Problems with precision loss in my cosine function

Question

This is my task:

Write a C function to evaluate the series // cos(x) = x-(x2 /2!)+(x4 /4!)-(x6 /6!)+... etc. Variable realNuber use radians instead of degrees

I lose precision, but I don't understand where. The answer with realNumber = 60 must be 0.500, but I've 0.501. Please help.

#include "stdio.h"
#include "inttypes.h"

double power(float N, uint32_t P){
    double buffer = 1;

    for (int i = 0; i < P; ++i) {
        buffer *= N;
    }

    return buffer;
}

float factorial(float number){
    float result = number;

    if (number == 0) {
        return 0;
    }

    for (int i = 0; i < number - 1; ++i) {
        result *= i + 1;
    }

    return result;
}

float cos(float x){
    float result = x * (3.14159265359 / 180.);
    float polar = -1;

    for (int i = 2; i < 10; i += 2) {
        result += power(result, i) / factorial(i) * polar;
        polar *= -1;
    }

    return result;
}

int main(void){
    float realNumber = 0;
    float result = 0;

    scanf("%f", &realNumber);

    result = cos(realNumber);

    printf("%.13f", result);
}

I tried making changes in function cos(); maybe the problem is in a different place?

Answer 1

You originally wrote:

Write a C function to evaluate the series // cos(x) = x-(x2 /2!)+(x4 /4!)-(x6 /6!)

But that is NOT the Taylor series for cos.

The proper formula is:

(Note the 1 in the first term not an x)
Source

With a correction to your Taylor series, and some other fix up, I got:

Output

Success #stdin #stdout 0s 5392KB
0.4999999701977

My Code:

#include "stdio.h"
#include "inttypes.h"

// No Changes
double power(float N, uint32_t P){
    double buffer = 1;

    for (int i = 0; i < P; ++i) {
        buffer *= N;
    }

    return buffer;
}

// No Changes
float factorial(float number){
    float result = number;

    if (number == 0) {
        return 0;
    }

    for (int i = 0; i < number - 1; ++i) {
        result *= i + 1;
    }

    return result;
}

// Minor changes, explained in comments
float cos(float x){
    x = x * (3.14159265359 / 180.); // Convert Degrees to Radians
    float result = 1;               // Taylor series starts with 1, not with x !!! 
    float polar = -1;

    for (int i = 2; i <= 10; i += 2) {
        result += power(x, i) / factorial(i) * polar;
        polar *= -1;
    }

    return result;
}

// Skipped the scanf in favor of hard-coded value, for simplicity.
int main(void){
    float realNumber = 60;
    float result = 0;

    result = cos(realNumber);

    printf("%.13f", result);
}

When I re-wrote the cos function to eliminate using power and factorial functions, I got this:

double cos(float x){
    x = x * (3.14159265359 / 180.); // Convert Degrees to Radians
    double num   = 1;  // Numerator of the fraction (x^2, x^4...)
    int    sgn   = +1; // Sign, alternating -1, +1
    uint64_t den = 1;  // Denominator: Factorials, 2!, 4!, 6!...
    float ans    = 1;  // Accumulated answer
    for (int i = 2; i <= 10; i += 2) {
        num *= x*x;
        den *= i*i-i;
        sgn *= -1;
        ans += num / den * sgn;
    }

    return ans;
}