Sinus function using Taylor expansion - c

Question

The teacher asks to remove the pi subtraction cycle in the main function. I don’t know how to write the program so that the correct results will come out for any values.
#include <stdio.h>
#include <math.h>
double sinus(double x);
int main(void) {
double a, x;
scanf("%le", & x);
a = x;
while (fabs(x) > 2 * (M_PI)) {
x = fabs(x) - 2 * (M_PI);
}
if (a > 0)
a = sinus(x);
else a = (-1) * sinus(x);
printf("%le", (double) a);
return 0;
}
double sinus(double x) {
double sum = 0, h, eps = 1.e-16;
int i = 2;
h = x;
do {
sum += h;
h *= -((x * x) / (i * (i + 1)));
i += 2;
}
while (fabs(h) > eps);
return sum;
return 0;
}

Answer 1

#include <stdio.h>
#include <math.h>
double sinus(double x);
int main(void)
{
double a,x;
scanf("%le",&x);
a=x;
x=fmod(fabs(x),2*(M_PI));
if(a>0)
a=sinus(x);
else a=(-1)*sinus(x);
printf("%le",(double)a);
return 0;}
double sinus(double x)
{
double sum=0, h, eps=1.e-16; int i=2;
h=x;
do{
sum+=h;
h*=-((x*x)/(i*(i+1)));
i+=2;}
while( fabs(h)>eps );
return sum;
return 0;
}

Answer 2

… how to write the program so that the correct results will come out for any values.
OP's loop is slow with large x and an infinfite loop with very large x:
while (fabs(x) > 2 * (M_PI)) {
x = fabs(x) - 2 * (M_PI);
}
A simple, though not high quality solution, is to use fmod() in the function itself. #Damien:
#ifndef M_PI
#define M_PI 3.1415926535897932384626433832795
#endif
double sinus(double x) {
x = fmod(x, 2*M_PI); // Reduce to [-2*M_PI ... 2*M_PI]
...
Although function fmod() is not expected to inject any error, the problem is that M_PI (a rational number) is an approximation of π, (an irrational number). Using that value approximation injects error especially x near multiplies of π. This is likely OK for modest quality code.
Good range reduction is a problem as challenging as the trigonometric functions themselves.
See K.C. Ng's "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit" .
OP's sinus() should use additional range reduction and trigonometric properties to get x in range [-M_PI/4 ... M_PI/4] (example) before attempting the power series solution. Otherwise, convergence is slow and errors accumulate.