sin/sin.c

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

#define PI 3.14159265358979323846264338327950288419716939937

#define x0 0.60725293500888125616944675250493
#define y0 0
static double atanlist[100] = {
    0.78539816339744830961566084581988, 0.46364760900080611621425623146121,
    0.24497866312686415417208248121128, 0.12435499454676143503135484916387,
    0.062418809995957348473979112985505, 0.031239833430268276253711744892491,
    0.01562372862047683080280152125657, 0.0078123410601011112964633918421993,
    0.0039062301319669718276286653114244, 0.0019531225164788186851214826250767,
    0.00097656218955931943040343019971729, 0.00048828121119489827546923962564485,
    0.00024414062014936176401672294325966, 0.00012207031189367020423905864611796,
    0.000061035156174208775021662569173829, 0.000030517578115526096861825953438536,
    0.000015258789061315762107231935812698, 7.6293945311019702633884823401051e-6,
    3.814697265606496282923075616373e-6, 1.9073486328101870353653693059172e-6,
    9.5367431640596087942067068992311e-7, 4.7683715820308885992758382144925e-7,
    2.3841857910155798249094797721893e-7, 1.1920928955078068531136849713792e-7,
    5.9604644775390554413921062141789e-8, 2.980232238769530367674013276771e-8,
    1.4901161193847655147092516595963e-8, 7.4505805969238279871365645744954e-9,
    3.7252902984619140452670705718119e-9, 1.8626451492309570290958838214765e-9,
    9.3132257461547851535573547768456e-10, 4.6566128730773925777884193471057e-10,
    2.3283064365386962890204274183882e-10, 1.1641532182693481445259909272985e-10,
    5.8207660913467407226496761591232e-11, 2.9103830456733703613273032698904e-11,
    1.4551915228366851806639597837363e-11, 7.2759576141834259033201841046704e-12,
    3.6379788070917129516601402005838e-12, 1.818989403545856475830076118823e-12,
    9.0949470177292823791503881172787e-13, 4.5474735088646411895751949990348e-13,
    2.2737367544323205947875976170669e-13, 1.1368683772161602973937988232271e-13,
    5.6843418860808014869689941345026e-14, 2.8421709430404007434844970695472e-14,
    1.4210854715202003717422485350606e-14, 7.1054273576010018587112426756617e-15,
    3.5527136788005009293556213378757e-15, 1.7763568394002504646778106689434e-15,
    8.8817841970012523233890533447242e-16, 4.440892098500626161694526672363e-16,
    2.2204460492503130808472633361816e-16, 1.1102230246251565404236316680908e-16,
    5.5511151231257827021181583404541e-17, 2.7755575615628913510590791702271e-17,
    1.3877787807814456755295395851135e-17, 6.9388939039072283776476979255676e-18,
    3.4694469519536141888238489627838e-18, 1.7347234759768070944119244813919e-18,
    8.6736173798840354720596224069595e-19, 4.3368086899420177360298112034798e-19,
    2.1684043449710088680149056017399e-19, 1.0842021724855044340074528008699e-19,
    5.4210108624275221700372640043497e-20, 2.7105054312137610850186320021749e-20,
    1.3552527156068805425093160010874e-20, 6.7762635780344027125465800054371e-21,
    3.3881317890172013562732900027186e-21, 1.6940658945086006781366450013593e-21,
    8.4703294725430033906832250067964e-22, 4.2351647362715016953416125033982e-22,
    2.1175823681357508476708062516991e-22, 1.0587911840678754238354031258496e-22,
    5.2939559203393771191770156292478e-23, 2.6469779601696885595885078146239e-23,
    1.3234889800848442797942539073119e-23, 6.6174449004242213989712695365597e-24,
    3.3087224502121106994856347682799e-24, 1.6543612251060553497428173841399e-24,
    8.2718061255302767487140869206996e-25, 4.1359030627651383743570434603498e-25,
    2.0679515313825691871785217301749e-25, 1.0339757656912845935892608650875e-25,
    5.1698788284564229679463043254373e-26, 2.5849394142282114839731521627186e-26,
    1.2924697071141057419865760813593e-26, 6.4623485355705287099328804067966e-27,
    3.2311742677852643549664402033983e-27, 1.6155871338926321774832201016991e-27,
    8.0779356694631608874161005084957e-28, 4.0389678347315804437080502542479e-28,
    2.0194839173657902218540251271239e-28, 1.009741958682895110927012563562e-28,
    5.0487097934144755546350628178098e-29, 2.5243548967072377773175314089049e-29,
    1.2621774483536188886587657044525e-29, 6.3108872417680944432938285222623e-30,
    3.1554436208840472216469142611311e-30, 1.5777218104420236108234571305656e-30
};

int sign(double alpha)
{
    return alpha > 0 ? 1 : (alpha < 0 ? -1 : 0);
    // return (int)copysign(1.0, alpha);
}

int trig(double theta, double *xy, double *dxy)
{
    int i = 0;
    double t = 1.0; // t=2^-i
    int d = 0;
    double alpha = theta;

    xy[0] = x0;
    xy[1] = y0;
    for (i = 0; i < 56; i++)
    {
        d = alpha > 0 ? 1 : (alpha < 0 ? -1 : 0);
        dxy[0] = -d * xy[1] * t;
        dxy[1] = d * xy[0] * t;
        xy[0] += dxy[0];
        xy[1] += dxy[1];
        alpha -= d * atanlist[i];
        t /= 2.0;
    }
    return i;
}

double sin1(double theta, int *it, double *dsin)
{
    if (theta < 0)
        return -sin1(theta, it, dsin);
    if (theta > 2 * PI)
        return sin1(theta - 2 * PI, it, dsin);
    if (theta > PI)
        return -sin1(theta - PI, it, dsin);
    if (theta > PI / 2)
        return sin1(PI - theta, it, dsin);

    double xy[2];
    double dxy[2];

    *it = trig(theta, xy, dxy);
    *dsin = dxy[1];
    return xy[1];
}

// just do sin, instead of trig()
double sin11(double theta, int *it, double *dsin)
{
    if (theta < 0)
        return -sin11(theta, it, dsin);
    if (theta > 2 * PI)
        return sin11(theta - 2 * PI, it, dsin);
    if (theta > PI)
        return -sin11(theta - PI, it, dsin);
    if (theta > PI / 2)
        return sin11(PI - theta, it, dsin);

    double xy[2];
    double dxy[2];

    int i = 0;
    double t = 1.0; // t=2^-i
    int d = 0;
    double alpha = theta;

    xy[0] = x0;
    xy[1] = y0;
    dxy[1] = 1;
    for (i = 0; i < 55; i++)
    {
        // d = sign(alpha);
        dxy[0] = -xy[1] * copysign(t,alpha);
        dxy[1] = xy[0] * copysign(t,alpha);
        xy[0] += dxy[0];
        xy[1] += dxy[1];
        alpha -= d * atanlist[i];
        t /= 2.0;
    }
    *dsin = dxy[1];
    *it = i;
    return xy[1];
}

double sin2(double theta, int *it, double *dsin)
{
    if (theta < 0)
        return -sin2(theta, it, dsin);
    if (theta > 2 * PI)
        return sin2(theta - 2 * PI, it, dsin);
    if (theta > PI)
        return -sin2(theta - PI, it, dsin);
    if (theta > PI / 2)
        return sin2(PI - theta, it, dsin);

    double result = 0;
    double term = theta;
    int i = 0;
    while (i < 100 && fabs(term) > 1e-16)
    {
        result += term;
        i += 1;
        term *= -1;
        term /= 2 * i;
        term /= (2 * i + 1);
        term *= theta * theta;
    }
    *it = i;
    *dsin = term;
    return result;
}

int main()
{
    int it;
    double thetalist[5] = {1e-4, 0.5, 1, 1.3, 1.7};
    double sin, dsin;
    clock_t start, end;

    for (int i = 0; i < 5; i++)
    {
        start = clock();
        for (int j = 0; j < 10000000; j++)
            sin = sin11(thetalist[i], &it, &dsin);
        end = clock();
        printf("sin11(%.16g) = %.16g, it = %d, dsin=%.16g, took %f s.\n", thetalist[i], sin, it, dsin, (double)(end - start) / CLOCKS_PER_SEC);

        start = clock();
        for (int j = 0; j < 10000000; j++)
            sin = sin2(thetalist[i], &it, &dsin);
        end = clock();
        printf("sin2(%.16g) = %.16g, it = %d, dsin=%.16g, took %f s.\n", thetalist[i], sin, it, dsin, (double)(end - start) / CLOCKS_PER_SEC);
    }

    return 0;
}