计算科学 - Computing sin(π/2)sin⁡(π/2) numerically - 吾爱随笔录

I'm trying to understand the types of numerical errors, to do this I want to calculate $\sin(\pi/2)=1$ numerically. To do this I use the Taylor series of $\sin(x)$ in 0:

\sin (x) = x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \frac{x^{7}}{5040} + \dots

$\sin(x)=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+ \ldots$

To make the calcultaions I did the following Fortran program:

program main

implicit none
real*8 pi, x, res, error

pi=3.14159265
x = pi/2
res = x - x**3/6 + x**5/120 - x**7/5040

error = 1-res
print*, "Result: ",res
print*, "Error: ", error

end

And I get the following results depending of the degreee of the Taylor series and in the decimals of pi.

If $\pi=3.14$

Taylor 3rd Degree: 0.92501782114084341
Taylor 5th Degree: 1.0045086615502121
Taylor 7th Degree: 0.99984349522599403
Taylor 9th Degree: 1.0000032059098956
Taylor 11thDegree: 0.99999962708361323

If $\pi=3.14159265$

Taylor 3rd Degree: 0.92483221907327295
Taylor 5th Degree: 1.0045248564076874
Taylor 7th Degree: 0.99984310136039689
Taylor 9th Degree: 1.0000035425853664
Taylor 11thDegree: 0.99999994374102952

And I observe that the error gets bigger if I put more decimals in my $\pi$ approximation. Why is this happening? Edit: Adding more terms to the Taylor expansion (11th degree) makes adding more $\pi$ digits better.

Also, How can I make a better approximation? What method will be better to a problem like this one?

How can I predict the size of these errors?