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Computing π with the Bailey-Borwein-Plouffe Formula
Also listed in…
Math
function binaryPow(b, e) {
let y = 1;
while (e > 0) {
if (e & 1) y *= b;
b *= b;
e >>= 1;
}
return y;
}
binaryPow(3, 17)
function modPow(b, e, m) {
b = b % m;
let y = 1;
while (e > 0) {
if (e & 1) y = (y * b) % m;
b = (b * b) % m;
e >>= 1;
}
return y;
}
modPow(3, 17, 50)
3 ** 17 % 50
function mod(x, n) {
x %= n;
return x < 0 ? x + n : x;
}
computePi = {
function S(j, n) {
// Lefthand summation. Add a number of terms equal
// to the starting digit. Most of the time is spent
// in this loop.
let left = 0;
for (let k = 0; k <= n; k++) {
let r = 8 * k + j;
left = mod(left + modPow(16, n - k, r) / r, 1);
}
// Righthand summation. Add a small number of terms
// until the result stops changing.
let right = 0;
for (let k = n + 1; ; k++) {
let rnew = right + 16 ** (n - k) / (8 * k + j);
if (right === rnew) break;
right = rnew;
}
return left + right;
}
// Start at digit d and compute n digits
return function piBBP(d, n) {
// Seems to be the convention for including the leading 3
d -= 1;
// Shift n digits to the left of the radix point to obtain our final
// result as a integer
return Math.floor(
16 ** n * mod(4 * S(1, d) - 2 * S(4, d) - S(5, d) - S(6, d), 1)
);
};
}
computePi(0, 10).toString(16)
BBPbigint = {
if (!supportsBigInt) return () => () => 0;
return new Function(`
function modPow(b, e, m) {
if (!e) return 1n;
b = b % m;
let y = 1n;
while (true) {
if (e & 1n) y = (y * b) % m;
e >>= 1n;
if (!e) return y;
b = (b * b) % m;
}
}
function S(j, n, d, mask) {
const shift = d << 2n;
let left = 0n;
for (let k = 0n; k <= n; k++) {
let r = k * 8n + j;
left = (left + (modPow(16n, n - k, r) << shift) / r) & mask;
}
let right = 0n;
for (let k = n + 1n; ; k++) {
let rnew = right + 16n ** (d + n - k) / (k * 8n + j);
if (right === rnew) break;
right = rnew;
}
return left + right;
}
return function computePi(dd, nn) {
const n = BigInt(nn);
const d = BigInt(dd) - 1n;
const mask = 16n ** n - 1n;
return (
4n * S(1n, d, n, mask) -
2n * S(4n, d, n, mask) -
S(5n, d, n, mask) -
S(6n, d, n, mask)
) & mask;
};`)();
}
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