extra-number
Version:
A collection for common number functions (queries, comparisons, rounding, divisors, etc).<br>
1,484 lines • 104 kB
JavaScript
//#region TYPES
/** Classification of floating point numbers. */ export var FloatClass = /*#__PURE__*/ function(FloatClass) {
/** Number is zero. */ FloatClass[FloatClass["ZERO"] = 0] = "ZERO";
/** Number is subnormal (denormal). */ FloatClass[FloatClass["SUBNORMAL"] = 16] = "SUBNORMAL";
/** Number is normal. */ FloatClass[FloatClass["NORMAL"] = 32] = "NORMAL";
/** Number is infinite. */ FloatClass[FloatClass["INFINITE"] = 48] = "INFINITE";
/** Number is NaN (not a number). */ FloatClass[FloatClass["NAN"] = 64] = "NAN";
return FloatClass;
}({});
//#endregion
//#region CONSTANTS
/** Smallest normal (not subnormal) 64-bit floating point number. */ export const MIN_NORMAL = 2.2250738585072014e-308;
/** Ratio of a circle's circumference to its diameter. */ export const TAU = 2 * Math.PI;
//#endregion
//#region ABOUT
/**
* Check if value is a number.
* @param v a value
* @returns is number?
* @example
* ```ts
* xnumber.is(3.14);
* // → true
*
* xnumber.is(NaN);
* // → true (an unexpressable number)
*
* xnumber.is('3.14');
* // → false
*
* xnumber.is({'value': 3.14});
* // → false
*
* xnumber.is(null);
* // → false
* ```
*/ export function is(v) {
return typeof v === "number";
}
/**
* Check if number is normal (not zero, subnormal, infinite, or NaN).
* @param x a number
* @returns is normal number?
* @example
* ```ts
* xnumber.isNormal(3.14);
* // → true
*
* xnumber.isNormal(0);
* // → false
*
* xnumber.isNormal(1e-320);
* // → false
*
* xnumber.isNormal(Infinity);
* // → false
*/ export function isNormal(x) {
if (x === 0) return false;
if (!Number.isFinite(x)) return false;
if (Math.abs(x) < MIN_NORMAL) return false;
return true;
}
/**
* Classify a number into its floating point category.
* @param x a number
* @returns floating point class of `x`
* @example
* ```ts
* xnumber.classify(0);
* // → FloatClass.ZERO
*
* xnumber.classify(1e-320);
* // → FloatClass.SUBNORMAL
*
* xnumber.classify(3.14);
* // → FloatClass.NORMAL
*
* xnumber.classify(Infinity);
* // → FloatClass.INFINITE
*
* xnumber.classify(NaN);
* // → FloatClass.NAN
*/ export function classify(x) {
if (x === 0) return FloatClass.ZERO;
if (Number.isNaN(x)) return FloatClass.NAN;
if (!Number.isFinite(x)) return FloatClass.INFINITE;
if (Math.abs(x) < MIN_NORMAL) return FloatClass.SUBNORMAL;
return FloatClass.NORMAL;
}
/**
* Examine if sum of divisors equals the number itself.
* @param x a number
* @returns sum of divisors(x) excluding x = x?
* @example
* ```ts
* xnumber.isPerfect(6);
* // → true (1+2+3=6)
*
* xnumber.isPerfect(28);
* // → true (1+2+4+7+14=28)
*
* xnumber.isPerfect(12);
* // → false (1+2+3+4+6=16)
*
* xnumber.isPerfect(1);
* // → false (no proper divisors)
* ```
*/ export function isPerfect(x) {
return aliquotSum(x) === x;
}
// TODO: isAlmostPerfect()?
// TODO: isSemiPerfect()?
// TODO: isQuasiPerfect()?
// TODO: isWeird()?
/**
* Examine if sum of divisors exceeds the number itself.
* @param x a number
* @returns sum of divisors(x) excluding x > x?
* @example
* ```ts
* xnumber.isAbundant(8);
* // → false (1+2+4=7 < 8)
*
* xnumber.isAbundant(15);
* // → false (1+3+5=9 < 15)
*
* xnumber.isAbundant(12);
* // → true (1+2+3+4+6=16 > 12)
*
* xnumber.isAbundant(6);
* // → false (1+2+3=6 = 6)
* ```
*/ export function isAbundant(x) {
return aliquotSum(x) > x;
}
export { isAbundant as isExcessive };
// TODO: isPrimitiveAbundant()?
// TODO: isSuperAbundant()?
/**
* Obtain the sum of divisors exceeding the number itself.
* @param x a number
* @returns sum of divisors(x) excluding x - x
* @example
* ```ts
* xnumber.abundance(12);
* // → 4 (1+2+3+4+6=16; 16-12=4)
*
* xnumber.abundance(15);
* // → -6 (1+3+5=9; 9-15=-6)
* ```
*/ export function abundance(x) {
return aliquotSum(x) - x;
}
/**
* Obtain the ratio of sum of divisors to the number itself.
* @param x a number
* @returns (sum of divisors(x)) / x
* @example
* ```ts
* xnumber.abundancyIndex(12);
* // → 2.3333... (1+2+3+4+6+12=28; 28/12=7/3)
*
* xnumber.abundancyIndex(15);
* // → 1.6 (1+3+5+15=24; 24/15=8/5)
* ```
*/ export function abundancyIndex(x) {
return (aliquotSum(x) + x) / x;
}
// TODO: areAmicable()?
// TODO: areFriendly()?
/**
* Count the number of significant digits in a number.
* @param x a number
* @returns |m - '.'| | x = m × 10ᵉ; m = mantissa, e = exponent
* @example
* ```ts
* xnumber.significantDigits(0.0034);
* // → 2
*
* xnumber.significantDigits(120.5e50);
* // → 4
*
* xnumber.significantDigits(120.5e-50);
* // → 4
* ```
*/ export function significantDigits(x) {
const a = x.toExponential();
return a.replace(/e[\+\-0-9]*$/, "").replace(/^0\.?0*|\./, "").length;
}
// - https://stackoverflow.com/questions/22884720/what-is-the-fastest-way-to-count-the-number-of-significant-digits-of-a-number
//#endregion
//#region COMPARE
/**
* Compare two numbers.
* @param x a number
* @param y another number
* @returns x<y: -ve, x=y: 0, x>y: +ve
* @example
* ```ts
* xnumber.compare(10, 12);
* // → -2 (-ve)
*
* xnumber.compare(12, 12);
* // → 0
*
* xnumber.compare(17, 12);
* // → 5 (+ve)
* ```
*/ export function compare(x, y) {
return x - y;
}
/**
* Check if two numbers are unordered (NaN involved).
* @param x a number
* @param y another number
* @returns x=NaN or y=NaN?
* @example
* ```ts
* xnumber.isUnordered(10, NaN);
* // → true
*
* xnumber.isUnordered(NaN, 12);
* // → true
*
* xnumber.isUnordered(10, 12);
* // → false
*
* xnumber.isUnordered(NaN, NaN);
* // → true
* ```
*/ export function isUnordered(x, y) {
return Number.isNaN(x) || Number.isNaN(y);
}
//#endregion
//#region SIGN
/**
* Copy the sign of a number to another number.
* @param mag number whose magnitude is to be copied
* @param sgn sign of the number to copy
* @returns `|mag|` with sign of `sgn`
* @example
* ```ts
* xnumber.copySign(3.14, -1);
* // → -3.14
*
* xnumber.copySign(-3.14, 1);
* // → 3.14
*
* xnumber.copySign(-6.28, -1);
* // → -6.28
*/ export function copySign(mag, sgn) {
return Math.sign(sgn) === Math.sign(mag) ? mag : -mag;
}
//#endregion
//#region ROUND
/**
* Round down a number to specific precision.
* @param x a number
* @param pre to precision [1]
* @example
* ```ts
* xnumber.floor(9.161, 1);
* // → 9
*
* xnumber.floor(9.161, 0.01);
* // → 9.16
*
* xnumber.floor(9.1617, 0.05);
* // → 9.15
* ```
*/ export function floor(x, pre = 1) {
return Math.floor(x / pre) * pre;
}
/**
* Round up a number to specific precision.
* @param x a number
* @param pre to precision [1]
* @example
* ```ts
* xnumber.ceil(9.121, 1);
* // → 10
*
* xnumber.ceil(9.121, 0.01);
* // → 9.13
*
* xnumber.ceil(9.1217, 0.05);
* // → 9.15
* ```
*/ export function ceil(x, pre = 1) {
return Math.ceil(x / pre) * pre;
}
/**
* Round a number to specific precision.
* @param x a number
* @param pre to precision [1]
* @example
* ```ts
* xnumber.round(9.135, 1);
* // → 9
*
* xnumber.round(9.135, 1e-2);
* // → 9.14
*
* xnumber.round(9.1357, 0.05);
* // → 9.15
*
* 0.1 + 0.2
* // → 0.30000000000000004 (why?)
*
* xnumber.round(0.1 + 0.2, 1e-12);
* // → 0.3 (nice!)
* ```
*/ export function round(x, pre = 1) {
return Math.round(x / pre) * pre;
}
/**
* Perform floor-divison of two numbers.
* @param x divisor
* @param y dividend
* @returns ⌊x/y⌋
* @example
* ```ts
* xnumber.floorDiv(15, 4);
* // → 3
*
* xnumber.floorDiv(2, 2);
* // → 1
*
* xnumber.floorDiv(-15, 4);
* // → -4
* ```
*/ export function floorDiv(x, y) {
return Math.floor(x / y);
}
/**
* Perform ceiling-divison of two numbers.
* @param x divisor
* @param y dividend
* @returns ⌈x/y⌉
* @example
* ```ts
* xnumber.ceilDiv(15, 4);
* // → 4
*
* xnumber.ceilDiv(2, 2);
* // → 1
*
* xnumber.ceilDiv(-15, 4);
* // → -3
* ```
*/ export function ceilDiv(x, y) {
return Math.ceil(x / y);
}
/**
* Perform rounded-divison of two numbers.
* @param x divisor
* @param y dividend
* @returns [x/y]
* @example
* ```ts
* xnumber.roundDiv(15, 4);
* // → 4
*
* xnumber.roundDiv(2, 2);
* // → 1
*
* xnumber.roundDiv(-15, 4);
* // → -4
* ```
*/ export function roundDiv(x, y) {
return Math.round(x / y);
}
//#endregion
//#region MODULO
/**
* Find the remainder of x/y with sign of x (truncated division).
* @param x dividend
* @param y divisor
* @returns trunc(x % y)
* @example
* ```ts
* xnumber.rem(1, 10);
* // → 1
*
* xnumber.rem(-1, 10);
* // → -1
*
* xnumber.rem(1, -10);
* // → 1
* ```
*/ 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)
* @example
* ```ts
* xnumber.mod(1, 10);
* // → 1
*
* xnumber.mod(-1, 10);
* // → 9
*
* xnumber.mod(1, -10);
* // → -9
* ```
*/ export function mod(x, y) {
return x - y * Math.floor(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 n>0: floor(x % y), n<0: ceil(x % y)
* @example
* ```ts
* xnumber.modp(1, 10);
* // → 1
*
* xnumber.modp(-1, 10);
* // → 9
*
* xnumber.modp(1, -10);
* // → 1
* ```
*/ export function modp(x, y) {
return x - Math.abs(y) * Math.floor(x / Math.abs(y));
}
// - https://en.wikipedia.org/wiki/Modulo_operation
//#endregion
//#region RANGE CONTROL
/**
* Constrain a number within a minimum and a maximum value.
* @param x a number
* @param min minimum value
* @param max maximum value
* @returns x<min: min, x>max: max, x
* @example
* ```ts
* xnumber.constrain(20, 0, 50);
* // → 20
*
* xnumber.constrain(-10, 0, 100);
* // → 0
*
* xnumber.constrain(120, 0, 100);
* // → 100
* ```
*/ export function constrain(x, min, max) {
return Math.min(Math.max(x, min), max);
}
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)
/**
* Normalize a number from its current range into a value between 0 and 1.
* @param x a number
* @param r lower bound of current range
* @param R upper bound of current range
* @returns ∈ [0, 1]
* @example
* ```ts
* xnumber.normalize(20, 0, 50);
* // → 0.4
*
* xnumber.normalize(-10, 0, 100);
* // → -0.1
* ```
*/ export function normalize(x, r, R) {
return (x - r) / (R - r);
}
export { normalize as norm };
// - https://processing.org/reference/norm_.html
/**
* Re-map a number from one range to another.
* @param x a number
* @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]
* @example
* ```ts
* xnumber.remap(25, 0, 100, 0, 1366);
* // → 341.5
*
* xnumber.remap(110, 0, 100, -20, -10);
* // → -9
* ```
*/ export function remap(x, r, R, t, T) {
return t + (x - r) / (R - r) * (T - t);
}
export { remap as map };
// - https://processing.org/reference/map_.html
/**
* Linearly interpolate a number between two numbers.
* @param x start number
* @param y stop number
* @param t interpolant ∈ [0, 1]
* @returns ∈ [x, y]
* @example
* ```ts
* xnumber.lerp(80, 320, 0.8);
* // → 272
*
* xnumber.lerp(80, 320, 0.20);
* // → 128
*
* xnumber.lerp(80, 320, 0.32);
* // → 156.8
* ```
*/ export function lerp(x, y, t) {
return x + t * (y - x);
}
// - https://processing.org/reference/lerp_.html
// - https://docs.unity3d.com/ScriptReference/Vector3.Lerp.html
//#endregion
//#region FRACTIONAL
/**
* Make a number purely fractional, by shifting the decimal point.
* @param x a number
* @param n maximum number of digits to shift (-1: all) [20]
* @returns fractional number
* @example
* ```ts
* xnumber.fractionize(123);
* // → 0.123
*
* xnumber.fractionize(-9876, 3);
* // → -0.988
*
* xnumber.fractionize(0.001234, 3);
* // → 0.001
* ```
*/ export function fractionize(x, n = 20) {
const xp = Math.abs(x);
const xdig = xp < 1 ? 0 : 1 + Math.floor(Math.log10(xp));
if (n < 0) return x * Math.pow(10, -xdig);
return Math.round(x * Math.pow(10, n - xdig)) * Math.pow(10, -n);
}
/**
* Make a number purely integer, by shifting the decimal point.
* @param x a number
* @param n maximum number of digits to shift (-1: all) [20]
* @returns integer number
* @example
* ```ts
* xnumber.defractionize(0.123);
* // → 123
*
* xnumber.defractionize(-0.988, 2);
* // → -99
*
* xnumber.defractionize(12.345, 2);
* // → 1235
* ```
*/ export function defractionize(x, n = 20) {
let xp = Math.abs(x), i = 0;
for(; n < 0 || i < n;){
xp *= 10;
++i;
if (xp % 1 === 0) break;
}
return Math.round(x * Math.pow(10, i));
}
//#endregion
//#region ARITHMETIC
/**
* Check if a number is a power-of-n.
* @param x a number
* @param n base
* @returns nⁱ = x? | i = +ve integer
* @example
* ```ts
* xnumber.isPow(32, 2);
* // → true
*
* xnumber.isPow(32, 4);
* // → false
*
* xnumber.isPow(64, 4);
* // → true
* ```
*/ export function isPow(x, n) {
if (n === 0) return x === 0;
const p = log(Math.abs(x), Math.abs(n));
if (p !== Math.floor(p)) return false;
return x < 0 ? n < 0 && (p & 1) === 1 : n > 0 || (p & 1) === 0;
}
/**
* Find largest power-of-n less than or equal to given number.
* @param x a number
* @param n base
* @returns nⁱ | nⁱ ≤ x and nⁱ > x/n
* @example
* ```ts
* xnumber.prevPow(32, 2);
* // → 32
*
* xnumber.prevPow(35, 2);
* // → 32
*
* xnumber.prevPow(35, 3);
* // → 27
* ```
*/ export function prevPow(x, n) {
if (x <= 1) return 0;
const p = Math.floor(Math.log(x) / Math.log(n));
return Math.pow(n, p);
}
/**
* Find smallest power-of-n greater than or equal to given number.
* @param x a number
* @param n base
* @returns nⁱ | nⁱ ≥ x and nⁱ < n*x
* @example
* ```ts
* xnumber.nextPow(32, 2);
* // → 32
*
* xnumber.nextPow(29, 2);
* // → 32
*
* xnumber.nextPow(29, 3);
* // → 81
* ```
*/ export function nextPow(x, n) {
if (x <= 0) return 1;
const p = Math.ceil(Math.log(x) / Math.log(n));
return Math.pow(n, p);
}
/**
* Find the nth root of a number (ⁿ√).
* @param x a number
* @param n root
* @returns ⁿ√x
* @example
* ```ts
* xnumber.root(25, 2);
* // → 5
*
* xnumber.root(-8, 3);
* // → -2
* ```
*/ export function root(x, n) {
if ((n & 1) === 0) return Math.pow(x, 1 / n);
return Math.sign(x) * Math.pow(Math.abs(x), 1 / n);
}
// - https://github.com/alawatthe/MathLib/blob/master/src/Functn/functions/root.ts
/**
* Find the logarithm of a number with a given base.
* @param x a number
* @param b logarithm base [e]
* @returns log_b (x)
* @example
* ```ts
* xnumber.log(Math.E);
* // → 1
*
* xnumber.log(10);
* // → 2.302585092994046
*
* xnumber.log(243, 3);
* // → 5
*
* xnumber.log(64, 2);
* // → 6
* ```
*/ export function log(x, b = Math.E) {
return Math.log(x) / Math.log(b);
}
// - https://en.wikipedia.org/wiki/Logarithm
//#endregion
//#region DIVISORS
/**
* List all divisors of a number, except itself.
* @param x a number
* @returns proper divisors (factors)
* @example
* ```ts
* xnumber.properDivisors(6);
* // → [1, 2, 3]
*
* xnumber.properDivisors(1);
* // → []
*
* xnumber.properDivisors(0);
* // → []
*
* xnumber.properDivisors(-24);
* // → [1, 2, 3, 4, 6, 8, 12]
* ```
*/ export function properDivisors(x) {
const a = [];
x = Math.abs(x);
for(let i = 1; i < x; i++)if (x % i === 0) 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 number.
* @param x a number
* @returns Σdᵢ | dᵢ is a divisor of x and ≠x
* @example
* ```ts
* xnumber.aliquotSum(6);
* // → 6 (1+2+3)
*
* xnumber.aliquotSum(1);
* // → 0
*
* xnumber.aliquotSum(0);
* // → 0
*
* xnumber.aliquotSum(-24);
* // → 36 (1+2+3+4+6+8+12)
* ```
*/ export function aliquotSum(x) {
let a = 0;
x = Math.abs(x);
for(let i = 0; i < x; i++)if (x % i === 0) a += i;
return a;
}
/**
* Find the least prime number which divides a number.
* @param x a number
* @returns least prime factor
* @example
* ```ts
* xnumber.minPrimeFactor(1);
* // → 0
*
* xnumber.minPrimeFactor(3);
* // → 3
*
* xnumber.minPrimeFactor(21);
* // → 3
*
* xnumber.minPrimeFactor(55);
* // → 5
*
* xnumber.minPrimeFactor(53);
* // → 53
* ```
*/ export function minPrimeFactor(x) {
x = Math.abs(x);
// 1. LPF of 2, 3 is the number itself.
if (x <= 1) return 0;
if (x <= 3) return x;
// 2. LPF for multiples of 2, 3.
if (x % 2 === 0) return 2;
if (x % 3 === 0) return 3;
// 3. LPF can be 6k-1 or 6k+1.
for(let i = 6, I = Math.sqrt(x) + 1; i <= I; i += 6){
if (x % (i - 1) === 0) return i - 1;
if (x % (i + 1) === 0) return i + 1;
}
return x;
}
export { minPrimeFactor as leastPrimeFactor };
// - https://mathworld.wolfram.com/LeastPrimeFactor.html
/**
* Find the greatest prime number which divides a number.
* @param x a number
* @returns greatest prime factor
* @example
* ```ts
* xnumber.maxPrimeFactor(1);
* // → 0
*
* xnumber.maxPrimeFactor(3);
* // → 3
*
* xnumber.maxPrimeFactor(21);
* // → 7
*
* xnumber.maxPrimeFactor(55);
* // → 11
*
* xnumber.maxPrimeFactor(53);
* // → 53
* ```
*/ export function maxPrimeFactor(x) {
let a = 0;
x = Math.abs(x);
// 1. GPF of 2, 3 is the number itself.
if (x <= 1) return 0;
if (x <= 3) return x;
// 2. Remove factors 2, 3.
for(; x % 2 === 0; a = 2)x /= 2;
for(; x % 3 === 0; a = 3)x /= 3;
// 3. Remove factors 6k-1, 6k+1.
for(let i = 6, I = Math.sqrt(x) + 1; x > 1 && i <= I; i += 6){
for(; x % (i - 1) === 0; a = i - 1)x /= i - 1;
for(; x % (i + 1) === 0; a = i + 1)x /= i + 1;
}
if (x <= 1) 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 number.
* @param x a number
* @returns [f₀, f₁, ...] | fᵢ divides x and is prime
* @example
* ```ts
* xnumber.primeFactors(1);
* // → []
*
* xnumber.primeFactors(3);
* // → [3]
*
* xnumber.primeFactors(21);
* // → [3, 7]
*
* xnumber.primeFactors(55);
* // → [5, 11]
*
* xnumber.primeFactors(53);
* // → [53]
* ```
*/ export function primeFactors(x) {
const a = [];
x = Math.abs(x);
if (x <= 1) return [];
if (x <= 3) return [
x
];
// 2. Try factors 2, 3.
x = pushPrimeFactorTo$(a, x, 2);
x = pushPrimeFactorTo$(a, x, 3);
// 3. Try factors 6k-1, 6k+1.
for(let i = 6, I = Math.sqrt(x) + 1; x > 1 && i <= I; i += 6){
x = pushPrimeFactorTo$(a, x, i - 1);
x = pushPrimeFactorTo$(a, x, i + 1);
}
if (x > 1) a.push(x);
return a;
}
function pushPrimeFactorTo$(a, x, f) {
if (x % f !== 0) return x;
do {
x /= f;
}while (x % f === 0)
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 number.
* @param x a number
* @returns [[f₀, e₀], [f₁, e₁], ...] | fᵢ is a prime factor of x and eᵢ is its exponent
* @example
* ```ts
* xnumber.primeExponentials(1);
* // → []
*
* xnumber.primeExponentials(9);
* // → [[3, 2]]
*
* xnumber.primeExponentials(63);
* // → [[3, 2], [7, 1]]
*
* xnumber.primeExponentials(605);
* // → [[5, 1], [11, 2]]
*
* xnumber.primeExponentials(53);
* // → [[53, 1]]
* ```
*/ export function primeExponentials(x) {
const a = [];
x = Math.abs(x);
if (x <= 1) return [];
if (x <= 3) return [
[
x,
1
]
];
// 2. Try factors 2, 3.
x = pushPrimeExponentialTo$(a, x, 2);
x = pushPrimeExponentialTo$(a, x, 3);
// 3. Try factors 6k-1, 6k+1.
for(let i = 6, I = Math.sqrt(x) + 1; x > 1 && i <= I; i += 6){
x = pushPrimeExponentialTo$(a, x, i - 1);
x = pushPrimeExponentialTo$(a, x, i + 1);
}
if (x > 1) a.push([
x,
1
]);
return a;
}
function pushPrimeExponentialTo$(a, x, f) {
if (x % f !== 0) return x;
let e = 0;
do {
x /= f;
++e;
}while (x % f === 0)
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
/**
* Examine if number is prime.
* @param x a number
* @returns is divisible by 1 and itself only?
* @example
* ```ts
* xnumber.isPrime(7);
* // → true
*
* xnumber.isPrime(53);
* // → true
*
* xnumber.isPrime(4);
* // → false
*
* xnumber.isPrime(1);
* // → false
*
* xnumber.isPrime(0);
* // → false
* ```
*/ export function isPrime(x) {
return x !== 0 && minPrimeFactor(x) === Math.abs(x);
}
/**
* Find the greatest common divisor of numbers.
* @param xs a list of numbers
* @returns gcd(x₁, x₂, ...)
* @example
* ```ts
* xnumber.gcd(6, 15);
* // → 3
*
* xnumber.gcd(6, 15, 21);
* // → 3
*
* xnumber.gcd(6, 15, 20);
* // → 1
* ```
*/ export function gcd(...xs) {
let a = xs[0] || 1;
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 numbers.
function gcdPair(x, y) {
while(y !== 0){
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 numbers.
* @param xs a list of numbers
* @returns lcm(x₁, x₂, ...)
* @example
* ```ts
* xnumber.lcm(2, 3);
* // → 6
*
* xnumber.lcm(2, 3, 4);
* // → 12
*
* xnumber.lcm(2, 3, 4, 5);
* // → 60
* ```
*/ export function lcm(...xs) {
let a = xs[0] || 1;
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
//#endregion
//#region ARRANGEMENTS
/**
* Find the factorial of a number.
* @param n a number
* @param k denominator factorial [0]
* @returns P(n, k); k=0: n!, k>0: n!/k!
* @example
* ```ts
* xnumber.factorial(5);
* // → 120
*
* xnumber.factorial(6);
* // → 720
*
* xnumber.factorial(7);
* // → 5040
* ```
*/ export function factorial(n, k = 0) {
if (n < 0) return 0;
let a = 1;
for(let i = k + 1; 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)
* @example
* ```ts
* xnumber.binomial(4, 1);
* // → 4
*
* xnumber.binomial(4, 2);
* // → 6
*
* xnumber.binomial(4, 3);
* // → 4
* ```
*/ export function binomial(n, k) {
// 1. Generalization to negative integers
if (k < 0 || k > Math.abs(n)) return 0;
if (n < 0) return Math.pow(-1, k) * binomial(-n, k);
// 2. Take advantage of symmetry
k = k > n - k ? n - k : k;
let a = 1;
for(let i = 1; i <= k; i++, n--)a *= n / i;
return a;
}
// - 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)
* @example
* ```ts
* xnumber.multinomial(1, 2);
* // → 3
*
* xnumber.multinomial(1, 2, 3);
* // → 60
*
* xnumber.multinomial(1, 2, 3, 4);
* // → 12600
* ```
*/ export function multinomial(...ks) {
let n = sum(...ks), a = 1;
for(let i = 0, j = 0, I = ks.length; i < I;){
if (j <= 0) j = ks[i++];
else a *= n-- / j--;
}
return a;
}
// - https://en.wikipedia.org/wiki/Multinomial_distribution
//#endregion
//#region GEOMETRY
/**
* Convert radians to degrees.
* @param x radians
* @returns 2π → 360
* @example
* ```ts
* xnumber.degrees(3);
* // → 171.88733853924697
*
* xnumber.degrees(1);
* // → 57.29577951308232
* ```
*/ export function degrees(x) {
return x * (180 / Math.PI);
}
// - https://processing.org/reference/degrees_.html
/**
* Convert degrees to radians.
* @param x degrees
* @returns 360 → 2π
* @example
* ```ts
* xnumber.radians(180);
* // → 3.141592653589793
*
* xnumber.radians(60);
* // → 1.0471975511965976
* ```
*/ export function radians(x) {
return x * (Math.PI / 180);
}
// - https://processing.org/reference/radians_.html
//#endregion
//#region STATISTICS
/**
* Find the sum of numbers (Σ).
* @param xs a list of numbers
* @returns Σxᵢ
* @example
* ```ts
* xnumber.sum(1, 2);
* // → 3
*
* xnumber.sum(1, 2, 3);
* // → 6
*
* xnumber.sum(1, 2, 3, 4);
* // → 10
* ```
*/ export function sum(...xs) {
let a = 0;
for (const x of xs)a += x;
return a;
}
/**
* Find the product of numbers (∏).
* @param xs a list of numbers
* @returns ∏xᵢ
* @example
* ```ts
* xnumber.product(1, 2);
* // → 2
*
* xnumber.product(1, 2, 3);
* // → 6
*
* xnumber.product(1, 2, 3, 4);
* // → 24
* ```
*/ export function product(...xs) {
let a = 1;
for (const x of xs)a *= x;
return a;
}
/**
* Find the value separating the higher and lower halves of numbers.
* @param xs a list of numbers
* @returns xₘ | sort(xs) = [..., xₘ, ...]
* @example
* ```ts
* xnumber.median(1, 7);
* // → 4
*
* xnumber.median(1, 7, 8);
* // → 7
*
* xnumber.median(1, 7, 8, 10);
* // → 7.5
* ```
*/ export function median(...xs) {
if (xs.length === 0) return 0;
xs.sort((a, b)=>a - b);
const i = xs.length >> 1;
if ((xs.length & 1) === 1) return xs[i];
return (xs[i - 1] + xs[i]) / 2;
}
// - 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 numbers
* @returns [xₘ₁, xₘ₂, ...] | count(xₘᵢ) ≥ count(xᵢ) ∀ xᵢ ∈ xs
* @example
* ```ts
* xnumber.modes(1, 2);
* // → [1, 2]
*
* xnumber.modes(1, 2, 2);
* // → [2]
*
* xnumber.modes(1, 2, 2, 3, 3);
* // → [2, 3]
* ```
*/ export function modes(...xs) {
xs.sort((a, b)=>a - b);
const r = maxRepeat(xs);
return getRepeats(xs, r);
}
// - https://en.wikipedia.org/wiki/Mode_(statistics)
// Get the maximum number of times any number 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 and largest values.
* @param xs a list of numbers
* @returns [min(xs), max(xs)]
* @example
* ```ts
* xnumber.range(1, 7);
* // → 6
*
* xnumber.range(1, 7, 6);
* // → 6
*
* xnumber.range(1, 7, 8, 6);
* // → 7
* ```
*/ export function range(...xs) {
return [
Math.min(...xs),
Math.max(...xs)
];
}
// - https://en.wikipedia.org/wiki/Range_(statistics)
/**
* Find the mean of squared deviation of numbers from its mean.
* @param xs a list of numbers
* @returns σ² = E[(xs - µ)²] | µ = mean(xs)
* @example
* ```ts
* xnumber.variance(1, 2);
* // → 0.25
*
* xnumber.variance(1, 2, 3);
* // → 0.6666666666666666
*
* xnumber.variance(1, 2, 3, 4);
* // → 1.25
* ```
*/ export function variance(...xs) {
if (xs.length === 0) return 0;
const m = arithmeticMean(...xs);
let a = 0;
for (const x of xs)a += (x - m) ** 2;
return a / xs.length;
}
// - https://en.wikipedia.org/wiki/Variance
//#endregion
//#region MEAN (STATISTICS)
/**
* Find the average of numbers.
* @param xs a list of numbers
* @returns Σxᵢ/n | n = size(xs)
* @example
* ```ts
* xnumber.arithmethicMean(1, 2);
* // → 1.5
*
* xnumber.arithmethicMean(1, 2, 3);
* // → 2
*
* xnumber.arithmethicMean(1, 2, 3, 4);
* // → 2.5
* ```
*/ export function arithmeticMean(...xs) {
if (xs.length === 0) return 0;
return sum(...xs) / xs.length;
}
export { arithmeticMean as mean };
/**
* Find the geometric mean of numbers.
* @param xs a list of numbers
* @returns ⁿ√(∏xᵢ) | n = size(xs)
* @example
* ```ts
* xnumber.geometricMean(1, 2);
* // → 1.4142135623730951 (Math.sqrt(2))
*
* xnumber.geometricMean(1, 2, 3);
* // → 1.8171205928321397 (Math.cbrt(6))
*
* xnumber.geometricMean(1, 2, 3, 4);
* // → 2.2133638394006434 (Math.pow(24, 1/4))
* ```
*/ export function geometricMean(...xs) {
const n = xs.length;
return root(product(...xs), n);
}
/**
* Find the harmonic mean of numbers.
* @param xs a list of numbers
* @returns n/Σ(1/xᵢ) | n = size(xs)
* @example
* ```ts
* xnumber.harmonicMean(1, 2);
* // → 1.3333333333333333 (4/3)
*
* xnumber.harmonicMean(1, 2, 3);
* // → 1.6363636363636365 (18/11)
*
* xnumber.harmonicMean(1, 2, 3, 4);
* // → 1.92 (48/25)
* ```
*/ export function harmonicMean(...xs) {
const n = xs.length;
const p = product(...xs);
let q = 0;
for (const x of xs)q += p / x;
return n * p / q;
}
/**
* Find the quadriatic mean of numbers.
* @param xs a list of numbers
* @returns √(Σxᵢ²)/n | n = size(xs)
* @example
* ```ts
* xnumber.quadriaticMean(1, 2);
* // → 1.5811388300841898 (Math.sqrt(5/2))
*
* xnumber.quadriaticMean(1, 2, 3);
* // → 2.160246899469287 (Math.sqrt(14/3))
*
* xnumber.quadriaticMean(1, 2, 3, 4);
* // → 2.7386127875258306 (Math.sqrt(30/4))
* ```
*/ export function quadriaticMean(...xs) {
const n = xs.length;
let a = 0;
for (const x of xs)a += x * x;
return Math.sqrt(a / n);
}
export { quadriaticMean as rootMeanSquare };
/**
* Find the cubic mean of numbers.
* @param xs a list of numbers
* @returns ³√(Σxᵢ³)/n | n = size(xs)
* @example
* ```ts
* xnumber.cubicMean(1, 2);
* // → 1.6509636244473134 (Math.cbrt(9/2))
*
* xnumber.cubicMean(1, 2, 3);
* // → 2.2894284851066637 (Math.cbrt(36/3))
*
* xnumber.cubicMean(1, 2, 3, 4);
* // → 2.924017738212866 (Math.cbrt(100/4))
* ```
*/ export function cubicMean(...xs) {
const n = xs.length;
let a = 0;
for (const x of xs)a += x ** 3;
return Math.cbrt(a / n);
}
//#endregion
//#region ROMAN NUMERALS
/** List of symbols in the Roman numeral system. */ const ROMAN_SYMBOLS = [
'I',
'IV',
'V',
'IX',
'X',
'XL',
'L',
'XC',
'C',
'CD',
'D',
'CM',
'M'
];
/** Respective values in the Roman numeral system. */ const ROMAN_VALUES = [
1,
4,
5,
9,
10,
40,
50,
90,
100,
400,
500,
900,
1000
];
/** Maximum power-of-10 representable in Roman numerals. */ const ROMAN_MAX_POW10 = 3;
/**
* Convert roman numerals to number.
* @param txt roman numerals
* @returns eg. XCV -> 95, IV.V -> 4.5, IXe+II -> 900
* @example
* ```ts
* xnumber.fromRomanNumerals('XCV');
* // → 95
*
* xnumber.fromRomanNumerals('IV.V');
* // → 4.5
*
* xnumber.fromRomanNumerals('IXe+II');
* // → 900
* ```
*/ export function fromRomanNumerals(txt) {
const m = /^\s*(-?[\sIVXLCDM]+)\s*(?:\.([\sIVXLCDM]+))?(?:E[-+]([\sIVXLCDM]+))?\s*$/i.exec(txt) || [];
const aint = fromRomanInteger(m[1]);
const afrc = m[2] ? fromRomanInteger(m[2]) : 0;
const aexp = m[3] ? fromRomanInteger(m[3]) : 0;
return Math.sign(aint) * (Math.abs(aint) + fractionize(afrc)) * Math.pow(10, aexp);
}
export { fromRomanNumerals as fromRoman };
function fromRomanInteger(txt) {
let s = ROMAN_SYMBOLS.length - 1;
const n = txt.search(/^\s*-/) >= 0;
let a = 0;
txt = txt.replace(/\W/g, '').toUpperCase();
for(let i = 0, I = txt.length; i < I; i += ROMAN_SYMBOLS[s].length){
while(s >= 0 && txt.substring(i, i + ROMAN_SYMBOLS[s].length) !== ROMAN_SYMBOLS[s])--s;
if (s < 0) break;
a += ROMAN_VALUES[s];
}
return n ? -a : a;
}
/**
* Convert number to roman numerals.
* @param x a number
* @param exp use exponential/scientific notation [false]
* @returns eg. 95 -> XCV, 4.5 -> IV.V, 900 -> IXe+II (scientific = true)
* @example
* ```ts
* xnumber.toRomanNumerals(95);
* // → 'XCV'
*
* xnumber.toRomanNumerals(4.5);
* // → 'IV.V'
*
* xnumber.toRomanNumerals(900, true);
* // → 'IXe+II'
* ```
*/ export function toRomanNumerals(x, exp = false) {
const xp = Math.abs(x);
// Let's get the exponent and mantissa.
const xexp = Math.floor(Math.log10(xp));
const dexp = exp || xexp > ROMAN_MAX_POW10 ? xexp : 0;
const dman = xp * Math.pow(10, -dexp);
const dfrc = defractionize(dman % 1, ROMAN_MAX_POW10);
const dint = Math.sign(x) * Math.floor(dman);
// Now, convert to roman numerals.
const aman = toRomanInteger(dint) + (dfrc ? '.' + toRomanInteger(dfrc) : '');
const aexp = dexp !== 0 ? 'e' + (dexp > 0 ? '+' : '') + toRomanInteger(dexp) : '';
return aman + aexp;
}
export { toRomanNumerals as toRoman };
export function toRomanInteger(x) {
let a = x < 0 ? '-' : '';
x = Math.abs(x);
for(let s = ROMAN_SYMBOLS.length - 1; s >= 0; --s)while(x >= ROMAN_VALUES[s]){
x -= ROMAN_VALUES[s];
a += ROMAN_SYMBOLS[s];
}
return a;
} //#endregion
// export {default as fromScientific} from './_fromScientific';
// export {default as toScientific} from './_toScientific';
//# sourceMappingURL=data:application/json;base64,{"version":3,"sources":["file:///home/runner/work/extra-number/extra-number/index.ts"],"sourcesContent":["//#region TYPES\n/** Classification of floating point numbers. */\nexport enum FloatClass {\n  /** Number is zero. */\n  ZERO      = 0x00,\n  /** Number is subnormal (denormal). */\n  SUBNORMAL = 0x10,\n  /** Number is normal. */\n  NORMAL    = 0x20,\n  /** Number is infinite. */\n  INFINITE  = 0x30,\n  /** Number is NaN (not a number). */\n  NAN       = 0x40,\n}\n//#endregion\n\n\n\n\n//#region CONSTANTS\n/** Smallest normal (not subnormal) 64-bit floating point number. */\nexport const MIN_NORMAL = 2.2250738585072014e-308;\n/** Ratio of a circle's circumference to its diameter. */\nexport const TAU = 2 * Math.PI;\n//#endregion\n\n\n\n\n//#region ABOUT\n/**\n * Check if value is a number.\n * @param v a value\n * @returns is number?\n * @example\n * ```ts\n * xnumber.is(3.14);\n * // → true\n *\n * xnumber.is(NaN);\n * // → true (an unexpressable number)\n *\n * xnumber.is('3.14');\n * // → false\n *\n * xnumber.is({'value': 3.14});\n * // → false\n *\n * xnumber.is(null);\n * // → false\n * ```\n */\nexport function is(v: unknown): v is number {\n  return typeof v === \"number\";\n}\n\n\n/**\n * Check if number is normal (not zero, subnormal, infinite, or NaN).\n * @param x a number\n * @returns is normal number?\n * @example\n * ```ts\n * xnumber.isNormal(3.14);\n * // → true\n *\n * xnumber.isNormal(0);\n * // → false\n *\n * xnumber.isNormal(1e-320);\n * // → false\n *\n * xnumber.isNormal(Infinity);\n * // → false\n */\nexport function isNormal(x: number): boolean {\n  if (x === 0) return false;\n  if (!Number.isFinite(x)) return false;\n  if (Math.abs(x) < MIN_NORMAL) return false;\n  return true;\n}\n\n\n/**\n * Classify a number into its floating point category.\n * @param x a number\n * @returns floating point class of `x`\n * @example\n * ```ts\n * xnumber.classify(0);\n * // → FloatClass.ZERO\n *\n * xnumber.classify(1e-320);\n * // → FloatClass.SUBNORMAL\n *\n * xnumber.classify(3.14);\n * // → FloatClass.NORMAL\n *\n * xnumber.classify(Infinity);\n * // → FloatClass.INFINITE\n *\n * xnumber.classify(NaN);\n * // → FloatClass.NAN\n */\nexport function classify(x: number): FloatClass {\n  if (x === 0)                  return FloatClass.ZERO;\n  if (Number.isNaN(x))          return FloatClass.NAN;\n  if (!Number.isFinite(x))      return FloatClass.INFINITE;\n  if (Math.abs(x) < MIN_NORMAL) return FloatClass.SUBNORMAL;\n  return FloatClass.NORMAL;\n}\n\n\n/**\n * Examine if sum of divisors equals the number itself.\n * @param x a number\n * @returns sum of divisors(x) excluding x = x?\n * @example\n * ```ts\n * xnumber.isPerfect(6);\n * // → true (1+2+3=6)\n *\n * xnumber.isPerfect(28);\n * // → true (1+2+4+7+14=28)\n *\n * xnumber.isPerfect(12);\n * // → false (1+2+3+4+6=16)\n *\n * xnumber.isPerfect(1);\n * // → false (no proper divisors)\n * ```\n */\nexport function isPerfect(x: number): boolean {\n  return aliquotSum(x) === x;\n}\n\n\n// TODO: isAlmostPerfect()?\n// TODO: isSemiPerfect()?\n// TODO: isQuasiPerfect()?\n// TODO: isWeird()?\n\n\n/**\n * Examine if sum of divisors exceeds the number itself.\n * @param x a number\n * @returns sum of divisors(x) excluding x > x?\n * @example\n * ```ts\n * xnumber.isAbundant(8);\n * // → false (1+2+4=7 < 8)\n *\n * xnumber.isAbundant(15);\n * // → false (1+3+5=9 < 15)\n *\n * xnumber.isAbundant(12);\n * // → true (1+2+3+4+6=16 > 12)\n *\n * xnumber.isAbundant(6);\n * // → false (1+2+3=6 = 6)\n * ```\n */\nexport function isAbundant(x: number): boolean {\n  return aliquotSum(x) > x;\n}\nexport {isAbundant as isExcessive};\n\n\n// TODO: isPrimitiveAbundant()?\n// TODO: isSuperAbundant()?\n\n\n/**\n * Obtain the sum of divisors exceeding the number itself.\n * @param x a number\n * @returns sum of divisors(x) excluding x - x\n * @example\n * ```ts\n * xnumber.abundance(12);\n * // → 4 (1+2+3+4+6=16; 16-12=4)\n *\n * xnumber.abundance(15);\n * // → -6 (1+3+5=9; 9-15=-6)\n * ```\n */\nexport function abundance(x: number): number {\n  return aliquotSum(x) - x;\n}\n\n\n/**\n * Obtain the ratio of sum of divisors to the number itself.\n * @param x a number\n * @returns (sum of divisors(x)) / x\n * @example\n * ```ts\n * xnumber.abundancyIndex(12);\n * // → 2.3333... (1+2+3+4+6+12=28; 28/12=7/3)\n *\n * xnumber.abundancyIndex(15);\n * // → 1.6 (1+3+5+15=24; 24/15=8/5)\n * ```\n */\nexport function abundancyIndex(x: number): number {\n  return (aliquotSum(x) + x) / x;\n}\n\n\n// TODO: areAmicable()?\n// TODO: areFriendly()?\n\n\n/**\n * Count the number of significant digits in a number.\n * @param x a number\n * @returns |m - '.'| | x = m × 10ᵉ; m = mantissa, e = exponent\n * @example\n * ```ts\n * xnumber.significantDigits(0.0034);\n * // → 2\n *\n * xnumber.significantDigits(120.5e50);\n * // → 4\n *\n * xnumber.significantDigits(120.5e-50);\n * // → 4\n * ```\n */\nexport function significantDigits(x: number): number {\n  const a = x.toExponential();\n  return a.replace(/e[\\+\\-0-9]*$/, \"\").replace(/^0\\.?0*|\\./, \"\").length;\n}\n// - https://stackoverflow.com/questions/22884720/what-is-the-fastest-way-to-count-the-number-of-significant-digits-of-a-number\n//#endregion\n\n\n\n\n//#region COMPARE\n/**\n * Compare two numbers.\n * @param x a number\n * @param y another number\n * @returns x<y: -ve, x=y: 0, x>y: +ve\n * @example\n * ```ts\n * xnumber.compare(10, 12);\n * // → -2  (-ve)\n *\n * xnumber.compare(12, 12);\n * // → 0\n *\n * xnumber.compare(17, 12);\n * // → 5  (+ve)\n * ```\n */\nexport function compare(x: number, y: number): number {\n  return x - y;\n}\n\n\n/**\n * Check if two numbers are unordered (NaN involved).\n * @param x a number\n * @param y another number\n * @returns x=NaN or y=NaN?\n * @example\n * ```ts\n * xnumber.isUnordered(10, NaN);\n * // → true\n *\n * xnumber.isUnordered(NaN, 12);\n * // → true\n *\n * xnumber.isUnordered(10, 12);\n * // → false\n *\n * xnumber.isUnordered(NaN, NaN);\n * // → true\n * ```\n */\nexport function isUnordered(x: number, y: number): boolean {\n  return Number.isNaN(x) || Number.isNaN(y);\n}\n//#endregion\n\n\n\n\n//#region SIGN\n/**\n * Copy the sign of a number to another number.\n * @param mag number whose magnitude is to be copied\n * @param sgn sign of the number to copy\n * @returns `|mag|` with sign of `sgn`\n * @example\n * ```ts\n * xnumber.copySign(3.14, -1);\n * // → -3.14\n *\n * xnumber.copySign(-3.14, 1);\n * // → 3.14\n *\n * xnumber.copySign(-6.28, -1);\n * // → -6.28\n */\nexport function copySign(mag: number, sgn: number): number {\n  return Math.sign(sgn) === Math.sign(mag) ? mag : -mag;\n}\n//#endregion\n\n\n\n\n//#region ROUND\n/**\n * Round down a number to specific precision.\n * @param x a number\n * @param pre to precision [1]\n * @example\n * ```ts\n * xnumber.floor(9.161, 1);\n * // → 9\n *\n * xnumber.floor(9.161, 0.01);\n * // → 9.16\n *\n * xnumber.floor(9.1617, 0.05);\n * // → 9.15\n * ```\n */\nexport function floor(x: number, pre: number = 1): number {\n  return Math.floor(x/pre) * pre;\n}\n\n\n/**\n * Round up a number to specific precision.\n * @param x a number\n * @param pre to precision [1]\n * @example\n * ```ts\n * xnumber.ceil(9.121, 1);\n * // → 10\n *\n * xnumber.ceil(9.121, 0.01);\n * // → 9.13\n *\n * xnumber.ceil(9.1217, 0.05);\n * // → 9.15\n * ```\n */\nexport function ceil(x: number, pre: number = 1): number {\n  return Math.ceil(x/pre) * pre;\n}\n\n\n/**\n * Round a number to specific precision.\n * @param x a number\n * @param pre to precision [1]\n * @example\n * ```ts\n * xnumber.round(9.135, 1);\n * // → 9\n *\n * xnumber.round(9.135, 1e-2);\n * // → 9.14\n *\n * xnumber.round(9.1357, 0.05);\n * // → 9.15\n *\n * 0.1 + 0.2\n * // → 0.30000000000000004 (why?)\n *\n * xnumber.round(0.1 + 0.2, 1e-12);\n * // → 0.3 (nice!)\n * ```\n */\nexport function round(x: number, pre: number = 1): number {\n  return Math.round(x/pre) * pre;\n}\n\n\n/**\n * Perform floor-divison of two numbers.\n * @param x divisor\n * @param y dividend\n * @returns ⌊x/y⌋\n * @example\n * ```ts\n * xnumber.floorDiv(15, 4);\n * // → 3\n *\n * xnumber.floorDiv(2, 2);\n * // → 1\n *\n * xnumber.floorDiv(-15, 4);\n * // → -4\n * ```\n */\nexport function floorDiv(x: number, y: number): number {\n  return Math.floor(x/y);\n}\n\n\n/**\n * Perform ceiling-divison of two numbers.\n * @param x divisor\n * @param y dividend\n * @returns ⌈x/y⌉\n * @example\n * ```ts\n * xnumber.ceilDiv(15, 4);\n * // → 4\n *\n * xnumber.ceilDiv(2, 2);\n * // → 1\n *\n * xnumber.ceilDiv(-15, 4);\n * // → -3\n * ```\n */\nexport function ceilDiv(x: number, y: number): number {\n  return Math.ceil(x/y);\n}\n\n\n/**\n * Perform rounded-divison of two numbers.\n * @param x divisor\n * @param y dividend\n * @returns [x/y]\n * @example\n * ```ts\n * xnumber.roundDiv(15, 4);\n * // → 4\n *\n * xnumber.roundDiv(2, 2);\n * // → 1\n *\n * xnumber.roundDiv(-15, 4);\n * // → -4\n * ```\n */\nexport function roundDiv(x: number, y: number): number {\n  return Math.round(x/y);\n}\n//#endregion\n\n\n\n\n//#region MODULO\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 * @example\n * ```ts\n * xnumber.rem(1, 10);\n * // → 1\n *\n * xnumber.rem(-1, 10);\n * // → -1\n *\n * xnumber.rem(1, -10);\n * // → 1\n * ```\n */\nexport function rem(x: number, y: number): number {\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 * @example\n * ```ts\n * xnumber.mod(1, 10);\n * // → 1\n *\n * xnumber.mod(-1, 10);\n * // → 9\n *\n * xnumber.mod(1, -10);\n * // → -9\n * ```\n */\nexport function mod(x: number, y: number): number {\n  return x - y * Math.floor(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 n>0: floor(x % y), n<0: ceil(x % y)\n * @example\n * ```ts\n * xnumber.modp(1, 10);\n * // → 1\n *\n * xnumber.modp(-1, 10);\n * // → 9\n *\n * xnumber.modp(1, -10);\n * // → 1\n * ```\n */\nexport function modp(x: number, y: number): number {\n  return x - Math.abs(y) * Math.floor(x / Math.abs(y));\n}\n// - https://en.wikipedia.org/wiki/Modulo_operation\n//#endregion\n\n\n\n\n//#region RANGE CONTROL\n/**\n * Constrain a number within a minimum and a maximum value.\n * @param x a number\n * @param min minimum value\n * @param max maximum value\n * @returns x<min: min, x>max: max, x\n * @example\n * ```ts\n * xnumber.constrain(20, 0, 50);\n * // → 20\n *\n * xnumber.constrain(-10, 0, 100);\n * // → 0\n *\n * xnumber.constrain(120, 0, 100);\n * // → 100\n * ```\n */\nexport function constrain(x: number, min: number, max: number): number {\n  return Math.min(Math.max(x, min), max);\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 * Normalize a number from its current range into a value between 0 and 1.\n * @param x a number\n * @param r lower bound of current range\n * @param R upper bound of current range\n * @returns ∈ [0, 1]\n * @example\n * ```ts\n * xnumber.normalize(20, 0, 50);\n * // → 0.4\n *\n * xnumber.normalize(-10, 0, 100);\n * // → -0.1\n * ```\n */\nexport function normalize(x: number, r: number, R: number): number {\n  return (x - r) / (R - r);\n}\nexport {normalize as norm};\n// - https://processing.org/reference/norm_.html\n\n\n/**\n * Re-map a number from one range to another.\n * @param x a number\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 * @example\n * ```ts\n * xnumber.remap(25, 0, 100, 0, 1366);\n * // → 341.5\n *\n * xnumber.remap(110, 0, 100, -20, -10);\n * // → -9\n * ```\n */\nexport function remap(x: number, r: number, R: number, t: number, T: number): number {\n  return t + ((x - r) / (R - r)) * (T - t);\n}\nexport {remap as map};\n// - https://processing.org/reference/map_.html\n\n\n/**\n * Linearly interpolate a number between two numbers.\n * @param x start number\n * @param y stop number\n * @param t interpolant ∈ [0, 1]\n * @returns ∈ [x, y]\n * @example\n * ```ts\n * xnumber.lerp(80, 320, 0.8);\n * // → 272\n *\n * xnumber.lerp(80, 320, 0.20);\n * // → 128\n *\n * xnumber.lerp(80, 320, 0.32);\n * // → 156.8\n * ```\n */\nexport function lerp(x: number, y: number, t: number): number {\n  return x + t * (y - x);\n}\n// - https://processing.org/reference/lerp_.html\n// - https://docs.unity3d.com/ScriptReference/Vector3.Lerp.html\n//#endregion\n\n\n\n\n//#region FRACTIONAL\n/**\n * Make a number purely fractional, by shifting the decimal point.\n * @param x a number\n * @param n maximum number of digits to shift (-1: all) [20]\n * @returns fractional number\n * @example\n * ```ts\n * xnumber.fractionize(123);\n * // → 0.123\n *\n * xnumber.fractionize(-9876, 3);\n * // → -0.988\n *\n * xnumber.fractionize(0.001234, 3);\n * // → 0.001\n * ```\n */\nexport function fractionize(x: number, n: number = 20): number {\n  const xp   = Math.abs(x);\n  const xdig = xp < 1? 0 : 1 + Math.floor(Math.log10(xp));\n  if (n < 0) return x * Math.pow(10, -xdig);\n  return Math.round(x * Math.pow(10, n - xdig)) * Math.pow(10, -n);\n}\n\n\n/**\n * Make a number purely integer, by shifting the decimal point.\n * @param x a number\n * @param n maximum number of digits to shift (-1: all) [20]\n * @returns integer number\n * @example\n * ```ts\n * xnumber.defractionize(0.123);\n * // → 123\n *\n * xnumber.defractionize(-0.988, 2);\n * // → -99\n *\n * xnumber.defractionize(12.345, 2);\n * // → 1235\n * ```\n */\nexport function defractionize(x: number, n: number = 20): number {\n  let xp = Math.abs(x), i = 0;\n  for (; n<0 || i<n;) {\n    xp *= 10; ++i;\n    if (xp % 1 === 0) break;\n  }\n  return Math.round(x * Math.pow(10, i));\n}\n//#endregion\n\n\n\n\n//#region ARITHMETIC\n/**\n * Check if a number is a power-of-n.\n * @param x a number\n * @param n base\n * @returns nⁱ = x? | i = +ve integer\n * @example\n * ```ts\n * xnumber.isPow(32, 2);\n * // → true\n *\n * xnumber.isPow(32, 4);\n * // → false\n *\n * xnumber.isPow(64, 4);\n * // → true\n * ```\n */\nexport function isPow(x: number, n: number): boolean {\n  if (n === 0) return x === 0;\n  const p = log(Math.abs(x), Math.abs(n));\n  if (p !== Math.floor(p)) return false;\n  return x < 0? (n < 0 && (p & 1) === 1) : (n > 0 || (p & 1) === 0);\n}\n\n\n/**\n * Find largest power-of-n less than or equal to given number.\n * @param x a number\n * @param n base\n * @returns nⁱ | nⁱ ≤ x and nⁱ > x/n\n * @example\n * ```ts\n * xnumber.prevPow(32, 2);\n * // → 32\n *\n * xnumber.prevPow(35, 2);\n * // → 32\n *\n * xnumber.prevPow(35, 3);\n * // → 27\n * ```\n */\nexport function prevPow(x: number, n: number): number {\n  if (x <= 1) return 0;\n  const p = Math.floor(Math.log(x) / Math.log(n));\n  return Math.pow(n, p);\n}\n\n\n/**\n * Find smallest power-of-n greater than or equal to given number.\n * @param x a number\n * @param n base\n * @returns nⁱ | nⁱ ≥ x and nⁱ < n*x\n * @example\n * ```ts\n * xnumber.nextPow(32, 2);\n * // → 32\n *\n * xnumber.nextPow(29, 2);\n * // → 32\n *\n * xnumber.nextPow(29, 3);\n * // → 81\n * ```\n */\nexport function nextPow(x: number, n: number): number {\n  if (x <= 0) return 1;\n  const p = Math.ceil(Math.log(x) / Math.log(n));\n  return Math.pow(n, p);\n}\n\n\n/**\n * Find the nth root of a number (ⁿ√).\n * @param x a number\n * @param n root\n * @returns ⁿ√x\n * @example\n * ```ts\n * xnumber.root(25, 2);\n * // → 5\n *\n * xnumber.root(-8, 3);\n * // → -2\n * ```\n */\nexport function root(x: number, n: number): number {\n  if ((n & 1) === 0) return Math.pow(x, 1/n);\n  return Math.sign(x) * Math.pow(Math.abs(x), 1/n);\n}\n// - https://github.com/alawatthe/MathLib/blob/master/src/Functn/functions/root.ts\n\n\n/**\n * Find the logarithm of a number with a given base.\n * @param x a number\n * @param b logarithm base [e]\n * @returns log_b (x)\n * @example\n * ```ts\n * xnumber.log(Math.E);\n * // → 1\n *\n * xnumber.log(10);\n * // → 2.302585092994046\n *\n * xnumber.log(243, 3);\n * // → 5\n *\n * xnumber.log(64, 2);\n * // → 6\n * ```\n */\nexport function log(x: number, b: number=Math.E): number {\n  return Math.log(x) / Math.log(b);\n}\n// - https://en.wikipedia.org/wiki/Logarithm\n//#endregion\n\n\n\n\n//#region DIVISORS\n/**\n * List all divisors of a number, except itself.\n * @param x a number\n * @returns proper divisors (factors)\n * @example\n * ```ts\n * xnumber.properDivisors(6);\n * // → [1, 2, 3]\n *\n * xnumber.properDivisors(1);\n * // → []\n *\n * xnumber.properDivisors(0);\n * // → []\n *\n * xnumber.properDivisors(-24);\n * // → [1, 2, 3, 4, 6, 8, 12]\n * ```\n */\nexport function properDivisors(x: number): number[] {\n  const a = []; x = Math.abs(x);\n  for (let i=1; i<x; i++)\n    if (x % i === 0) 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 number.\n * @param x a number\n * @returns Σdᵢ | dᵢ is a divisor of x and ≠x\n * @example\n * ```ts\n * xnumber.aliquotSum(6);\n * // → 6 (1+2+3)\n *\n * xnumber.aliquotSum(1);\n * // → 0\n *\n * xnumber.aliquotSum(0);\n * // → 0\n *\n * xnumber.aliquotSum(-24);\n * // → 36 (1+2+3+4+6+8+12)\n * ```\n */\nexport function aliquotSum(x: number): number {\n  let a = 0; x = Math.abs(x);\n  for (let i=0; i<x; i++)\n    if (x % i === 0) a += i;\n  return a;\n}\n\n\n/**\n * Find the least prime number which divides a number.\n * @param x a number\n * @returns least prime factor\n * @example\n * ```ts\n * xnumber.minPrimeFactor(1);\n * // → 0\n *\n * xnumber.minPrimeFactor(3);\n * // → 3\n *\n * xnumber.minPrimeFactor(21);\n * // → 3\n *\n * xnumber.minPrimeFactor(55);\n * // → 5\n *\n * xnumber.minPrimeFactor(53);\n * // → 53\n * ```\n */\nexport function minPrimeFactor(x: number): number {\n  x = Math.abs(x);\n  // 1. LPF of 2, 3 is the number itself.\n  if (x <= 1) return 0;\n  if (x <= 3) return x;\n  // 2. LPF for multiples of 2, 3.\n  if (x % 2 === 0) return 2;\n  if (x % 3 === 0) return 3;\n  // 3. LPF can be 6k-1 or 6k+1.\n  for (let i=6, I=Math.sqrt(x)+1; i<=I; i+=6) {\n    if (x % (i-1) === 0) return i-1;\n    if (x % (i+1) === 0) return i+1;\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 number.\n * @param x a number\n * @returns greatest prime factor\n * @example\n * ```ts\n * xnumber.maxPrimeFactor(1);\n * // → 0\n *\n * xnumber.maxPrimeFactor(3);\n * // → 3\n *\n * xnumber.maxPrimeFactor(21);\n * // → 7\n *\n * xnumber.maxPrimeFactor(55);\n * // → 11\n *\n * xnumber.maxPrimeFactor(53);\n * // → 53\n * ```\n */\nexport function maxPrimeFactor(x: number): number {\n  let a = 0; x = Math.abs(x);\n  // 1. GPF of 2, 3 is the number itself.\n  if (x <= 1) return 0;\n  if (x <= 3) return x;\n  // 2. Remove factors 2, 3.\n  for (; x % 2 === 0; a=2)\n    x /= 2;\n  for (; x % 3 === 0; a=3)\n    x /= 3;\n  // 3. Remove factors 6k-1, 6k+1.\n  for (let i=6, I=Math.sqrt(x)+1; x>1 && i<=I; i+=6) {\n    for (; x % (i-1) === 0; a=i-1)\n      x /= i-1;\n    for (; x % (i+1) === 0; a=i+1)\n      x /= i+1;\n  }\n  if (x <= 1) 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 number.\n * @param x a number\n * @returns [f₀, f₁, ...] | fᵢ divides x and is prime\n * @example\n * ```ts\n * xnumber.primeFactors(1);\n * // → []\n *\n * xnumber.primeFactors(3);\n * // → [3]\n *\n * xnumber.primeFactors(21);\n * // → [3, 7]\n *\n * xnumber.primeFactors(55);\n * // → [5, 11]\n *\n * xnumber.primeFactors(53);\n * // → [53]\n * ```\n */\nexport function primeFactors(x: number): number[] {\n  const a: number[] = []; x = Math.abs(x);\n  if (x <= 1) return [];\n  if (x <= 3) return [x];\n  // 2. Try factors 2, 3.\n  x = pushPrimeFactorTo$(a, x, 2);\n  x = pushPrimeFactorTo$(a, x, 3);\n  // 3. Try factors 6k-1, 6k+1.\n  for (let i=6, I=Math.sqrt(x)+1; x>1 && i<=I; i+=6) {\n    x = pushPrimeFactorTo$(a, x, i-1);\n    x = pushPrimeFactorTo$(a, x, i+1);\n  }\n  if (x > 1) a.push(x);\n  return a;\n}\n\nfunction pushPrimeFactorTo$(a: number[], x: number, f: number): number {\n  if (x % f !== 0) return x;\n  do {\n    x /= f;\n  } while (x % f === 0);\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 number.\n * @param x a number\n * @returns [[f₀, e₀], [f₁, e₁], ...] | fᵢ is a prime factor of x and eᵢ is its exponent\n * @example\n * ```ts\n * xnumber.primeExponentials(1);\n * // → []\n *\n * xnumber.primeExponentials(9);\n * // → [[3, 2]]\n *\n * xnumber.primeExponentials(63);\n * // → [[3, 2], [7, 1]]\n *\n * xnumber.primeExponentials(605);\n * // → [[5, 1], [11, 2]]\n *\n * xnumber.primeExponentials(53);\n * // → [[53, 1]]\n * ```\n */\nexport function primeExponentials(x: number): [number, number][] {\n  const a: [number, number][] = []; x = Math.abs(x);\n  if (x <= 1) return [];\n  if (x <= 3) return [[x, 1]];\n  // 2. Try factors 2, 3.\n  x = pushPrimeExponentialTo$(a, x, 2);\n  x = pushPrimeExponentialTo$(a, x, 3);\n  // 3. Try factors 6k-1, 6k+1.\n  for (let i=6, I=Math.sqrt(x)+1; x>1 && i<=I; i+=6) {\n    x = pushPrimeExponentialTo$(a, x, i-1);\n    x = pushPrimeExponentialTo$(a, x, i+1);\n  }\n  if (x > 1) a.push([x, 1]);\n  return a;\n}\n\nfunction pushPrimeExponentialTo$(a: [number, number][], x: number, f: number): number {\n  if (x % f !== 0) return x;\n  let e = 0;\n  do {\n    x /= f; ++e;\n  } while (x % f === 0);\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 * Examine if number is prime.\n * @param x a number\n * @returns is divisible by 1 and itself only?\n * @example\n * ```ts\n * xnumber.isPrime(7);\n * // → true\n *\n * xnumber.isPrime(53);\n * // → true\n *\n * xnumber.isPrime(4);\n * // → false\n *\n * xnumber.isPrime(1);\n * // → false\n *\n * xnumber.isPrime(0);\n * // → false\n * ```\n */\nexport function isPrime(x: number): boolean {\n  return x !== 0 && minPrimeFactor(x) === Math.abs(x);\n}\n\n\n/**\n * Find the greatest common divisor of numbers.\n * @param xs a list of numbers\n * @returns gcd(x₁, x₂, ...)\n * @example\n * ```ts\n * xnumber.gcd(6, 15);\n * // → 3\n *\n * xnumber.gcd(6, 15, 21);\n * // → 3\n *\n * xnumber.gcd(6, 15, 20);\n * // → 1\n * ```\n */\nexport function gcd(...xs: number[]): number {\n  let a = xs[0] || 1;\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 numbers.\nfunction gcdPair(x: number, y: number): number {\n  while (y !== 0) {\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 numbers.\n * @param xs a list of numbers\n * @returns lcm(x₁, x₂, ...)\n * @example\n * ```ts\n * xnumber.lcm(2, 3);\n * // → 6\n *\n * xnumber.lcm(2, 3, 4);\n * // → 12\n *\n * xnumber.lcm(2, 3, 4, 5);\n * // → 60\n * ```\n */\nexport function lcm(...xs: number[]): number {\n  let a = xs[0] || 1;\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//#endregion\n\n\n\n\n//#region ARRANGEMENTS\n/**\n * Find the factorial of a number.\n * @param n a number\n * @param k denominator factorial [0]\n * @returns P(n, k); k=0: n!, k>0: n!/k!\n * @example\n * ```ts\n * xnumber.factorial(5);\n * // → 120\n *\n * xnumber.factorial(6);\n * // → 720\n *\n * xnumber.factorial(7);\n * // → 5040\n * ```\n */\nexport function factorial(n: number, k: number = 0): number {\n  if (n < 0) return 0;\n  let a = 1;\n  for (let i=k+1; 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 * @example\n * ```ts\n * xnumber.binomial(4, 1);\n * // → 4\n *\n * xnumber.binomial(4, 2);\n * // → 6\n *\n * xnumber.binomial(4, 3);\n * // → 4\n * ```\n */\nexport function binomial(n: number, k: number): number {\n  // 1. Generalization to negative integers\n  if (k < 0 || k > Math.abs(n)) return 0;\n  if (n < 0) return Math.pow(-1, k) * binomial(-n, k);\n  // 2. Take advantage of symmetry\n  k = k > n - k? n - k : k;\n  let a = 1;\n  for (let i=1; i<=k; i++, n--)\n    a *= n/i;\n  return a;\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 * @example\n * ```ts\n * xnumber.multinomial(1, 2);\n * // → 3\n *\n * xnumber.multinomial(1, 2, 3);\n * // → 60\n *\n * xnumber.multinomial(1, 2, 3, 4);\n * // → 12600\n * ```\n */\nexport function multinomial(...ks: number[]): number {\n  let n = sum(...ks), a = 1;\n  for (let i=0, j=0, I=ks.length; i<I;) {\n    if (j <= 0) j = ks[i++];\n    else a *= n-- / j--;\n  }\n  return a;\n}\n// - https://en.wikipedia.org/wiki/Multinomial_distribution\n//#endregion\n\n\n\n\n//#region GEOMETRY\n/**\n * Convert radians to degrees.\n * @param x radians\n * @returns 2π → 360\n * @example\n * ```ts\n * xnumber.degrees(3);\n * // → 171.88733853924697\n *\n * xnumber.degrees(1);\n * // → 57.29577951308232\n * ```\n */\nexport function degrees(x: number): number {\n  return x * (180 / Math.PI);\n}\n// - https://processing.org/reference/degrees_.html\n\n\n/**\n * Convert degrees to radians.\n * @param x degrees\n * @returns 360 → 2π\n * @example\n * ```ts\n * xnumber.radians(180);\n * // → 3.141592653589793\n *\n * xnumber.radians(60);\n * // → 1.0471975511965976\n * ```\n */\nexport function radians(x: number): number {\n  return x * (Math.PI / 180);\n}\n// - https://processing.org/reference/radians_.html\n//#endregion\n\n\n\n\n//#region STATISTICS\n/**\n * Find the sum of numbers (Σ).\n * @param xs a list of numbers\n * @returns Σxᵢ\n * @example\n * ```ts\n * xnumber.sum(1, 2);\n * // → 3\n *\n * xnumber.sum(1, 2, 3);\n * // → 6\n *\n * xnumber.sum(1, 2, 3, 4);\n * // → 10\n * ```\n */\nexport function sum(...xs: number[]): number {\n  let a = 0;\n  for (const x of xs)\n    a += x;\n  return a;\n}\n\n\n/**\n * Find the product of numbers (∏).\n * @param xs a list of numbers\n * @returns ∏xᵢ\n * @example\n * ```ts\n * xnumber.product(1, 2);\n * // → 2\n *\n * xnumber.product(1, 2, 3);\n * // → 6\n *\n * xnumber.product(1, 2, 3, 4);\n * // → 24\n * ```\n */\nexport function product(...xs: number[]): number {\n  let a = 1;\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 numbers.\n * @param xs a list of numbers\n * @returns xₘ | sort(xs) = [..., xₘ, ...]\n * @example\n * ```ts\n * xnumber.median(1, 7);\n * // → 4\n *\n * xnumber.median(1, 7, 8);\n * // → 7\n *\n * xnumber.median(1, 7, 8, 10);\n * // → 7.5\n * ```\n */\nexport function median(...xs: number[]): number {\n  if (xs.length === 0) return 0;\n  xs.sort((a, b) => a - b);\n  const i = xs.length >> 1;\n  if ((xs.length & 1) === 1) return xs[i];\n  return (xs[i - 1] + xs[i]) / 2;\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 numbers\n * @returns [xₘ₁, xₘ₂, ...] | count(xₘᵢ) ≥ count(xᵢ) ∀ xᵢ ∈ xs\n * @example\n * ```ts\n * xnumber.modes(1, 2);\n * // → [1, 2]\n *\n * xnumber.modes(1, 2, 2);\n * // → [2]\n *\n * xnumber.modes(1, 2, 2, 3, 3);\n * // → [2, 3]\n * ```\n */\nexport function modes(...xs: number[]): number[] {\n  xs.sort((a, b) => a - b);\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 number has repeated in a sorted array.\nfunction maxRepeat(xs: number[]): 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: number[], r: number): number[] {\n  const a: number[] = []; 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 and largest values.\n * @param xs a list of numbers\n * @returns [min(xs), max(xs)]\n * @example\n * ```ts\n * xnumber.range(1, 7);\n * // → 6\n *\n * xnumber.range(1, 7, 6);\n * // → 6\n *\n * xnumber.range(1, 7, 8, 6);\n * // → 7\n * ```\n */\nexport function range(...xs: number[]): [number, number] {\n  return [Math.min(...xs), Math.max(...xs)];\n}\n// - https://en.wikipedia.org/wiki/Range_(statistics)\n\n\n/**\n * Find the mean of squared deviation of numbers from its mean.\n * @param xs a list of numbers\n * @returns σ² = E[(xs - µ)²] | µ = mean(xs)\n * @example\n * ```ts\n * xnumber.variance(1, 2);\n * // → 0.25\n *\n * xnumber.variance(1, 2, 3);\n * // → 0.6666666666666666\n *\n * xnumber.variance(1, 2, 3, 4);\n * // → 1.25\n * ```\n */\nexport function variance(...xs: number[]): number {\n  if (xs.length === 0) return 0;\n  const m = arithmeticMean(...xs); let a = 0;\n  for (const x of xs)\n    a += (x - m) ** 2;\n  return a / xs.length;\n}\n// - https://en.wikipedia.org/wiki/Variance\n//#endregion\n\n\n\n\n\n//#region MEAN (STATISTICS)\n/**\n * Find the average of numbers.\n * @param xs a list of numbers\n * @returns Σxᵢ/n | n = size(xs)\n * @example\n * ```ts\n * xnumber.arithmethicMean(1, 2);\n * // → 1.5\n *\n * xnumber.arithmethicMean(1, 2, 3);\n * // → 2\n *\n * xnumber.arithmethicMean(1, 2, 3, 4);\n * // → 2.5\n * ```\n */\nexport function arithmeticMean(...xs: number[]): number {\n  if (xs.length === 0) return 0;\n  return sum(...xs) / xs.length;\n}\nexport {arithmeticMean as mean};\n\n\n/**\n * Find the geometric mean of numbers.\n * @param xs a list of numbers\n * @returns ⁿ√(∏xᵢ) | n = size(xs)\n * @example\n * ```ts\n * xnumber.geometricMean(1, 2);\n * // → 1.4142135623730951  (Math.sqrt(2))\n *\n * xnumber.geometricMean(1, 2, 3);\n * // → 1.8171205928321397  (Math.cbrt(6))\n *\n * xnumber.geometricMean(1, 2, 3, 4);\n * // → 2.2133638394006434  (Math.pow(24, 1/4))\n * ```\n */\nexport function geometricMean(...xs: number[]): number {\n  const n = xs.length;\n  return root(product(...xs), n);\n}\n\n\n/**\n * Find the harmonic mean of numbers.\n * @param xs a list of numbers\n * @returns n/Σ(1/xᵢ) | n = size(xs)\n * @example\n * ```ts\n * xnumber.harmonicMean(1, 2);\n * // → 1.3333333333333333  (4/3)\n *\n * xnumber.harmonicMean(1, 2, 3);\n * // → 1.6363636363636365  (18/11)\n *\n * xnumber.harmonicMean(1, 2, 3, 4);\n * // → 1.92  (48/25)\n * ```\n */\nexport function harmonicMean(...xs: number[]): number {\n  const n = xs.length;\n  const p = product(...xs); let q = 0;\n  for (const x of xs)\n    q += p/x;\n  return n * p/q;\n}\n\n\n/**\n * Find the quadriatic mean of numbers.\n * @param xs a list of numbers\n * @returns √(Σxᵢ²)/n | n = size(xs)\n * @example\n * ```ts\n * xnumber.quadriaticMean(1, 2);\n * // → 1.5811388300841898  (Math.sqrt(5/2))\n *\n * xnumber.quadriaticMean(1, 2, 3);\n * // → 2.160246899469287  (Math.sqrt(14/3))\n *\n * xnumber.quadriaticMean(1, 2, 3, 4);\n * // → 2.7386127875258306  (Math.sqrt(30/4))\n * ```\n */\nexport function quadriaticMean(...xs: number[]): number {\n  const n = xs.length; let a = 0;\n  for (const x of xs)\n    a += x*x;\n  return Math.sqrt(a/n);\n}\nexport {quadriaticMean as rootMeanSquare};\n\n\n/**\n * Find the cubic mean of numbers.\n * @param xs a list of numbers\n * @returns ³√(Σxᵢ³)/n | n = size(xs)\n * @example\n * ```ts\n * xnumber.cubicMean(1, 2);\n * // → 1.6509636244473134  (Math.cbrt(9/2))\n *\n * xnumber.cubicMean(1, 2, 3);\n * // → 2.2894284851066637  (Math.cbrt(36/3))\n *\n * xnumber.cubicMean(1, 2, 3, 4);\n * // → 2.924017738212866  (Math.cbrt(100/4))\n * ```\n */\nexport function cubicMean(...xs: number[]): number {\n  const n = xs.length; let a = 0;\n  for (const x of xs)\n    a += x**3;\n  return Math.cbrt(a/n);\n}\n//#endregion\n\n\n\n\n//#region ROMAN NUMERALS\n/** List of symbols in the Roman numeral system. */\nconst ROMAN_SYMBOLS = ['I', 'IV', 'V', 'IX', 'X', 'XL', 'L', 'XC', 'C', 'CD', 'D', 'CM', 'M'];\n/** Respective values in the Roman numeral system. */\nconst ROMAN_VALUES  = [1, 4, 5, 9, 10, 40, 50, 90, 100, 400, 500, 900, 1000];\n/** Maximum power-of-10 representable in Roman numerals. */\nconst ROMAN_MAX_POW10 = 3;\n\n\n/**\n * Convert roman numerals to number.\n * @param txt roman numerals\n * @returns eg. XCV -> 95, IV.V -> 4.5, IXe+II -> 900\n * @example\n * ```ts\n * xnumber.fromRomanNumerals('XCV');\n * // → 95\n *\n * xnumber.fromRomanNumerals('IV.V');\n * // → 4.5\n *\n * xnumber.fromRomanNumerals('IXe+II');\n * // → 900\n * ```\n */\nexport function fromRomanNumerals(txt: string): number {\n  const m  = /^\\s*(-?[\\sIVXLCDM]+)\\s*(?:\\.([\\sIVXLCDM]+))?(?:E[-+]([\\sIVXLCDM]+))?\\s*$/i.exec(txt) || [];\n  const aint = fromRomanInteger(m[1]);\n  const afrc = m[2]? fromRomanInteger(m[2]) : 0;\n  const aexp = m[3]? fromRomanInteger(m[3]) : 0;\n  return Math.sign(aint) * (Math.abs(aint) + fractionize(afrc)) * Math.pow(10, aexp);\n}\nexport {fromRomanNumerals as fromRoman};\n\n\nfunction fromRomanInteger(txt: string): number {\n  let   s = ROMAN_SYMBOLS.length - 1;\n  const n = txt.search(/^\\s*-/) >= 0; let a = 0;\n  txt = txt.replace(/\\W/g, '').toUpperCase();\n  for (let i=0, I=txt.length; i<I;  i += ROMAN_SYMBOLS[s].length) {\n    while (s >= 0 && txt.substring(i, i +  ROMAN_SYMBOLS[s].length) !== ROMAN_SYMBOLS[s]) --s;\n    if (s < 0) break;\n    a += ROMAN_VALUES[s];\n  }\n  return n? -a : a;\n}\n\n\n/**\n * Convert number to roman numerals.\n * @param x a number\n * @param exp use exponential/scientific notation [false]\n * @returns eg. 95 -> XCV, 4.5 -> IV.V, 900 -> IXe+II (scientific = true)\n * @example\n * ```ts\n * xnumber.toRomanNumerals(95);\n * // → 'XCV'\n *\n * xnumber.toRomanNumerals(4.5);\n * // → 'IV.V'\n *\n * xnumber.toRomanNumerals(900, true);\n * // → 'IXe+II'\n * ```\n */\nexport function toRomanNumerals(x: number, exp: boolean = false): string {\n  const xp = Math.abs(x);\n  // Let's get the exponent and mantissa.\n  const xexp = Math.floor(Math.log10(xp));\n  const dexp = exp || (xexp > ROMAN_MAX_POW10)? xexp : 0;\n  const dman = xp * Math.pow(10, -dexp);\n  const dfrc = defractionize(dman % 1, ROMAN_MAX_POW10);\n  const dint = Math.sign(x) * Math.floor(dman);\n  // Now, convert to roman numerals.\n  const aman = toRomanInteger(dint) + (dfrc? '.' + toRomanInteger(dfrc) : '');\n  const aexp = dexp !== 0? 'e' + (dexp > 0? '+' : '') + toRomanInteger(dexp) : '';\n  return aman + aexp;\n}\nexport {toRomanNumerals as toRoman};\n\n\nexport function toRomanInteger(x: number): string {\n  let a = x < 0? '-' : '';\n  x = Math.abs(x);\n  for (let s=ROMAN_SYMBOLS.length-1; s>=0; --s)\n    while (x >= ROMAN_VALUES[s]) {\n      x -= ROMAN_VALUES[s];\n      a += ROMAN_SYMBOLS[s];\n    }\n  return a;\n}\n//#endregion\n\n\n// export {default as fromScientific} from './_fromScientific';\n// export {default as toScientific} from 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