extra-bigint
Version:
A [BigInt] can represent whole numbers larger than 2⁵³ - 1 [(1)].<br>
683 lines • 61.4 kB
JavaScript
// METHODS
// =======
// ABOUT
// -----
/**
* Check if value is a bigint.
* @param x a value
* @returns is bigint?
*/ export function is(x) {
return typeof x === "bigint";
}
// COMPARE
// -------
/**
* Compare two bigints.
* @param x a bigint
* @param y another bigint
* @returns x<y: -ve, x=y: 0, x>y: +ve
*/ export function compare(x, y) {
return Number(x - y);
}
// SIGN
// ----
/**
* Get the absolute of a bigint.
* @param x a bigint
* @returns |x|
*/ export function abs(x) {
return x < 0n ? -x : x;
}
/**
* Get the sign of a bigint.
* @param x a bigint
* @returns x>0: 1n, x<0: -1n, x=0: 0n
*/ export function sign(x) {
return x < 0n ? -1n : x > 0n ? 1n : 0n;
}
// ROUNDED DIVISION
// ----------------
/**
* Perform floor-divison of two bigints (\\).
* @param x dividend
* @param y divisor
* @returns ⌊x/y⌋
*/ export function floorDiv(x, y) {
if (y < 0n) {
x = -x;
y = -y;
}
return x >= 0n ? x / y : (x - y + 1n) / y;
}
// - https://python-reference.readthedocs.io/en/latest/docs/operators/floor_division.html
/**
* Perform ceiling-divison of two bigints.
* @param x dividend
* @param y divisor
* @returns ⌈x/y⌉
*/ export function ceilDiv(x, y) {
if (y < 0n) {
x = -x;
y = -y;
}
return x >= 0n ? (x + y - 1n) / y : x / y;
}
/**
* Perform rounded-divison of two bigints.
* @param x divisor
* @param y dividend
* @returns [x/y]
*/ export function roundDiv(x, y) {
if (y < 0n) {
x = -x;
y = -y;
}
return x >= 0n ? (x + y / 2n) / y : (x - y / 2n) / y;
}
// MODULO
// ------
/**
* Find the remainder of x/y with sign of x (truncated division).
* @param x dividend
* @param y divisor
* @returns trunc(x % y)
*/ export function rem(x, y) {
return x % y;
}
// - https://en.wikipedia.org/wiki/Modulo_operation
/**
* Find the remainder of x/y with sign of y (floored division).
* @param x dividend
* @param y divisor
* @returns floor(x % y)
*/ export function mod(x, y) {
return x - y * floorDiv(x, y);
}
// - https://en.wikipedia.org/wiki/Modulo_operation
/**
* Find the remainder of x/y with +ve sign (euclidean division).
* @param x dividend
* @param y divisor
* @returns x/y>0: floor(x % y), x/y<0: ceil(x % y)
*/ export function modp(x, y) {
return x - abs(y) * floorDiv(x, abs(y));
}
// - https://en.wikipedia.org/wiki/Modulo_operation
// RANGE CONTROL
// -------------
/**
* Constrain a bigint within a minimum and a maximum value.
* @param x a bigint
* @param minv minimum value
* @param maxv maximum value
* @returns x<min: min, x>max: max, x
*/ export function constrain(x, minv, maxv) {
return min(max(x, minv), maxv);
}
export { constrain as clamp };
// - https://processing.org/reference/constrain_.html
// - https://www.npmjs.com/package/clamp
// - https://en.cppreference.com/w/cpp/algorithm/clamp
// - https://dlang.org/library/std/algorithm/comparison/clamp.html
// - https://www.rdocumentation.org/packages/raster/versions/3.0-12/topics/clamp
// - https://docs.microsoft.com/en-us/dotnet/api/system.math.clamp?view=netcore-3.1
// - https://docs.microsoft.com/en-us/windows/win32/direct3dhlsl/dx-graphics-hlsl-clamp
// - https://www.khronos.org/registry/OpenGL-Refpages/gl4/html/clamp.xhtml
// - https://docs.unity3d.com/ScriptReference/Mathf.Clamp.html
// - https://en.wikipedia.org/wiki/Clamping_(graphics)
/**
* Re-map a bigint from one range to another.
* @param x a bigint
* @param r lower bound of current range
* @param R upper bound of current range
* @param t lower bound of target range
* @param T upper bound of target range
* @returns ∈ [ymin, ymax]
*/ export function remap(x, r, R, t, T) {
return t + (x - r) * (T - t) / (R - r);
}
export { remap as map };
// - https://processing.org/reference/map_.html
/**
* Linearly interpolate a bigint between two bigints.
* @param x start bigint
* @param y stop bigint
* @param t interpolant ∈ [0, 1]
* @returns ∈ [x, y]
*/ export function lerp(x, y, t) {
return x + BigInt(Math.floor(t * Number(y - x)));
}
// - https://processing.org/reference/lerp_.html
// - https://docs.unity3d.com/ScriptReference/Vector3.Lerp.html
// - https://stackoverflow.com/questions/53970655/how-to-convert-bigint-to-number-in-javascript
// POWER / LOGARITHM
// -----------------
/**
* Check if bigint is a power-of-2.
* @param x a bigint
* @returns 2ⁱ = x? | i = +ve integer
*/ export function isPow2(x) {
return /^10*$/.test(x.toString(2));
}
/**
* Check if bigint is a power-of-10.
* @param x a bigint
* @returns 10ⁱ = x? | i = +ve integer
*/ export function isPow10(x) {
return /^10*$/.test(x.toString());
}
/**
* Find largest power-of-2 less than or equal to given bigint.
* @param x a bigint
* @returns 2ⁱ | 2ⁱ ≤ x and 2ⁱ > x/2
*/ export function prevPow2(x) {
if (x <= 1n) return 0n;
const n = x.toString(2).length;
return BigInt("0b1" + "0".repeat(n - 1));
}
/**
* Find largest power-of-10 less than or equal to given bigint.
* @param x a bigint
* @returns 10ⁱ | 10ⁱ ≤ x and 10ⁱ > x/10
*/ export function prevPow10(x) {
if (x <= 1n) return 0n;
const n = x.toString(10).length;
return BigInt("1" + "0".repeat(n - 1));
}
/**
* Find smallest power-of-2 greater than or equal to given bigint.
* @param x a bigint
* @returns 2ⁱ | 2ⁱ ≥ x and 2ⁱ < 2x
*/ export function nextPow2(x) {
if (x <= 0n) return 1n;
const n = (x - 1n).toString(2).length;
return BigInt("0b1" + "0".repeat(n));
}
/**
* Find smallest power-of-10 greater than or equal to given bigint.
* @param x a bigint
* @returns 10ⁱ | 10ⁱ ≥ x and 10ⁱ < 10x
*/ export function nextPow10(x) {
if (x <= 0n) return 1n;
const n = (x - 1n).toString(10).length;
return BigInt("1" + "0".repeat(n));
}
/**
* Find the base-2 logarithm of a bigint.
* @param x a bigint
* @returns log₂(x)
*/ export function log2(x) {
const n = x.toString(2).length - 1;
return x <= 0n ? 0n : BigInt(n);
}
/**
* Find the base-10 logarithm of a bigint.
* @param x a bigint
* @returns log₁₀(x)
*/ export function log10(x) {
const n = x.toString().length - 1;
return x <= 0n ? 0n : BigInt(n);
}
// TODO: isPow() based on accelerated search
// TODO: prevPow() based on accelerated search
// TODO: nextPow() based on accelerated search
// TODO: log() based on accelerated search
// ROOT
// ----
/**
* Find the square root of a bigint.
* @param x a bigint
* @returns √x
*/ export function sqrt(x) {
if (x === 0n) return 0n;
return x > 0n ? sqrtPos(x) : null;
}
function sqrtPos(x) {
let a = 1n << log2(x) / 2n + 1n; // initial guess
let b = 1n + a;
while(a < b){
b = a;
a = (b + x / b) / 2n;
}
return b;
}
/**
* Find the cube root of a bigint.
* @param x a bigint
* @returns ³√x
*/ export function cbrt(x) {
return root(x, 3n);
}
/**
* Find the nth root of a bigint.
* @param x a bigint
* @param n nth root (1n)
* @returns ⁿ√x
*/ export function root(x, n = 1n) {
if (x === 0n) return 0n;
else if (x > 0n) return rootPos(x, n);
return n % 2n !== 0n ? -rootPos(-x, n) : null;
}
function rootPos(x, n) {
let a = 1n << log2(x) / n + 1n; // initial guess
let b = 1n + a;
const m = n - 1n;
if (a === 2n) return 1n;
while(a < b){
b = a;
a = (m * b + x / b ** m) / n;
}
return b;
}
// DIVISORS
// --------
/**
* List all divisors of a bigint, except itself.
* @param x a bigint
* @returns proper divisors (factors)
*/ export function properDivisors(x) {
x = abs(x);
const a = [];
for(let i = 1n; i < x; i++)if (x % i === 0n) a.push(i);
return a;
}
export { properDivisors as aliquotParts };
// - https://mathworld.wolfram.com/ProperDivisor.html
// - https://en.wikipedia.org/wiki/Divisor#Further_notions_and_facts
/**
* Sum all proper divisors of a bigint.
* @param x a bigint
* @returns Σdᵢ | dᵢ is a divisor of x and ≠x
*/ export function aliquotSum(x) {
x = abs(x);
let a = 0n;
for(let i = 1n; i < x; i++)if (x % i === 0n) a += i;
return a;
}
/**
* Find the least prime number which divides a bigint.
* @param x a bigint
* @returns least prime factor
*/ export function minPrimeFactor(x) {
x = abs(x);
// 1. LPF of 2, 3 is the number itself.
if (x <= 1n) return 0n;
if (x <= 3n) return x;
// 2. LPF for multiples of 2, 3.
if (x % 2n === 0n) return 2n;
if (x % 3n === 0n) return 3n;
// 3. LPF can be 6k-1 or 6k+1.
for(let i = 6n, I = sqrt(x) + 1n; i <= I; i += 6n){
if (x % (i - 1n) === 0n) return i - 1n;
if (x % (i + 1n) === 0n) return i + 1n;
}
return x;
}
export { minPrimeFactor as leastPrimeFactor };
// - https://mathworld.wolfram.com/LeastPrimeFactor.html
/**
* Find the greatest prime number which divides a bigint.
* @param x a bigint
* @returns greatest prime factor
*/ export function maxPrimeFactor(x) {
x = abs(x);
let a = 0n;
// 1. GPF of 2, 3 is the number itself.
if (x <= 1n) return 0n;
if (x <= 3n) return x;
// 2. Remove factors 2, 3.
for(; x % 2n === 0n; a = 2n)x /= 2n;
for(; x % 3n === 0n; a = 3n)x /= 3n;
// 3. Remove factors 6k-1, 6k+1.
for(let i = 6n, I = sqrt(x) + 1n; x > 1n && i <= I; i += 6n){
for(; x % (i - 1n) == 0n; a = i - 1n)x /= i - 1n;
for(; x % (i + 1n) == 0n; a = i + 1n)x /= i + 1n;
}
if (x <= 1n) return a;
return x;
}
export { maxPrimeFactor as greatestPrimeFactor };
// - https://mathworld.wolfram.com/GreatestPrimeFactor.html
// - https://www.geeksforgeeks.org/find-largest-prime-factor-number/
/**
* Find the prime factors of a bigint.
* @param x a bigint
* @returns [f₀, f₁, ...] | fᵢ divides x and is prime
*/ export function primeFactors(x) {
x = abs(x);
const a = [];
if (x <= 1n) return [];
if (x <= 3n) return [
x
];
// 2. Try factors 2, 3.
x = pushPrimeFactorTo$(a, x, 2n);
x = pushPrimeFactorTo$(a, x, 3n);
// 3. Try factors 6k-1, 6k+1.
for(let i = 6n, I = sqrt(x) + 1n; x > 1n && i <= I; i += 6n){
x = pushPrimeFactorTo$(a, x, i - 1n);
x = pushPrimeFactorTo$(a, x, i + 1n);
}
if (x > 1n) a.push(x);
return a;
}
function pushPrimeFactorTo$(a, x, f) {
if (x % f !== 0n) return x;
do {
x /= f;
}while (x % f === 0n)
a.push(f);
return x;
}
// - https://www.geeksforgeeks.org/prime-factors-big-number/
// - https://mathworld.wolfram.com/PrimeFactor.html
/**
* Find the prime factors and respective exponents of a bigint.
* @param x a bigint
* @returns [[f₀, e₀], [f₁, e₁], ...] | fᵢ is a prime factor of x and eᵢ is its exponent
*/ export function primeExponentials(x) {
x = abs(x);
const a = [];
if (x <= 1n) return [];
if (x <= 3n) return [
[
x,
1n
]
];
// 2. Try factors 2, 3.
x = pushPrimeExponentialTo$(a, x, 2n);
x = pushPrimeExponentialTo$(a, x, 3n);
// 3. Try factors 6k-1, 6k+1.
for(let i = 6n, I = sqrt(x) + 1n; x > 1n && i <= I; i += 6n){
x = pushPrimeExponentialTo$(a, x, i - 1n);
x = pushPrimeExponentialTo$(a, x, i + 1n);
}
if (x > 1n) a.push([
x,
1n
]);
return a;
}
function pushPrimeExponentialTo$(a, x, f) {
if (x % f !== 0n) return x;
let e = 0n;
do {
x /= f;
++e;
}while (x % f === 0n)
a.push([
f,
e
]);
return x;
}
// - https://www.geeksforgeeks.org/prime-factors-big-number/
// - https://reference.wolfram.com/language/ref/FactorInteger.html
// - https://mathworld.wolfram.com/PrimeFactorization.html
// - https://mathworld.wolfram.com/PrimeFactor.html
/**
* Check if bigint is prime.
* @param x a bigint
* @returns is divisible by 1n and itself only?
*/ export function isPrime(x) {
return x !== 0n && minPrimeFactor(x) === abs(x);
}
/**
* Find the greatest common divisor of bigints.
* @param xs a list of bigints
* @returns gcd(x₁, x₂, ...)
*/ export function gcd(...xs) {
let a = xs[0] || 1n;
for(let i = 1, I = xs.length; i < I; i++)a = gcdPair(a, xs[i]);
return a;
}
export { gcd as hcf };
// Find the greatest common divisor of a pair of bigints.
function gcdPair(x, y) {
while(y !== 0n){
const t = y;
y = x % y;
x = t;
}
return x;
}
// - https://lemire.me/blog/2013/12/26/fastest-way-to-compute-the-greatest-common-divisor/
// - https://en.wikipedia.org/wiki/Euclidean_algorithm
/**
* Find the least common multiple of bigints.
* @param xs a list of bigints
* @returns lcm(x₁, x₂, ...)
*/ export function lcm(...xs) {
let a = xs[0] || 1n;
for(let i = 1, I = xs.length; i < I; i++)a = a * xs[i] / gcdPair(a, xs[i]);
return a;
}
// - https://en.wikipedia.org/wiki/Least_common_multiple
// ARRANGEMENTS
// ------------
/**
* Find the factorial of a bigint.
* @param n a bigint
* @param k denominator factorial [0]
* @returns P(n, k); k=0: n!, k>0: n!/k!
*/ export function factorial(n, k = 0n) {
let a = 1n;
if (n < 0n) return 0n;
for(let i = k + 1n; i <= n; i++)a *= i;
return a;
}
// - https://github.com/alawatthe/MathLib/blob/master/src/Functn/functions/factorial.ts
// - https://en.wikipedia.org/wiki/Permutation
/**
* Find the number of ways to choose k elements from a set of n elements.
* @param n elements in source set
* @param k elements in choose set
* @returns C(n, k)
*/ export function binomial(n, k) {
// 1. Generalization to negative integers
if (k < 0n || k > abs(n)) return 0n;
if (n < 0n) return (-1n) ** k * binomial(-n, k);
// 2. Take advantage of symmetry
k = k > n - k ? n - k : k;
let a = 1n, b = 1n;
for(let i = 1n; i <= k; i++, n--){
a *= n;
b *= i;
}
return a / b;
}
// - https://github.com/alawatthe/MathLib/blob/master/src/Functn/functions/binomial.ts
// - https://en.wikipedia.org/wiki/Binomial_coefficient
/**
* Find the number of ways to put n objects in m bins (n=sum(kᵢ)).
* @param ks objects per bin (kᵢ)
* @returns n!/(k₁!k₂!...) | n=sum(kᵢ)
*/ export function multinomial(...ks) {
let n = sum(...ks), a = 1n, b = 1n;
for(let i = 0, j = 0n, I = ks.length; i < I;){
if (j <= 0n) j = ks[i++];
else {
a *= n--;
b *= j--;
}
}
return a / b;
}
// - https://en.wikipedia.org/wiki/Multinomial_distribution
// GEOMETRY
// --------
/**
* Find the length of hypotenuse.
* @param xs lengths of perpendicular sides (x, y, z, ...)
* @returns √(x² + y² + z²)
*/ export function hypot(...xs) {
let a = 0n;
for (const x of xs)a += x * x;
return sqrt(a);
}
// STATISTICS
// ----------
/**
* Find the sum of bigints (Σ).
* @param xs bigints
* @returns Σxᵢ
*/ export function sum(...xs) {
let a = 0n;
for (const x of xs)a += x;
return a;
}
/**
* Find the product of bigints (∏).
* @param xs bigints
* @returns ∏xᵢ
*/ export function product(...xs) {
let a = 1n;
for (const x of xs)a *= x;
return a;
}
/**
* Find the value separating the higher and lower halves of bigints.
* @param xs a list of bigints
* @returns xₘ | sort(xs) = [..., xₘ, ...]
*/ export function median(...xs) {
if (xs.length === 0) return 0n;
xs.sort(compare);
const i = xs.length >> 1;
if ((xs.length & 1) === 1) return xs[i];
return (xs[i - 1] + xs[i]) / 2n;
}
// - https://stackoverflow.com/questions/45309447/calculating-median-javascript
// - https://en.wikipedia.org/wiki/Median
/**
* Find the values that appear most often.
* @param xs a list of bigints
* @returns [xₘ₁, xₘ₂, ...] | count(xₘᵢ) ≥ count(xᵢ) ∀ xᵢ ∈ xs
*/ export function modes(...xs) {
xs.sort(compare);
const r = maxRepeat(xs);
return getRepeats(xs, r);
}
// - https://en.wikipedia.org/wiki/Mode_(statistics)
// Get the maximum number of times any bigint has repeated in a sorted array.
function maxRepeat(xs) {
let count = Math.min(xs.length, 1), max = count;
for(let i = 1, I = xs.length; i < I; i++){
if (xs[i - 1] === xs[i]) count++;
else {
max = Math.max(max, count);
count = 1;
}
}
return Math.max(max, count);
}
// Get the numbers which have been repeated atleast given number of times.
function getRepeats(xs, r) {
const a = [];
r--;
for(let i = 0, I = xs.length - r; i < I; i++)if (xs[i] === xs[i + r]) a.push(xs[i += r]);
return a;
}
/**
* Find the smallest bigint.
* @param xs bigints
* @returns min(xs)
*/ export function min(...xs) {
if (xs.length === 0) return 0n;
let a = xs[0];
for (const x of xs)a = x < a ? x : a;
return a;
}
/**
* Find the largest bigint.
* @param xs bigints
* @returns max(xs)
*/ export function max(...xs) {
if (xs.length === 0) return 0n;
let a = xs[0];
for (const x of xs)a = x > a ? x : a;
return a;
}
/**
* Find the minimum and maximum bigint.
* @param xs bigints
* @returns [min, max]
*/ export function range(...xs) {
if (xs.length === 0) return [
0n,
0n
];
let a = xs[0], b = a;
for (const x of xs){
a = x < a ? x : a;
b = x > b ? x : b;
}
return [
a,
b
];
}
/**
* Find the mean of squared deviation of bigints from its mean.
* @param xs a list of bigints
* @returns σ² = E[(xs - µ)²] | µ = mean(xs)
*/ export function variance(...xs) {
if (xs.length === 0) return 0n;
const m = arithmeticMean(...xs);
let a = 0n;
for (const x of xs)a += (x - m) ** 2n;
return a / BigInt(xs.length);
}
// - https://en.wikipedia.org/wiki/Variance
// MEAN (STATISTICS)
// -----------------
/**
* Find the arithmetic mean of bigints (µ).
* @param xs bigints
* @returns µ = (Σxᵢ)/n | n = size(xs)
*/ export function arithmeticMean(...xs) {
const n = BigInt(xs.length);
return sum(...xs) / n;
}
export { arithmeticMean as mean };
/**
* Find the geometric mean of bigints.
* @param xs bigints
* @returns ⁿ√(∏xᵢ) | n = size(xs)
*/ export function geometricMean(...xs) {
const n = BigInt(xs.length);
return root(product(...xs), n);
}
/**
* Find the harmonic mean of bigints.
* @param xs bigints
* @returns n/Σ(1/xᵢ) | n = size(xs)
*/ export function harmonicMean(...xs) {
const n = BigInt(xs.length);
const p = product(...xs);
let q = 0n;
for (const x of xs)q += p / x;
return n * p / q;
}
/**
* Find the quadriatic mean of bigints.
* @param xs bigints
* @returns √(Σxᵢ²)/n | n = size(xs)
*/ export function quadriaticMean(...xs) {
const n = BigInt(xs.length);
let a = 0n;
for (const x of xs)a += x * x;
return sqrt(a / n);
}
export { quadriaticMean as rootMeanSquare };
/**
* Find the cubic mean of bigints.
* @param xs bigints
* @returns ³√(Σxᵢ³)/n | n = size(xs)
*/ export function cubicMean(...xs) {
const n = BigInt(xs.length);
let a = 0n;
for (const x of xs)a += x ** 3n;
return cbrt(a / n);
}
//# sourceMappingURL=data:application/json;base64,{"version":3,"sources":["file:///home/runner/work/extra-bigint/extra-bigint/index.ts"],"sourcesContent":["// METHODS\n// =======\n\n// ABOUT\n// -----\n\n/**\n * Check if value is a bigint.\n * @param x a value\n * @returns is bigint?\n */\nexport function is(x: unknown): x is bigint {\n  return typeof x===\"bigint\";\n}\n\n\n\n\n// COMPARE\n// -------\n\n/**\n * Compare two bigints.\n * @param x a bigint\n * @param y another bigint\n * @returns x<y: -ve, x=y: 0, x>y: +ve\n */\nexport function compare(x: bigint, y: bigint): number {\n  return Number(x-y);\n}\n\n\n\n\n// SIGN\n// ----\n\n/**\n * Get the absolute of a bigint.\n * @param x a bigint\n * @returns |x|\n */\nexport function abs(x: bigint): bigint {\n  return x<0n? -x : x;\n}\n\n\n/**\n * Get the sign of a bigint.\n * @param x a bigint\n * @returns x>0: 1n, x<0: -1n, x=0: 0n\n */\nexport function sign(x: bigint): bigint {\n  return x<0n? -1n : (x>0n? 1n : 0n);\n}\n\n\n\n\n// ROUNDED DIVISION\n// ----------------\n\n/**\n * Perform floor-divison of two bigints (\\\\).\n * @param x dividend\n * @param y divisor\n * @returns ⌊x/y⌋\n */\nexport function floorDiv(x: bigint, y: bigint): bigint {\n  if (y<0n) { x = -x; y = -y; }\n  return x>=0n? x/y : (x-y+1n)/y;\n}\n// - https://python-reference.readthedocs.io/en/latest/docs/operators/floor_division.html\n\n\n/**\n * Perform ceiling-divison of two bigints.\n * @param x dividend\n * @param y divisor\n * @returns ⌈x/y⌉\n */\nexport function ceilDiv(x: bigint, y: bigint): bigint {\n  if (y<0n) { x = -x; y = -y; }\n  return x>=0n? (x+y-1n)/y : x/y;\n}\n\n\n/**\n * Perform rounded-divison of two bigints.\n * @param x divisor\n * @param y dividend\n * @returns [x/y]\n */\nexport function roundDiv(x: bigint, y: bigint): bigint {\n  if (y<0n) { x = -x; y = -y; }\n  return x>=0n? (x+y/2n)/y : (x-y/2n)/y;\n}\n\n\n\n\n// MODULO\n// ------\n\n/**\n * Find the remainder of x/y with sign of x (truncated division).\n * @param x dividend\n * @param y divisor\n * @returns trunc(x % y)\n */\nexport function rem(x: bigint, y: bigint): bigint {\n  return x % y;\n}\n// - https://en.wikipedia.org/wiki/Modulo_operation\n\n\n/**\n * Find the remainder of x/y with sign of y (floored division).\n * @param x dividend\n * @param y divisor\n * @returns floor(x % y)\n */\nexport function mod(x: bigint, y: bigint): bigint {\n  return x - y*floorDiv(x, y);\n}\n// - https://en.wikipedia.org/wiki/Modulo_operation\n\n\n/**\n * Find the remainder of x/y with +ve sign (euclidean division).\n * @param x dividend\n * @param y divisor\n * @returns x/y>0: floor(x % y), x/y<0: ceil(x % y)\n */\nexport function modp(x: bigint, y: bigint): bigint {\n  return x - abs(y)*floorDiv(x, abs(y));\n}\n// - https://en.wikipedia.org/wiki/Modulo_operation\n\n\n\n\n// RANGE CONTROL\n// -------------\n\n/**\n * Constrain a bigint within a minimum and a maximum value.\n * @param x a bigint\n * @param minv minimum value\n * @param maxv maximum value\n * @returns x<min: min, x>max: max, x\n */\nexport function constrain(x: bigint, minv: bigint, maxv: bigint): bigint {\n  return min(max(x, minv), maxv);\n}\nexport {constrain as clamp};\n// - https://processing.org/reference/constrain_.html\n// - https://www.npmjs.com/package/clamp\n// - https://en.cppreference.com/w/cpp/algorithm/clamp\n// - https://dlang.org/library/std/algorithm/comparison/clamp.html\n// - https://www.rdocumentation.org/packages/raster/versions/3.0-12/topics/clamp\n// - https://docs.microsoft.com/en-us/dotnet/api/system.math.clamp?view=netcore-3.1\n// - https://docs.microsoft.com/en-us/windows/win32/direct3dhlsl/dx-graphics-hlsl-clamp\n// - https://www.khronos.org/registry/OpenGL-Refpages/gl4/html/clamp.xhtml\n// - https://docs.unity3d.com/ScriptReference/Mathf.Clamp.html\n// - https://en.wikipedia.org/wiki/Clamping_(graphics)\n\n\n/**\n * Re-map a bigint from one range to another.\n * @param x a bigint\n * @param r lower bound of current range\n * @param R upper bound of current range\n * @param t lower bound of target range\n * @param T upper bound of target range\n * @returns ∈ [ymin, ymax]\n */\nexport function remap(x: bigint, r: bigint, R: bigint, t: bigint, T: bigint): bigint {\n  return t + (x - r)*(T - t)/(R - r);\n}\nexport {remap as map};\n// - https://processing.org/reference/map_.html\n\n\n/**\n * Linearly interpolate a bigint between two bigints.\n * @param x start bigint\n * @param y stop bigint\n * @param t interpolant ∈ [0, 1]\n * @returns ∈ [x, y]\n */\nexport function lerp(x: bigint, y: bigint, t: number): bigint {\n  return x + BigInt(Math.floor(t*Number(y - x)));\n}\n// - https://processing.org/reference/lerp_.html\n// - https://docs.unity3d.com/ScriptReference/Vector3.Lerp.html\n// - https://stackoverflow.com/questions/53970655/how-to-convert-bigint-to-number-in-javascript\n\n\n\n\n// POWER / LOGARITHM\n// -----------------\n\n/**\n * Check if bigint is a power-of-2.\n * @param x a bigint\n * @returns 2ⁱ = x? | i = +ve integer\n */\nexport function isPow2(x: bigint): boolean {\n  return /^10*$/.test(x.toString(2));\n}\n\n\n/**\n * Check if bigint is a power-of-10.\n * @param x a bigint\n * @returns 10ⁱ = x? | i = +ve integer\n */\nexport function isPow10(x: bigint): boolean {\n  return /^10*$/.test(x.toString());\n}\n\n\n/**\n * Find largest power-of-2 less than or equal to given bigint.\n * @param x a bigint\n * @returns 2ⁱ | 2ⁱ ≤ x and 2ⁱ > x/2\n */\nexport function prevPow2(x: bigint): bigint {\n  if (x<=1n) return 0n;\n  const n = x.toString(2).length;\n  return BigInt(\"0b1\" + \"0\".repeat(n-1));\n}\n\n\n/**\n * Find largest power-of-10 less than or equal to given bigint.\n * @param x a bigint\n * @returns 10ⁱ | 10ⁱ ≤ x and 10ⁱ > x/10\n */\nexport function prevPow10(x: bigint): bigint {\n  if (x<=1n) return 0n;\n  const n = x.toString(10).length;\n  return BigInt(\"1\" + \"0\".repeat(n-1));\n}\n\n\n/**\n * Find smallest power-of-2 greater than or equal to given bigint.\n * @param x a bigint\n * @returns 2ⁱ | 2ⁱ ≥ x and 2ⁱ < 2x\n */\nexport function nextPow2(x: bigint): bigint {\n  if (x<=0n) return 1n;\n  const n = (x-1n).toString(2).length;\n  return BigInt(\"0b1\" + \"0\".repeat(n));\n}\n\n\n/**\n * Find smallest power-of-10 greater than or equal to given bigint.\n * @param x a bigint\n * @returns 10ⁱ | 10ⁱ ≥ x and 10ⁱ < 10x\n */\nexport function nextPow10(x: bigint): bigint {\n  if (x<=0n) return 1n;\n  const n = (x-1n).toString(10).length;\n  return BigInt(\"1\" + \"0\".repeat(n));\n}\n\n\n/**\n * Find the base-2 logarithm of a bigint.\n * @param x a bigint\n * @returns log₂(x)\n */\nexport function log2(x: bigint): bigint {\n  const n = x.toString(2).length - 1;\n  return x<=0n? 0n : BigInt(n);\n}\n\n\n/**\n * Find the base-10 logarithm of a bigint.\n * @param x a bigint\n * @returns log₁₀(x)\n */\nexport function log10(x: bigint): bigint {\n  const n = x.toString().length - 1;\n  return x<=0n? 0n : BigInt(n);\n}\n\n\n// TODO: isPow() based on accelerated search\n// TODO: prevPow() based on accelerated search\n// TODO: nextPow() based on accelerated search\n// TODO: log() based on accelerated search\n\n\n\n\n// ROOT\n// ----\n\n/**\n * Find the square root of a bigint.\n * @param x a bigint\n * @returns √x\n */\nexport function sqrt(x: bigint): bigint | null {\n  if (x===0n) return 0n;\n  return x>0n? sqrtPos(x) : null;\n}\n\nfunction sqrtPos(x: bigint): bigint {\n  let a = 1n << (log2(x)/2n + 1n);  // initial guess\n  let b = 1n + a;\n  while (a<b) {\n    b = a;\n    a = (b + x/b)/2n;\n  }\n  return b;\n}\n\n\n/**\n * Find the cube root of a bigint.\n * @param x a bigint\n * @returns ³√x\n */\nexport function cbrt(x: bigint): bigint | null {\n  return root(x, 3n);\n}\n\n\n/**\n * Find the nth root of a bigint.\n * @param x a bigint\n * @param n nth root (1n)\n * @returns ⁿ√x\n */\nexport function root(x: bigint, n: bigint=1n): bigint | null {\n  if (x===0n) return 0n;\n  else if (x>0n) return rootPos(x, n);\n  return n % 2n!==0n?  -rootPos(-x, n) : null;\n}\n\nfunction rootPos(x: bigint, n: bigint): bigint {\n  let a = 1n << (log2(x)/n + 1n);  // initial guess\n  let b = 1n + a; const m = n - 1n;\n  if (a===2n) return 1n;\n  while (a<b) {\n    b = a;\n    a = (m*b + x/b**m)/n;\n  }\n  return b;\n}\n\n\n\n\n// DIVISORS\n// --------\n\n/**\n * List all divisors of a bigint, except itself.\n * @param x a bigint\n * @returns proper divisors (factors)\n */\nexport function properDivisors(x: bigint): bigint[] {\n  x = abs(x); const a = [];\n  for (let i=1n; i<x; i++)\n    if (x % i===0n) a.push(i);\n  return a;\n}\nexport {properDivisors as aliquotParts};\n// - https://mathworld.wolfram.com/ProperDivisor.html\n// - https://en.wikipedia.org/wiki/Divisor#Further_notions_and_facts\n\n\n/**\n * Sum all proper divisors of a bigint.\n * @param x a bigint\n * @returns Σdᵢ | dᵢ is a divisor of x and ≠x\n */\nexport function aliquotSum(x: bigint): bigint {\n  x = abs(x); let a = 0n;\n  for (let i=1n; i<x; i++)\n    if (x % i===0n) a += i;\n  return a;\n}\n\n\n/**\n * Find the least prime number which divides a bigint.\n * @param x a bigint\n * @returns least prime factor\n */\nexport function minPrimeFactor(x: bigint): bigint {\n  x = abs(x);\n  // 1. LPF of 2, 3 is the number itself.\n  if (x<=1n) return 0n;\n  if (x<=3n) return x;\n  // 2. LPF for multiples of 2, 3.\n  if (x % 2n===0n) return 2n;\n  if (x % 3n===0n) return 3n;\n  // 3. LPF can be 6k-1 or 6k+1.\n  for (let i=6n, I=(sqrt(x) as bigint)+1n; i<=I; i+=6n) {\n    if (x % (i-1n)===0n) return i-1n;\n    if (x % (i+1n)===0n) return i+1n;\n  }\n  return x;\n}\nexport {minPrimeFactor as leastPrimeFactor};\n// - https://mathworld.wolfram.com/LeastPrimeFactor.html\n\n\n/**\n * Find the greatest prime number which divides a bigint.\n * @param x a bigint\n * @returns greatest prime factor\n */\nexport function maxPrimeFactor(x: bigint): bigint {\n  x = abs(x); let a = 0n;\n  // 1. GPF of 2, 3 is the number itself.\n  if (x<=1n) return 0n;\n  if (x<=3n) return x;\n  // 2. Remove factors 2, 3.\n  for (; x % 2n===0n; a=2n)\n    x /= 2n;\n  for (; x % 3n===0n; a=3n)\n    x /= 3n;\n  // 3. Remove factors 6k-1, 6k+1.\n  for (let i=6n, I=(sqrt(x) as bigint)+1n; x>1n && i<=I; i+=6n) {\n    for (; x % (i-1n)==0n; a=i-1n)\n      x /= i-1n;\n    for (; x % (i+1n)==0n; a=i+1n)\n      x /= i+1n;\n  }\n  if (x<=1n) return a;\n  return x;\n}\nexport {maxPrimeFactor as greatestPrimeFactor};\n// - https://mathworld.wolfram.com/GreatestPrimeFactor.html\n// - https://www.geeksforgeeks.org/find-largest-prime-factor-number/\n\n\n/**\n * Find the prime factors of a bigint.\n * @param x a bigint\n * @returns [f₀, f₁, ...] | fᵢ divides x and is prime\n */\nexport function primeFactors(x: bigint): bigint[] {\n  x = abs(x); const a: bigint[] = [];\n  if (x<=1n) return [];\n  if (x<=3n) return [x];\n  // 2. Try factors 2, 3.\n  x = pushPrimeFactorTo$(a, x, 2n);\n  x = pushPrimeFactorTo$(a, x, 3n);\n  // 3. Try factors 6k-1, 6k+1.\n  for (let i=6n, I=(sqrt(x) as bigint)+1n; x>1n && i<=I; i+=6n) {\n    x = pushPrimeFactorTo$(a, x, i-1n);\n    x = pushPrimeFactorTo$(a, x, i+1n);\n  }\n  if (x>1n) a.push(x);\n  return a;\n}\n\nfunction pushPrimeFactorTo$(a: bigint[], x: bigint, f: bigint): bigint {\n  if (x % f!==0n) return x;\n  do {\n    x /= f;\n  } while (x % f===0n);\n  a.push(f);\n  return x;\n}\n// - https://www.geeksforgeeks.org/prime-factors-big-number/\n// - https://mathworld.wolfram.com/PrimeFactor.html\n\n\n/**\n * Find the prime factors and respective exponents of a bigint.\n * @param x a bigint\n * @returns [[f₀, e₀], [f₁, e₁], ...] | fᵢ is a prime factor of x and eᵢ is its exponent\n */\nexport function primeExponentials(x: bigint): [bigint, bigint][] {\n  x = abs(x); const a: [bigint, bigint][] = [];\n  if (x<=1n) return [];\n  if (x<=3n) return [[x, 1n]];\n  // 2. Try factors 2, 3.\n  x = pushPrimeExponentialTo$(a, x, 2n);\n  x = pushPrimeExponentialTo$(a, x, 3n);\n  // 3. Try factors 6k-1, 6k+1.\n  for (let i=6n, I=(sqrt(x) as bigint)+1n; x>1n && i<=I; i+=6n) {\n    x = pushPrimeExponentialTo$(a, x, i-1n);\n    x = pushPrimeExponentialTo$(a, x, i+1n);\n  }\n  if (x>1n) a.push([x, 1n]);\n  return a;\n}\n\nfunction pushPrimeExponentialTo$(a: [bigint, bigint][], x: bigint, f: bigint): bigint {\n  if (x % f!==0n) return x;\n  let e = 0n;\n  do {\n    x /= f; ++e;\n  } while (x % f===0n);\n  a.push([f, e]);\n  return x;\n}\n// - https://www.geeksforgeeks.org/prime-factors-big-number/\n// - https://reference.wolfram.com/language/ref/FactorInteger.html\n// - https://mathworld.wolfram.com/PrimeFactorization.html\n// - https://mathworld.wolfram.com/PrimeFactor.html\n\n\n/**\n * Check if bigint is prime.\n * @param x a bigint\n * @returns is divisible by 1n and itself only?\n */\nexport function isPrime(x: bigint): boolean {\n  return x!==0n && minPrimeFactor(x) === abs(x);\n}\n\n\n/**\n * Find the greatest common divisor of bigints.\n * @param xs a list of bigints\n * @returns gcd(x₁, x₂, ...)\n */\nexport function gcd(...xs: bigint[]): bigint {\n  let a = xs[0] || 1n;\n  for(let i=1, I=xs.length; i<I; i++)\n    a = gcdPair(a, xs[i]);\n  return a;\n}\nexport {gcd as hcf};\n\n// Find the greatest common divisor of a pair of bigints.\nfunction gcdPair(x: bigint, y: bigint): bigint {\n  while (y!==0n) {\n    const t = y;\n    y = x % y;\n    x = t;\n  }\n  return x;\n}\n// - https://lemire.me/blog/2013/12/26/fastest-way-to-compute-the-greatest-common-divisor/\n// - https://en.wikipedia.org/wiki/Euclidean_algorithm\n\n\n/**\n * Find the least common multiple of bigints.\n * @param xs a list of bigints\n * @returns lcm(x₁, x₂, ...)\n */\nexport function lcm(...xs: bigint[]): bigint {\n  let a = xs[0] || 1n;\n  for (let i=1, I=xs.length; i<I; i++)\n      a = a*xs[i] / gcdPair(a, xs[i]);\n  return a;\n}\n// - https://en.wikipedia.org/wiki/Least_common_multiple\n\n\n\n\n// ARRANGEMENTS\n// ------------\n\n/**\n * Find the factorial of a bigint.\n * @param n a bigint\n * @param k denominator factorial [0]\n * @returns P(n, k); k=0: n!, k>0: n!/k!\n */\nexport function factorial(n: bigint, k: bigint=0n): bigint {\n  let a = 1n;\n  if (n<0n) return 0n;\n  for (let i=k+1n; i<=n; i++)\n    a *= i;\n  return a;\n}\n// - https://github.com/alawatthe/MathLib/blob/master/src/Functn/functions/factorial.ts\n// - https://en.wikipedia.org/wiki/Permutation\n\n\n/**\n * Find the number of ways to choose k elements from a set of n elements.\n * @param n elements in source set\n * @param k elements in choose set\n * @returns C(n, k)\n */\nexport function binomial(n: bigint, k: bigint): bigint {\n  // 1. Generalization to negative integers\n  if (k<0n || k>abs(n)) return 0n;\n  if (n<0n) return ((-1n)**k)*binomial(-n, k);\n  // 2. Take advantage of symmetry\n  k = k>n-k? n-k:k;\n  let a = 1n, b = 1n;\n  for (let i=1n; i<=k; i++, n--) {\n    a *= n;\n    b *= i;\n  }\n  return a/b;\n}\n// - https://github.com/alawatthe/MathLib/blob/master/src/Functn/functions/binomial.ts\n// - https://en.wikipedia.org/wiki/Binomial_coefficient\n\n\n/**\n * Find the number of ways to put n objects in m bins (n=sum(kᵢ)).\n * @param ks objects per bin (kᵢ)\n * @returns n!/(k₁!k₂!...) | n=sum(kᵢ)\n */\nexport function multinomial(...ks: bigint[]): bigint {\n  let n = sum(...ks), a = 1n, b = 1n;\n  for (let i=0, j=0n, I=ks.length; i<I;) {\n    if (j<=0n) j = ks[i++];\n    else { a *= n--; b*= j--; }\n  }\n  return a/b;\n}\n// - https://en.wikipedia.org/wiki/Multinomial_distribution\n\n\n\n\n// GEOMETRY\n// --------\n\n/**\n * Find the length of hypotenuse.\n * @param xs lengths of perpendicular sides (x, y, z, ...)\n * @returns √(x² + y² + z²)\n */\nexport function hypot(...xs: bigint[]): bigint {\n  let a = 0n;\n  for (const x of xs)\n    a += x*x;\n  return sqrt(a) as bigint;\n}\n\n\n\n\n// STATISTICS\n// ----------\n\n/**\n * Find the sum of bigints (Σ).\n * @param xs bigints\n * @returns Σxᵢ\n */\nexport function sum(...xs: bigint[]): bigint {\n  let a = 0n;\n  for (const x of xs)\n    a += x;\n  return a;\n}\n\n\n/**\n * Find the product of bigints (∏).\n * @param xs bigints\n * @returns ∏xᵢ\n */\nexport function product(...xs: bigint[]): bigint {\n  let a = 1n;\n  for (const x of xs)\n    a *= x;\n  return a;\n}\n\n\n/**\n * Find the value separating the higher and lower halves of bigints.\n * @param xs a list of bigints\n * @returns xₘ | sort(xs) = [..., xₘ, ...]\n */\nexport function median(...xs: bigint[]): bigint {\n  if (xs.length===0) return 0n;\n  xs.sort(compare);\n  const i = xs.length>>1;\n  if ((xs.length & 1)===1) return xs[i];\n  return (xs[i-1] + xs[i])/2n;\n}\n// - https://stackoverflow.com/questions/45309447/calculating-median-javascript\n// - https://en.wikipedia.org/wiki/Median\n\n\n/**\n * Find the values that appear most often.\n * @param xs a list of bigints\n * @returns [xₘ₁, xₘ₂, ...] | count(xₘᵢ) ≥ count(xᵢ) ∀ xᵢ ∈ xs\n */\nexport function modes(...xs: bigint[]): bigint[] {\n  xs.sort(compare);\n  const r = maxRepeat(xs);\n  return getRepeats(xs, r);\n}\n// - https://en.wikipedia.org/wiki/Mode_(statistics)\n\n// Get the maximum number of times any bigint has repeated in a sorted array.\nfunction maxRepeat(xs: bigint[]): number {\n  let count = Math.min(xs.length, 1), max = count;\n  for (let i=1, I=xs.length; i<I; i++) {\n    if (xs[i-1]===xs[i]) count++;\n    else { max = Math.max(max, count); count = 1; }\n  }\n  return Math.max(max, count);\n}\n\n// Get the numbers which have been repeated atleast given number of times.\nfunction getRepeats(xs: bigint[], r: number): bigint[] {\n  const a: bigint[] = []; r--;\n  for (let i=0, I=xs.length-r; i<I; i++)\n    if (xs[i]===xs[i+r]) a.push(xs[i+=r]);\n  return a;\n}\n\n\n/**\n * Find the smallest bigint.\n * @param xs bigints\n * @returns min(xs)\n */\nexport function min(...xs: bigint[]): bigint {\n  if (xs.length===0) return 0n;\n  let a = xs[0];\n  for (const x of xs)\n    a = x<a? x : a;\n  return a;\n}\n\n\n/**\n * Find the largest bigint.\n * @param xs bigints\n * @returns max(xs)\n */\nexport function max(...xs: bigint[]): bigint {\n  if (xs.length===0) return 0n;\n  let a = xs[0];\n  for (const x of xs)\n    a = x>a? x : a;\n  return a;\n}\n\n\n/**\n * Find the minimum and maximum bigint.\n * @param xs bigints\n * @returns [min, max]\n */\nexport function range(...xs: bigint[]): [bigint, bigint] {\n  if (xs.length===0) return [0n, 0n];\n  let a = xs[0], b = a;\n  for (const x of xs) {\n    a = x<a? x : a;\n    b = x>b? x : b;\n  }\n  return [a, b];\n}\n\n\n/**\n * Find the mean of squared deviation of bigints from its mean.\n * @param xs a list of bigints\n * @returns σ² = E[(xs - µ)²] | µ = mean(xs)\n */\nexport function variance(...xs: bigint[]): bigint {\n  if (xs.length===0) return 0n;\n  const m = arithmeticMean(...xs); let a = 0n;\n  for (const x of xs)\n    a += (x-m)**2n;\n  return a/BigInt(xs.length);\n}\n// - https://en.wikipedia.org/wiki/Variance\n\n\n\n\n// MEAN (STATISTICS)\n// -----------------\n\n/**\n * Find the arithmetic mean of bigints (µ).\n * @param xs bigints\n * @returns µ = (Σxᵢ)/n | n = size(xs)\n */\nexport function arithmeticMean(...xs: bigint[]): bigint {\n  const n = BigInt(xs.length);\n  return sum(...xs)/n;\n}\nexport {arithmeticMean as mean};\n\n\n/**\n * Find the geometric mean of bigints.\n * @param xs bigints\n * @returns ⁿ√(∏xᵢ) | n = size(xs)\n */\nexport function geometricMean(...xs: bigint[]): bigint | null {\n  const n = BigInt(xs.length);\n  return root(product(...xs), n);\n}\n\n\n/**\n * Find the harmonic mean of bigints.\n * @param xs bigints\n * @returns n/Σ(1/xᵢ) | n = size(xs)\n */\nexport function harmonicMean(...xs: bigint[]): bigint {\n  const n = BigInt(xs.length);\n  const p = product(...xs); let q = 0n;\n  for (const x of 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