/* Module "math.wren" */
/* Math supplements the Num class with various other operations on numbers. */
class Math {
// Constants.
static e { 2.71828182845904523536 } // base of natural logarithms
static phi { 1.6180339887498948482 } // golden ratio
static ln2 { 0.69314718055994530942 } // natural logarithm of 2
static ln10 { 2.30258509299404568402 } // natural logarithm of 10
// Log function.
static log10(x) { x.log/ln10 } // Base 10 logarithm
// Hyperbolic trig functions.
static sinh(x) { (x.exp - (-x).exp)/2 } // sine
static cosh(x) { (x.exp + (-x).exp)/2 } // cosine
static tanh(x) { sinh(x)/cosh(x) } // tangent
// Inverse hyperbolic trig functions.
static asinh(x) { (x + (x*x + 1).sqrt).log } // sine
static acosh(x) { (x + (x*x - 1).sqrt).log } // cosine
static atanh(x) { ((1+x)/(1-x)).log/2 } // tangent
// Angle conversions.
static radians(d) { d * Num.pi / 180}
static degrees(r) { r * 180 / Num.pi }
// Returns the square root of 'x' squared + 'y' squared.
static hypot(x, y) { (x*x + y*y).sqrt }
// Returns the integer and fractional parts of 'x'. Both values have the same sign as 'x'.
static modf(x) { [x.truncate, x.fraction] }
// Returns the IEEE 754 floating-point remainder of 'x/y'.
static rem(x, y) {
if (x.isNan || y.isNan || x.isInfinity || y == 0) return nan
if (!x.isInfinity && y.isInfinity) return x
var nf = modf(x/y)
if (nf[1] != 0.5) {
return x - (x/y).round * y
} else {
var n = nf[0]
if (n%2 == 1) n = (n > 0) ? n + 1 : n - 1
return x - n * y
}
}
// Round away from zero.
static roundUp(x) { (x >= 0) ? x.ceil : x.floor }
// Round to 'p' decimal places, maximum 14.
// Mode parameter specifies the rounding mode:
// < 0 towards zero, == 0 nearest, > 0 away from zero.
static toPlaces(x, p, mode) {
if (p < 0) p = 0
if (p > 14) p = 14
var pw = 10.pow(p)
var nf = modf(x)
x = nf[1] * pw
x = (mode < 0) ? x.truncate : (mode == 0) ? x.round : roundUp(x)
return nf[0] + x/pw
}
// Convenience version of above method which uses 0 for the 'mode' parameter.
static toPlaces(x, p) { toPlaces(x, p, 0) }
// Gamma function using Lanczos approximation.
static gamma(x) {
var p = [
0.99999999999980993,
676.5203681218851,
-1259.1392167224028,
771.32342877765313,
-176.61502916214059,
12.507343278686905,
-0.13857109526572012,
9.9843695780195716e-6,
1.5056327351493116e-7
]
var t = x + 6.5
var sum = p[0]
for (i in 0..7) sum = sum + p[i+1]/(x + i)
return 2.sqrt * Num.pi.sqrt * t.pow(x-0.5) * (-t).exp * sum
}
// Static alternatives to instance methods in Num class.
// Clearer when both arguments are complex expressions.
static min(x, y) { (x < y) ? x : y }
static max(x, y) { (x > y) ? x : y }
static atan(x, y) { x.atan(y) }
}
/* Int contains various routines which are only applicable to integers. */
class Int {
// Truncated integer division (consistent with % operator).
static quo(x, y) { (x/y).truncate }
// Floored integer division (consistent with 'mod' method below).
static div(x, y) { (x/y).floor }
// Floored integer division modulus.
static mod(x, y) { ((x % y) + y) % y }
// Returns whether or not 'n' is a perfect square.
static isSquare(n) {
var s = n.sqrt.floor
return s * s == n
}
// Returns whether or not 'n' is a perfect cube.
static isCube(n) {
var c = n.cbrt.truncate
return c * c * c == n
}
// Returns the greatest common divisor of 'x' and 'y'.
static gcd(x, y) {
while (y != 0) {
var t = y
y = x % y
x = t
}
return x.abs
}
// Returns the least common multiple of 'x' and 'y'.
static lcm(x, y) {
if (x == 0 && y == 0) return 0
return (x*y).abs / gcd(x, y)
}
// Returns the greatest common divisor of a list of integers 'a'.
static gcd(a) {
if (!(a is List) || a.count < 2) {
Fiber.abort("Argument must be a list of at least two integers.")
}
var g = gcd(a[0], a[1])
if (a.count == 2) return g
return gcd(a[2..-1] + [g])
}
// Returns the least common multiple of a list of integers 'a'.
static lcm(a) {
if (!(a is List) || a.count < 2) {
Fiber.abort("Argument must be a list of at least two integers.")
}
var l = lcm(a[0], a[1])
if (a.count == 2) return l
return lcm(a[2..-1] + [l])
}
// Returns the remainder when 'b' raised to the power 'e' is divided by 'm'.
static modPow(b, e, m) {
if (m == 1) return 0
var r = 1
b = b % m
while (e > 0) {
if (e%2 == 1) r = (r*b) % m
e = e >> 1
b = (b*b) % m
}
return r
}
// Returns the factorial of 'n'.
static factorial(n) {
if (!(n is Num && n >= 0 && n < 19)) {
Fiber.abort("Argument must be a non-negative integer < 19.")
}
if (n < 2) return 1
var fact = 1
for (i in 2..n) fact = fact * i
return fact
}
// Returns the multinomial coefficient of n over a list f where sum(f) == n.
static multinomial(n, f) {
if (!(n is Num && n >= 0 && n < 19)) {
Fiber.abort("First argument must be a non-negative integer < 19.")
}
if (!(f is List)) Fiber.abort("Second argument must be a list.")
var sum = f.reduce { |acc, i| acc + i }
if (n != sum) {
Fiber.abort("The elements of the list must sum to 'n'.")
}
var prod = 1
for (e in f) {
if (e < 0) Fiber.abort("The elements of the list must be non-negative integers.")
if (e > 1) prod = prod * factorial(e)
}
return factorial(n)/prod
}
// Returns the binomial coefficent of n over k.
static binomial(n, k) { multinomial(n, [k, n-k]) }
// Determines whether 'n' is prime using a wheel with basis [2, 3].
static isPrime(n) {
if (!n.isInteger || n < 2) return false
if (n%2 == 0) return n == 2
if (n%3 == 0) return n == 3
var d = 5
while (d*d <= n) {
if (n%d == 0) return false
d = d + 2
if (n%d == 0) return false
d = d + 4
}
return true
}
// Returns the next prime number greater than 'n'.
static nextPrime(n) {
n = (n%2 == 0) ? n + 1 : n + 2
while (true) {
if (Int.isPrime(n)) return n
n = n + 2
}
}
// Sieves for primes up to and including 'n'.
// If primesOnly is true returns a list of the primes found.
// If primesOnly is false returns a bool list 'c' of size (n + 1) where:
// c[i] is false if 'i' is prime or true if 'i' is composite.
static primeSieve(n, primesOnly) {
if (n < 2) return []
var primes = [2]
var k = ((n-3)/2).floor + 1
var marked = List.filled(k, true)
var limit = ((n.sqrt.floor - 3)/2).floor + 1
limit = limit.max(0)
for (i in 0...limit) {
if (marked[i]) {
var p = 2*i + 3
var s = ((p*p - 3)/2).floor
var j = s
while (j < k) {
marked[j] = false
j = j + p
}
}
}
for (i in 0...k) {
if (marked[i]) primes.add(2*i + 3)
}
if (primesOnly) return primes
var c = List.filled(n+1, true)
for (p in primes) c[p] = false
return c
}
// Convenience version of above method which uses true for the primesOnly parameter.
static primeSieve(n) { primeSieve(n, true) }
// Sieves for primes up to and including 'limit' using a segmented approach
// and returns a list of the primes found in order.
// Second argument needs to be the cache size (L1) of your machine in bytes.
// Translated from public domain C++ code at:
// https://gist.github.com/kimwalisch/3dc39786fab8d5b34fee
static segmentedSieve(limit, cacheSize) {
cacheSize = (cacheSize/8).floor // 8 bytes per list element
var sqroot = limit.sqrt.floor
var segSize = sqroot.max(cacheSize)
if (limit < 2) return []
var allPrimes = [2]
var isPrime = List.filled(sqroot + 1, true)
var primes = []
var multiples = []
var i = 3
var n = 3
var s = 3
var low = 0
while (low <= limit) {
var sieve = List.filled(segSize, true)
var high = limit.min(low + segSize - 1)
while (i * i <= high) {
if (isPrime[i]) {
var j = i * i
while (j <= sqroot) {
isPrime[j] = false
j = j + i
}
}
i = i + 2
}
while (s * s <= high) {
if (isPrime[s]) {
primes.add(s)
multiples.add(s * s - low)
}
s = s + 2
}
for (ii in 0...primes.count) {
var j = multiples[ii]
var k = primes[ii] * 2
while (j < segSize) {
sieve[j] = false
j = j + k
}
multiples[ii] = j - segSize
}
while (n <= high) {
if (sieve[n - low]) allPrimes.add(n)
n = n + 2
}
low = low + segSize
}
return allPrimes
}
// Private helper method which counts and returns how many primes there are
// up to and including 'n' using the Legendre method.
static legendre_(n, primes, memoPhi) {
if (n < 2) return 0
var phi
phi = Fn.new { |x, a|
if (a == 0) return x
var key = cantorPair(x, a)
if (memoPhi.containsKey(key)) return memoPhi[key]
var pa = primes[a-1]
return memoPhi[key] = phi.call(x, a-1) - phi.call((x/pa).floor, a-1)
}
var a = legendre_(n.sqrt.floor, primes, memoPhi)
return phi.call(n, a) + a - 1
}
// Private helper method which sieves for primes up to and including 'n'
// and returns how many there are. Faster than Legendre method for small n.
static primeCountSmall_(n) {
if (n < 2) return 0
var count = 1
var k = ((n-3)/2).floor + 1
var marked = List.filled(k, true)
var limit = ((n.sqrt.floor - 3)/2).floor + 1
limit = limit.max(0)
for (i in 0...limit) {
if (marked[i]) {
var p = 2*i + 3
var s = ((p*p - 3)/2).floor
var j = s
while (j < k) {
marked[j] = false
j = j + p
}
}
}
for (i in 0...k) {
if (marked[i]) count = count + 1
}
return count
}
// Computes, using an appropriate method, and returns how many primes there are
// up to and including 'n'.
static primeCount(n) {
if (n < 1e4) return primeCountSmall_(n)
var limit = n.sqrt.floor
var primes = primeSieve(limit)
var memoPhi = {}
return legendre_(n, primes, memoPhi)
}
// Returns the prime factors of 'n' in order using a wheel with basis [2, 3, 5].
static primeFactors(n) {
if (!n.isInteger || n < 2) return []
var inc = [4, 2, 4, 2, 4, 6, 2, 6]
var factors = []
while (n%2 == 0) {
factors.add(2)
n = (n/2).truncate
}
while (n%3 == 0) {
factors.add(3)
n = (n/3).truncate
}
while (n%5 == 0) {
factors.add(5)
n = (n/5).truncate
}
var k = 7
var i = 0
while (k * k <= n) {
if (n%k == 0) {
factors.add(k)
n = (n/k).truncate
} else {
k = k + inc[i]
i = (i + 1) % 8
}
}
if (n > 1) factors.add(n)
return factors
}
// Returns all the divisors of 'n' including 1 and 'n' itself.
static divisors(n) {
if (!n.isInteger || n < 1) return []
var divisors = []
var divisors2 = []
var i = 1
var k = (n%2 == 0) ? 1 : 2
while (i <= n.sqrt) {
if (n%i == 0) {
divisors.add(i)
var j = (n/i).floor
if (j != i) divisors2.add(j)
}
i = i + k
}
if (!divisors2.isEmpty) divisors = divisors + divisors2[-1..0]
return divisors
}
// Returns all the divisors of 'n' excluding 'n'.
static properDivisors(n) {
var d = divisors(n)
var c = d.count
return (c <= 1) ? [] : d[0..-2]
}
// As 'divisors' method but uses a different algorithm.
// Better for large numbers with a small number of prime factors.
static divisors2(n) {
if (n <= 0) return []
var factors = Int.primeFactors(n)
var divs = [1]
for (p in factors) {
for (i in 0...divs.count) divs.add(divs[i]*p)
}
divs.sort()
var c = divs.count
if (c > 1) {
for (i in c-1..1) {
if (divs[i-1] == divs[i]) divs.removeAt(i)
}
}
return divs
}
// As 'properDivisors' but uses 'divisors2' method.
static properDivisors2(n) {
var d = divisors2(n)
var c = d.count
return (c <= 1) ? [] : d[0..-2]
}
// Private helper method which checks a number and base for validity.
static check_(n, b) {
if (!(n is Num && n.isInteger && n >= 0)) {
Fiber.abort("Number must be a non-negative integer.")
}
if (!(b is Num && b.isInteger && b >= 2 && b < 64)) {
Fiber.abort("Base must be an integer between 2 and 63.")
}
}
// Returns a list of an integer n's digits in base b. Optionally checks n and b are valid.
static digits(n, b, check) {
if (check) check_(n, b)
if (n == 0) return [0]
var digs = []
while (n > 0) {
digs.add(n%b)
n = (n/b).floor
}
return digs[-1..0]
}
// Returns the sum of an integer n's digits in base b. Optionally checks n and b are valid.
static digitSum(n, b, check) {
if (check) check_(n, b)
var sum = 0
while (n > 0) {
sum = sum + (n%b)
n = (n/b).floor
}
return sum
}
// Returns the digital root and additive persistence of an integer n in base b.
// Optionally checks n and b are valid.
static digitalRoot(n, b, check) {
if (check) check_(n, b)
var ap = 0
while (n > b - 1) {
n = digitSum(n, b)
ap = ap + 1
}
return [n, ap]
}
// Convenience versions of the above methods which never check for validity
// and/or use base 10 by default.
static digits(n, b) { digits(n, b, false) }
static digits(n) { digits(n, 10, false) }
static digitSum(n, b) { digitSum(n, b, false) }
static digitSum(n) { digitSum(n, 10, false) }
static digitalRoot(n, b) { digitalRoot(n, b, false) }
static digitalRoot(n) { digitalRoot(n, 10, false) }
// Returns the unique non-negative integer that is associated with a pair
// of non-negative integers 'x' and 'y' according to Cantor's pairing function.
static cantorPair(x, y) {
if (x.type != Num || !x.isInteger || x < 0) {
Fiber.abort("Arguments must be non-negative integers.")
}
if (y.type != Num || !y.isInteger || y < 0) {
Fiber.abort("Arguments must be non-negative integers.")
}
return (x*x + 3*x + 2*x*y + y + y*y) / 2
}
// Returns the pair of non-negative integers that are associated with a single
// non-negative integer 'z' according to Cantor's pairing function.
static cantorUnpair(z) {
if (z.type != Num || !z.isInteger || z < 0) {
Fiber.abort("Argument must be a non-negative integer.")
}
var i = (((1 + 8*z).sqrt-1)/2).floor
return [z - i*(1+i)/2, i*(3+i)/2 - z]
}
}
/*
Nums contains various routines applicable to lists or ranges of numbers
many of which are useful for statistical purposes.
*/
class Nums {
// Methods to calculate sum, various means, product and maximum/minimum element of 'a'.
// The sum and product of an empty list are considered to be 0 and 1 respectively.
static sum(a) { a.reduce(0) { |acc, x| acc + x } }
static mean(a) { sum(a)/a.count }
static geometricMean(a) { a.reduce { |prod, x| prod * x}.pow(1/a.count) }
static harmonicMean(a) { a.count / a.reduce { |acc, x| acc + 1/x } }
static quadraticMean(a) { (a.reduce(0) { |acc, x| acc + x*x }/a.count).sqrt }
static prod(a) { a.reduce(1) { |acc, x| acc * x } }
static max(a) { a.reduce { |acc, x| (x > acc) ? x : acc } }
static min(a) { a.reduce { |acc, x| (x < acc) ? x : acc } }
// As above methods but applying a function 'f' to each element of 'a'
// before performing the operation.
// 'f' should take a single Num parameter and return a Num.
static sum(a, f) { a.reduce(0) { |acc, x| acc + f.call(x) } }
static mean(a, f) { sum(a, f)/a.count }
static geometricMean(a, f) { a.reduce { |prod, x| prod * f.call(x)}.pow(1/a.count) }
static harmonicMean(a, f) { a.count / a.reduce { |acc, x| acc + 1/f.call(x) } }
static quadraticMean(a, f) { (a.reduce(0) { |acc, x| acc + f.call(x).pow(2) }/a.count).sqrt }
static prod(a, f) { a.reduce(1) { |acc, x| acc * f.call(x) } }
static max(a, f) { a.reduce { |acc, x|
var fx = f.call(x)
return (fx > acc) ? fx : acc
} }
static min(a, f) { a.reduce { |acc, x|
var fx = f.call(x)
return (fx < acc) ? fx : acc
} }
// Returns the median of a sorted list 'a'.
static median(a) {
var c = a.count
if (c == 0) {
Fiber.abort("An empty list cannot have a median")
} else if (c%2 == 1) {
return a[(c/2).floor]
} else {
var d = (c/2).floor
return (a[d] + a[d-1])/2
}
}
// Returns a list whose first element is a list of the mode(s) of 'a'
// and whose second element is the number of times the mode(s) occur.
static modes(a) {
var m = {}
for (e in a) m[e] = (!m[e]) ? 1 : m[e] + 1
var max = 0
for (e in a) if (m[e] > max) max = m[e]
var res = []
for (k in m.keys) if (m[k] == max) res.add(k)
return [max, res]
}
// Returns the sample variance of 'a'.
static variance(a) {
var m = mean(a)
var c = a.count
return (a.reduce(0) { |acc, x| acc + x*x } - m*m*c) / (c-1)
}
// Returns the population variance of 'a'.
static popVariance(a) {
var m = mean(a)
return (a.reduce(0) { |acc, x| acc + x*x }) / a.count - m*m
}
// Returns the sample standard deviation of 'a'.
static stdDev(a) { variance(a).sqrt }
// Returns the population standard deviation of 'a'.
static popStdDev(a) { popVariance(a).sqrt }
// Returns the mean deviation of 'a'.
static meanDev(a) {
var m = mean(a)
return a.reduce { |acc, x| acc + (x - m).abs } / a.count
}
// Converts a numeric list to a corresponding string list.
static fromStrings(a) { a.map { |s| Num.fromString(s) }.toList }
// Converts a string list to a corresponding numeric list.
static toStrings(a) { a.map { |n| n.toString }.toList }
}
/* Boolean supplements the Bool class with bitwise operations on boolean values. */
class Boolean {
// Private helper method to convert a boolean to an integer.
static btoi_(b) { b ? 1 : 0 }
// Private helper method to convert an integer to a boolean.
static itob_(i) { i != 0 }
// Private helper method to check its arguments are both booleans.
static check_(b1, b2) {
if (!((b1 is Bool) && (b2 is Bool))) Fiber.abort("Both arguments must be booleans.")
}
// Returns the logical 'and' of its boolean arguments.
static and(b1, b2) {
check_(b1, b2)
return itob_(btoi_(b1) & btoi_(b2))
}
// Returns the logical 'or' of its boolean arguments.
static or(b1, b2) {
check_(b1, b2)
return itob_(btoi_(b1) | btoi_(b2))
}
// Returns the logical 'xor' of its boolean arguments.
static xor(b1, b2) {
check_(b1, b2)
return itob_(btoi_(b1) ^ btoi_(b2))
}
}