how do I replace pow()? - c++

Question

how do I replace the pow() function in two cases in my code ?
I think this can be done with a for loop
#include <iostream>
#include <cmath>
using namespace std;
int main(){
double a, b, h, PI = 3.141592;
int n;
cin >> a >> b >> h >> n;
for (double x = a; x <= b; x += h) {
double ans = 1, y;
for (int k = 0; k <= n; k++) {
ans *= cos(k * PI / 4) * pow(x, k);
for (int i = 2; i <= k; i++) {
ans /= i;
}
}
y = pow(exp(cos(x * sin(PI / 4))), x * cos(PI / 4));
cout << ans << " " << y << " " << fabs(y-ans) << endl;
}
return 0;
}

Answer 1

Do not write everything in main.
Define double S(double x, int n) and double U(double x).
each element of sum can be calculated based on previous element.
cos(k * M_PI / 4) has repeating values so it can be stored in table.
double S(double x, int n)
{
double a = 1;
double s = a;
constexpr double q = std::cos(M_PI / 4);
constexpr double cos_val[]{ 1, q, 0, -q, -1, -q, 0, q };
for (int k = 1; k <= n; ++k) {
a *= x / k;
s += cos_val[k & 7] * a
}
return s;
}

Answer 2

For the inner loop, you need not calculate the power in each iteration if you consider that on the previous iteration you already calculated pow(x,k-1) and that pow(x,k) == pow(x,k-1)*x:
double pow_x = 1; // on first iteration pow(x,0) == 1
for (int k = 0; k <= n; k++) {
ans *= cos(k * PI / 4) * pow_x;
// ...
pow_x *= x; // pow(x,k) -> pow(x,k+1)
}
The second use of pow in your code cannot be easily replaced, because of the floating point exponent. You would have to rewrite pow to get the same result. However, your code does not match the formula in the image. The image says (pseudo maths notation):
e ^ ( x * C1 ) * C2
your code is calculating
y = pow(exp(cos(x * sin(PI / 4))), x * cos(PI / 4));
( e^(C2) ) ^ (x * C1)
change it to
y = exp(x * cos(PI / 4)) * cos(x * sin(PI / 4))