Colony

#include "fp_math.h"

#include <avr/pgmspace.h>

#include <math.h>

#include <serial.h>

#define ABS(x) (x > 0 ? x : -x)

#define FP_PI_OVER_TWO 102944

#define FP_TWO_PI 411775

#define TABLE_LENGTH 64

#define TABLE_STEP 1635//.160757

#define EXP_MAGICONSTANT 726817 //ln(2^16) * 2^16

//#define DEBUG

//TODO This can probably be removed at some point.

static int32_t linspace[TABLE_LENGTH] PROGMEM = {

        0, 1634, 3268, 4902, 6536, 8170, 9804, 11438,

        13072, 14706, 16340, 17974, 19608, 21242, 22876, 24510,

        26144, 27778, 29412, 31047, 32681, 34315, 35949, 37583,

        39217, 40851, 42485, 44119, 45753, 47387, 49021, 50655,

        52289, 53923, 55557, 57191, 58825, 60459, 62093, 63727,

        65361, 66995, 68629, 70263, 71897, 73531, 75165, 76799,

        78433, 80067, 81701, 83335, 84969, 86603, 88237, 89871,

        91506, 93140, 94774, 96408, 98042, 99676, 101310, 102944,

};

static int32_t fp_cos_table[TABLE_LENGTH] PROGMEM = {

        65536, 65516, 65455, 65353, 65210, 65027, 64804, 64540,

        64237, 63893, 63509, 63087, 62624, 62123, 61584, 61006,

        60390, 59736, 59046, 58319, 57555, 56756, 55921, 55052,

        54148, 53211, 52241, 51238, 50203, 49138, 48041, 46915,

        45760, 44576, 43364, 42126, 40861, 39571, 38256, 36918,

        35556, 34173, 32768, 31343, 29898, 28435, 26954, 25456,

        23943, 22415, 20872, 19317, 17750, 16171, 14583, 12986,

        11380, 9768, 8149, 6525, 4898, 3267, 1634, 0,

};

//FIXME Lazy implementations of tangent and sine.

int32_t fp_tan(int32_t theta) {

        return fp_div(fp_sin(theta), fp_cos(theta));

}

int32_t fp_sin(int32_t theta) {

        return fp_cos(theta + FP_PI_OVER_TWO);

}

int32_t fp_exp(int32_t x) {

        //Try with partial fractions for now.

        int32_t xsquare = fp_mult(x,x);

        int32_t result, i;

        if(x > EXP_MAGICONSTANT) {

                usb_puts("Output value is out of range.\n\r");

                return -1;

        }

        //This is again, massive amounts of lazyness.

        //The approximation fails outside this range (approximately).

        if(x < (-17l << 16))

                return 0;

        result = xsquare;

        for(i= (22l << 16); i > (2l << 16); i-= (4l << 16)) {

                result += i;

                result = fp_div(xsquare, result);

        }

        result += (2l << 16) - x;

        return (1l << 16) + fp_div(fp_mult((2l << 16), x), result);

}

//Quadratic interpolation.

int32_t fp_cos(int32_t theta) {

        uint8_t n;

        int8_t negative;

        int32_t x_n, x_np1, x_np2;

        int32_t y_n, y_np1, y_np2;

        int32_t dd_n, dd_np1, second_dd, result;

#ifdef DEBUG

        char buf[128];

#endif

        //Move theta into [0, pi/2] w/ appropriate sign.

        theta = ABS(theta) % FP_TWO_PI;

        //Reflecting into [0, pi] doesn't change sign.

        if(theta > FP_PI)

                theta = FP_TWO_PI - theta;

        //Reflecting into [0, pi/2] does.

        if(theta > FP_PI_OVER_TWO) {

                theta = FP_PI - theta;

                negative = 1;

        }

        //Find n such that theta is between x_{n} and x_{n+1}

        n = theta / TABLE_STEP;

        while( n < TABLE_LENGTH - 1

                && pgm_read_dword(&linspace[n+1]) < theta)

                n++;

        //Edge case

        if(n == TABLE_LENGTH - 1) {

                //Perform linear interpolation, since we're close to zero anyway.

                x_n = pgm_read_dword(&linspace[TABLE_LENGTH - 1]);

                y_n = pgm_read_dword(&fp_cos_table[TABLE_LENGTH - 1]);

                result = fp_mult(

                        fp_div(FP_PI_OVER_TWO - x_n, 0 - y_n), theta - x_n) + y_n;

                return negative ? -result : result;

        }

        //Perform quadratic interpolation.

        //Load in the necessary values.

        x_n = pgm_read_dword(&linspace[n]);

        x_np1 = pgm_read_dword(&linspace[n + 1]);

        x_np2  = pgm_read_dword(&linspace[n + 2]);

        y_n = pgm_read_dword(&fp_cos_table[n]);

        y_np1 = pgm_read_dword(&fp_cos_table[n + 1]);

        y_np2 = pgm_read_dword(&fp_cos_table[n + 2]);

#ifdef DEBUG

        sprintf(buf, "x_n = %ld, theta = %ld, x_{n+1} = %ld", x_n, theta, x_np1);

        usb_puts(buf);

#endif

        //Calculate first divided differences.

        dd_n = fp_div(y_np1 - y_n, x_np1 - x_n);

        dd_np1 = fp_div(y_np2 - y_np1, x_np2 - x_np1);

        //Calculate the second divided difference.

        second_dd = fp_div(dd_np1 - dd_n, x_np2 - x_n);

        result = fp_mult(fp_mult(second_dd, theta - x_n), theta - x_np1)

                           + fp_mult(dd_n, theta - x_n) + y_n;

        return negative ? -result : result;

}

//FIXME I didn't write these functions very carefully...

int32_t fp_mult(int32_t a, int32_t b) {

        return (int32_t) (((int64_t)a) * ((int64_t)b)) >>  16;

}

int32_t fp_div(int32_t a, int32_t b) {

        return (int32_t) ((int64_t)a << 16) / (int64_t)b;

}