Officially, I'm one day behind, but here's how you can calculate the value of π using the Leibniz formula.
Python
import math
sum = 0
estimate = 0
i = 0
epsilon = 0.0001
while abs(estimate - math.pi) > epsilon:
sum += (-1) ** i / (2 * i + 1)
estimate = sum * 4
i += 1
print(
f"After {i} iterations, the estimate is {estimate} and the real pi is {math.pi} "
f"(difference of {abs(estimate - math.pi)})"
)
Outputs:
After 10000 iterations, the estimate is 3.1414926535900345 and the real pi is 3.141592653589793 (difference of 9.99999997586265e-05)
JavaScript
let sum = 0;
let estimate = 0;
let i = 0;
const epsilon = 0.0001;
while (Math.abs(estimate - Math.PI) > epsilon) {
sum += (-1) ** i / (2 * i + 1);
estimate = sum * 4;
i += 1;
}
console.log(
`After ${i} iterations, the estimate is ${estimate} and the real pi is ${Math.PI} ` +
`(difference of ${Math.abs(estimate - Math.PI)})`
);
Outputs
After 10000 iterations, the estimate is 3.1414926535900345 and the real pi is 3.141592653589793 (difference of 0.0000999999997586265)
Ruby
sum = 0
estimate = 0
i = 0
epsilon = 0.0001
while (estimate - Math::PI).abs > epsilon
sum += ((-1) ** i / (2.0 * i + 1))
estimate = sum * 4
i += 1
end
print(
"After #{i} iterations, the estimate is #{estimate} and the real pi is #{Math::PI} "+
"(difference of #{(estimate - Math::PI).abs})"
)
Outputs
After 10000 iterations, the estimate is 3.1414926535900345 and the real pi is 3.141592653589793 (difference of 9.99999997586265e-05)
Backwards
Technically, these little snippets are checking that it works since each language already has access to a value of π as a standard library constant.
If you don't have that, you can decide on a number of iterations, for example 1,000, and use that.
Python
sum = 0
for i in range(1000):
sum += (-1) ** i / (2 * i + 1)
print(sum * 4)
JavaScript
let sum = 0;
for (const i of [...Array(10000).keys()]) {
sum += (-1) ** i / (2 * i + 1);
}
console.log(sum * 4);
Ruby
sum = 0
for i in 0..10000
sum += ((-1) ** i / (2.0 * i + 1))
end
puts sum * 4
Performance test
Perhaps a bit silly but also a fun thing to play with. Pull out hyperfine and compare Python 3.12, Node 20.11, Ruby 3.2, and Bun 1.0.30:
❯ hyperfine --warmup 10 "python3.12 ~/pi.py" "node ~/pi.js" "ruby ~/pi.rb" "bun run ~/pi.js"
Benchmark 1: python3.12 ~/pi.py
Time (mean ± σ): 53.4 ms ± 7.5 ms [User: 31.9 ms, System: 12.3 ms]
Range (min … max): 41.5 ms … 64.8 ms 44 runs
Benchmark 2: node ~/pi.js
Time (mean ± σ): 57.5 ms ± 10.6 ms [User: 43.3 ms, System: 11.0 ms]
Range (min … max): 46.2 ms … 82.6 ms 35 runs
Benchmark 3: ruby ~/pi.rb
Time (mean ± σ): 242.1 ms ± 11.6 ms [User: 68.4 ms, System: 37.2 ms]
Range (min … max): 227.3 ms … 265.3 ms 11 runs
Benchmark 4: bun run ~/pi.js
Time (mean ± σ): 32.9 ms ± 6.3 ms [User: 14.1 ms, System: 10.0 ms]
Range (min … max): 17.1 ms … 41.9 ms 60 runs
Summary
bun run ~/pi.js ran
1.62 ± 0.39 times faster than python3.12 ~/pi.py
1.75 ± 0.46 times faster than node ~/pi.js
7.35 ± 1.45 times faster than ruby ~/pi.rb
Comparing Pythons
Just because I have a couple of these installed:
❯ hyperfine --warmup 10 "python3.8 ~/pi.py" "python3.9 ~/pi.py" "python3.10 ~/pi.py" "python3.11 ~/pi.py" "python3.12 ~/pi.py"
Benchmark 1: python3.8 ~/pi.py
Time (mean ± σ): 54.6 ms ± 8.1 ms [User: 33.0 ms, System: 11.4 ms]
Range (min … max): 40.0 ms … 69.7 ms 56 runs
Benchmark 2: python3.9 ~/pi.py
Time (mean ± σ): 54.9 ms ± 8.0 ms [User: 32.2 ms, System: 12.3 ms]
Range (min … max): 42.3 ms … 70.1 ms 38 runs
Benchmark 3: python3.10 ~/pi.py
Time (mean ± σ): 54.7 ms ± 7.5 ms [User: 33.0 ms, System: 11.8 ms]
Range (min … max): 42.3 ms … 78.1 ms 44 runs
Benchmark 4: python3.11 ~/pi.py
Time (mean ± σ): 53.8 ms ± 6.0 ms [User: 32.7 ms, System: 13.0 ms]
Range (min … max): 44.8 ms … 70.3 ms 42 runs
Benchmark 5: python3.12 ~/pi.py
Time (mean ± σ): 53.0 ms ± 6.4 ms [User: 31.8 ms, System: 12.3 ms]
Range (min … max): 43.8 ms … 63.5 ms 42 runs
Summary
python3.12 ~/pi.py ran
1.02 ± 0.17 times faster than python3.11 ~/pi.py
1.03 ± 0.20 times faster than python3.8 ~/pi.py
1.03 ± 0.19 times faster than python3.10 ~/pi.py
1.04 ± 0.20 times faster than python3.9 ~/pi.py
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