Approximations

atan_approx.c

// Atan approximation
// https://www.dsprelated.com/showarticle/1052.php#tabs1-comments

#include <math.h>
#include <stdio.h>

double atan_approx(double z)
{
     double m = z<0? -1.0:1.0;
     z *= m;
     double y = 0.141499*z*z*z*z + 
          -0.343315*z*z*z +
          -0.016224*z*z +
          1.003839*z +
          -0.000158;
     return m*y;
}

double atan2_approx(double y, double x)
{
     double ay = fabs(y), ax = fabs(x);
     int invert = ay > ax;
     double z = invert ? ax/ay : ay/ax; // [0,1]
     double th = atan_approx(z);        // [0,π/4]
     if(invert) th = M_PI_2 - th;       // [0,π/2]
     if(x < 0) th = M_PI - th;          // [0,π]
     th = copysign(th, y);              // [-π,π]
     return th;
}

int main(int argc, char **argv)
{
    float x0 = -1.0f;
    float x1 = x0 + 2.f;
    float xinc = (x1 - x0)/50.0f;

    double err_max = 0.0;
    double lsqerr = 0.0;
    int lsqerr_n = 0;

    printf("x\tatan\tatan_approx\terr\n");
    for (float x = x0; x <= x1; x += xinc) {
        double y = atan(x);
        double ay = atan_approx(x);
        double err = fabs(ay - y);
        err_max = err_max<err? err:err_max;
        printf("%f\t%f\t%f\t%f\n", x, y, ay, err);
        lsqerr += err*err;
        ++lsqerr_n;
    }
    printf("LSQERR: %ef\n", lsqerr/lsqerr_n);
    printf("MAXERR: %f\n", err_max);
    printf("PI_4: %f\n", M_PI/4.0);
    
    return 0;   
}

/*
x	atan	atan_approx	err
-1.000000	-0.785398	-0.785641	0.000243
-0.960000	-0.764993	-0.765014	0.000021
-0.920000	-0.743756	-0.743675	0.000080
-0.880000	-0.721655	-0.721553	0.000102
-0.840000	-0.698660	-0.698583	0.000077
-0.800000	-0.674741	-0.674711	0.000030
-0.760000	-0.649870	-0.649889	0.000018
-0.720000	-0.624023	-0.624080	0.000057
-0.680000	-0.597177	-0.597256	0.000079
-0.640000	-0.569313	-0.569395	0.000082
-0.600000	-0.540419	-0.540487	0.000067
-0.560000	-0.510488	-0.510528	0.000040
-0.520000	-0.479519	-0.479524	0.000005
-0.480000	-0.447520	-0.447490	0.000030
-0.440000	-0.414507	-0.414449	0.000058
-0.400000	-0.380506	-0.380432	0.000074
-0.360000	-0.345555	-0.345480	0.000075
-0.320000	-0.309703	-0.309643	0.000060
-0.280000	-0.273009	-0.272978	0.000030
-0.240000	-0.235545	-0.235552	0.000007
-0.200000	-0.197395	-0.197441	0.000045
-0.160000	-0.158655	-0.158727	0.000072
-0.120000	-0.119429	-0.119505	0.000076
-0.080000	-0.079830	-0.079875	0.000045
-0.040000	-0.039979	-0.039948	0.000031
0.000000	0.000000	-0.000158	0.000158
0.040000	0.039979	0.039948	0.000031
0.080000	0.079830	0.079875	0.000045
0.120000	0.119429	0.119505	0.000076
0.160000	0.158655	0.158728	0.000072
0.200000	0.197396	0.197441	0.000045
0.240000	0.235545	0.235552	0.000007
0.280000	0.273009	0.272978	0.000030
0.320000	0.309703	0.309643	0.000060
0.360000	0.345556	0.345480	0.000075
0.400000	0.380506	0.380432	0.000074
0.440000	0.414507	0.414449	0.000058
0.480000	0.447520	0.447490	0.000030
0.520000	0.479519	0.479524	0.000005
0.560000	0.510488	0.510528	0.000040
0.600000	0.540420	0.540487	0.000068
0.640000	0.569313	0.569395	0.000082
0.680000	0.597177	0.597256	0.000079
0.720000	0.624023	0.624080	0.000057
0.760000	0.649871	0.649889	0.000018
0.800000	0.674741	0.674711	0.000030
0.840000	0.698660	0.698583	0.000077
0.880000	0.721655	0.721553	0.000102
0.920000	0.743756	0.743676	0.000080
0.960000	0.764993	0.765014	0.000021
LSQERR: 4.990835e-09f
MAXERR: 0.000243
PI_4: 0.785398
*/

waldorf_sin.c

// Pseudo Sine function.
//
// Originally from Stefan Stenzel, CEO of Waldorf
// @url: https://namoseley.wordpress.com/2017/01/03/pseudo-sinusoidal-waveform-ii/
// https://twitter.com/5tenzel/status/816225338330140673

#include <math.h>
#include <stdio.h>

/* x in [-1,1] -> [-1,1] */
float waldorf_cycle(float x)
{
    float abs_x = x < 0.f? -x:x;
    return x*4.f*(1 - abs_x);
}

/* x in [-1,1] -> [-1,1] */
float cycle(float x)
{
    return sin(x*3.14159265358979323846);
}

int main(int argc, char **argv)
{
    float x0 = -1.0f;
    float x1 = x0 + 2.f;
    float xinc = (x1 - x0)/50.0f;

    float err_max = 0.0;
    float lsqerr = 0.0;
    int lsqerr_n = 0;

    printf("x\tcycle\twaldorf_cycle\terr\n");
    for (float x = x0; x <= x1; x += xinc) {
        float y = cycle(x);
        float wy = waldorf_cycle(x);
        float err = fabs(wy - y);
        err_max = err_max<err? err:err_max;
        printf("%f\t%f\t%f\t%f\n", x, wy, y, err);
        lsqerr += err*err;
        ++lsqerr_n;
    }
    printf("LSQERR: %f\n", lsqerr/lsqerr_n);
    printf("MAXERR: %f\n", err_max);
    
    return 0;   
}

/*
clang waldorf_sin.c -o waldorf_sin && ./waldorf_sin
x	cycle	waldorf_cycle	err
-1.000000	-0.000000	-0.000000	0.000000
-0.960000	-0.153600	-0.125333	0.028267
-0.920000	-0.294400	-0.248690	0.045710
-0.880000	-0.422400	-0.368125	0.054275
-0.840000	-0.537600	-0.481754	0.055846
-0.800000	-0.640000	-0.587786	0.052215
-0.760000	-0.729600	-0.684547	0.045053
-0.720000	-0.806400	-0.770514	0.035887
-0.680000	-0.870400	-0.844328	0.026072
-0.640000	-0.921600	-0.904827	0.016773
-0.600000	-0.960000	-0.951057	0.008943
-0.560000	-0.985600	-0.982287	0.003313
-0.520000	-0.998400	-0.998027	0.000373
-0.480000	-0.998400	-0.998027	0.000373
-0.440000	-0.985600	-0.982287	0.003313
-0.400000	-0.960000	-0.951056	0.008944
-0.360000	-0.921600	-0.904827	0.016773
-0.320000	-0.870400	-0.844328	0.026072
-0.280000	-0.806400	-0.770513	0.035887
-0.240000	-0.729600	-0.684547	0.045053
-0.200000	-0.640000	-0.587785	0.052215
-0.160000	-0.537600	-0.481753	0.055846
-0.120000	-0.422399	-0.368124	0.054275
-0.080000	-0.294399	-0.248689	0.045710
-0.040000	-0.153599	-0.125333	0.028267
0.000000	0.000001	0.000001	0.000000
0.040000	0.153601	0.125334	0.028267
0.080000	0.294401	0.248690	0.045710
0.120000	0.422401	0.368125	0.054275
0.160000	0.537600	0.481754	0.055846
0.200000	0.640000	0.587786	0.052215
0.240000	0.729600	0.684547	0.045053
0.280000	0.806400	0.770514	0.035887
0.320000	0.870400	0.844328	0.026072
0.360000	0.921600	0.904827	0.016773
0.400000	0.960000	0.951057	0.008943
0.440000	0.985600	0.982287	0.003313
0.480000	0.998400	0.998027	0.000373
0.520000	0.998400	0.998027	0.000373
0.560000	0.985600	0.982287	0.003313
0.600000	0.960000	0.951056	0.008943
0.640000	0.921600	0.904827	0.016773
0.680000	0.870400	0.844328	0.026072
0.720000	0.806400	0.770513	0.035887
0.760000	0.729600	0.684547	0.045053
0.800000	0.639999	0.587785	0.052215
0.840000	0.537599	0.481753	0.055846
0.880000	0.422399	0.368124	0.054275
0.920000	0.294399	0.248689	0.045710
0.960000	0.153599	0.125332	0.028267
LSQERR: 0.001284
MAXERR: 0.055846
*/