# Matthew Esaia, 3/12/2025, Homework 14pi

Help:=proc(): print(`JesusG(N), RealFrac(), Chudnovsky(E)`): end:

## PROBLEM 1
#JesusG(n): uses N terms of Jesus Guillera's formula to compute an
#approximation to 128/pi^2
JesusG:=proc(N) local n:
	evalf(add((-1)^n * (binomial(2*n, n))^5 * (820*n^2 + 180*n + 13) / (2^(20*n)), n=0..N), 500):
end:
# JesusG(10) has an error of 1.048 x 10^-32
# JesusG(100) has an error of 4.23 x 10^-304

#RealFrac(): true value of 128/pi^2
RealFrac:=proc():
	return evalf(128 / Pi^2, 500):
end:

## PROBLEM 2
#Chudnovsky(E): approximates pi to E digits of precision
Chudnovsky:=proc(E) local i,err,sum,const:
	sum := 0:
	err := evalf(Pi - sum):
	const := 1 / (426880 * sqrt(10005)):
	for i from 0 while err > 10^(-E) do
		sum := sum + ( (-1)^i * (6*i)! * (545140134*i + 13591409) / ((3*i)! * (i!)^3 * (640320)^(3*i))):
		err := evalf(abs(Pi - (1 /(const*sum))), E + 10):
		#print(i, evalf(1 /(const*sum), E)):
	od:
	return evalf(1 /(const*sum), E):
end: