Pi via Three-Term Arctangent Formula
Approximate pi using the three-term Machin-like formula pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8). Choose the number of series terms and see each arctan value, the error against true pi, the correct digits, and the convergence table.
Input
Approximate pi with a three-term Machin-like formula. Enter how many series terms to use for each arctangent.
pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
Enter an integer from 1 to 200. More terms bring the approximation closer to true pi.
Result
Approximation of pi with 5 terms each
3.141739328007
True pi
3.14159265359
Absolute error
1.4667e-4
Correct digits
4
Value of each arctangent term
arctan(1/2)
0.463684275794
arctan(1/5)
0.197395561651
arctan(1/8)
0.124354994557
Convergence by term count
Pi approximation and absolute error against true pi as the term count increases from 1.
| Terms | Pi approximation | Absolute error |
|---|---|---|
| 1 | 3.3 | 1.5841e-1 |
| 2 | 3.1200625 | 2.1530e-2 |
| 3 | 3.145342914062 | 3.7503e-3 |
| 4 | 3.140871041584 | 7.2161e-4 |
| 5 | 3.141739328007 | 1.4667e-4 |
Each arctan is computed from the series arctan(x) = x − x^3/3 + x^5/5 − x^7/7 + …, and four times the sum of the three terms gives pi. Smaller arguments converge faster.
How it works
- Uses the three-term Machin-like formula pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8).
- Each arctan is evaluated with the power series arctan(x) = x − x^3/3 + x^5/5 − x^7/7 + … up to the chosen number of terms.
- The three arctan values are summed and multiplied by 4 to approximate pi.
- Because the arguments 1/2, 1/5, and 1/8 are small, the series converges quickly even with few terms.
- The absolute error against true pi (Math.PI) gives the approximate number of correct significant digits.
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Pi via Three-Term Arctangent Formula