Approximating e - Getting as many precision digits as possible

Question

I have been trying to approximate e using series representation to get as many precision digits as possible using the code below, but no matter how many terms I compute, the number of precision digits seems to remain the same. ie:

2.71828198432922363281250000000000000000000000000000

Is it my approach that's wrong? Here is the code:

  1 #include <stdio.h>
  2 #include <iostream>
  3 #include <math.h>
  4 using namespace std;
  5 
  6 float factorial (float a)
  7 {
  8         if (a > 1)
  9         {
 10                 return (a * factorial (a-1));
 11         } else
 12         {
 13                 return 1;
 14         }
 15 }
 16 
 17 int main()
 18 {
 19         float sum  = 0;
 20         int range=100000;
 21 
 22         for (int i=0; i<=range;i++)
 23         {
 24                 sum += pow(-1,i)/factorial(i);
 25         }
 26         sum = pow(sum,-1);
 27         printf("%4.50f\n", sum);
 28 } 

Answer 1

To get more exact digits, you should write your on data class which store more digit, say, 1000 digits. The hardest part is to wirte the +, -, *, / operations.

If what you want is just to experiment with the math formula, you can choose another language, such as Python. It has data types like Decimal, Fraction that can do more precise calculating.

I love math so I do write a python script to test the formula:

from decimal import Decimal, getcontext
prec = 100
getcontext().prec = prec

fac = Decimal(1)/2
sum = Decimal(0)
eps = Decimal(0.1)
eps = eps**prec

realE = '2.71828182845904523536028747135266249775724709369995'

i = 3
while 1:
    ds = fac - fac/i
    sum += ds
    if ds < eps: break
    fac /= i * (i+1)
    i += 2

print 'After %d iteration:' % i
print realE
print str(1/sum)

Here's the result:

After 71 iteration:
2.71828182845904523536028747135266249775724709369995
2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166429