a maximum error of 0.1620 degrees

float normalized_atan2( float y, float x )
{
    static const uint32_t sign_mask = 0x80000000;
    static const float b = 0.596227f;

    // Extract the sign bits
    uint32_t ux_s  = sign_mask & (uint32_t &)x;
    uint32_t uy_s  = sign_mask & (uint32_t &)y;

    // Determine the quadrant offset
    float q = (float)( ( ~ux_s & uy_s ) >> 29 | ux_s >> 30 ); 

    // Calculate the arctangent in the first quadrant
    float bxy_a = ::fabs( b * x * y );
    float num = bxy_a + y * y;
    float atan_1q =  num / ( x * x + bxy_a + num );

    // Translate it to the proper quadrant
    uint32_t uatan_2q = (ux_s ^ uy_s) | (uint32_t &)atan_1q;
    return q + (float &)uatan_2q;
} 

In case you need more precision, there is a 3rd order rational function:

normalized_atan(x) ~ ( c x + x^2 + x^3) / ( 1 + (c + 1) x + (c + 1) x^2 + x^3)

where c = (1 + sqrt(17)) / 8

which has a maximum approximation error of 0.00811º