# Category talk:Wren-big

### Arbitrary precision arithmetic

Many tasks on RC require the use of very large numbers (usually integers) often outside the 64-bit range.

Wren is unable to deal natively and precisely with integers outside a 53-bit range and so either has to limit such tasks to integers which it can deal with or is unable to offer a realistic solution at all.

This module aims to remedy that situation by providing classes for arbitrary precision arithmetic (integers, rational and decimal numbers) in an easy to use form. It is based on or guided by Peter Olson's public domain BigInteger.js and BigRational.js libraries for Javascript, a language which until the recent addition of native BigInt support was in the same boat as Wren regarding the 53-bit limitation.

BigInt class

Some changes or simplifications have been made to fit Wren's ethos more closely. In particular:

• There is no separate SmallInt class; instead integers with a magnitude of less than 2^53 are implemented as a subset of BigInt objects whose value is represented by a simple Num rather than a list of integers.
• There is no support for bases below 2 or (as far as the public interface is concerned) above 36 as practical applications for such bases are very limited. Similarly, support for user-defined digit alphabets has also been excluded.
• As currently written, this class always uses the Random module for random number generation. The latter is an optional part of the Wren distribution and can be used by either embedded or CLI scripts. Although it would not be difficult to allow for other RNGs to be 'plugged in', this would probably only be useful if Wren were being used for secure cryptographic purposes which, realistically, seems unlikely to be the case.

The BigInt.new constructor accepts either a string or a safe integer (i.e. within the normal 53-bit limitation). I couldn't see much point in accepting larger numbers (or floats) as this would inevitably mean that the BigInt object would be imprecise from the outset. Passing an empty string will also produce an error (rather than a zero object) as it's just as easy to pass 0 and an empty string may well be a mistake in any case.

If you need to generate a BigInt from a string in a non-decimal base or from another BigInt, you can use the fromBaseString or copy methods, respectively.

BigRat class

This follows the lines of the Rat class in the Wren-rat module though I have been guided by BigRational.js when coding some of the trickier parts.

The BigRat.new constructor accepts a BigInt numerator and denominator (or just a numerator on its own). However, you can pass in Nums or Strings if a BigInt can be built from them

There are also separate methods to enable you to to generate a BigRat from a Rat object, from a string in rational form (i.e. "n/d"), from a mixed string (e.g. "1_2/3") from a decimal string or value (e.g. "1.234") or from a Num.

BigDec class

As rational numbers can easily be created from or displayed as decimal numbers, at first glance there seems little point in adding a BigDec class to the module as well.

However, a problem with BigRat is that it always works to maximum precision which uses a lot of memory and can be very slow in certain circumstances.

BigDec aims to fix this problem by working to a user defined precision, where precision is measured in digits after the decimal point. One can therefore limit precision having regard to the accuracy needed for a particular task though, to dove-tail with the maximum accuracy of ordinary Nums, a minimum (and default) precision of 16 is imposed.

Otherwise, BigDec is simply a thin wrapper over BigRat and defers to it for the implementation of most of its methods. Where all you need is decimal numbers, it is also somewhat easier to use than BigRat as the user never has to deal with the latter directly.

General observations

The original Javascript libraries and this module are easy to use as they try to mimic native arithmetic by always producing immutable objects and by implicitly converting suitable operands into BigInts, BigRats or BigDecs. This is accentuated further by Wren's support for operator overloading which means that all the familiar operations on numbers can be used.

Of course, such a system inevitably keeps the garbage collector busy and is not going to be as fast and efficient as systems where BigInts, BigRats or BigDecs are mutable and where memory usage can therefore be managed by the user. On the other hand the latter systems can be difficult or unpleasant to use and it is therefore felt that the present approach is a good fit for Wren where ease of use is paramount.

### Source code

```/* Module "big.wren" */

import "./trait" for Comparable
import "random" for Random

/*
BigInt represents an arbitrary length integer allowing arithmetic operations on integers of
unlimited size. Internally, there are two kinds: 'small' (up to a magnitude of 2^53 -1)
and 'big' (magnitude >= 2^53). The former are stored as ordinary Nums and the latter as a
little-endian list of integers using a base of 1e7. In both cases, there is an additional Bool
to indicate whether the integer is signed or not. BigInt objects are immutable.
*/
class BigInt is Comparable {
// Constants.
static minusOne { BigInt.small_(-1) }
static zero     { BigInt.small_( 0) }
static one      { BigInt.small_( 1) }
static two      { BigInt.small_( 2) }
static three    { BigInt.small_( 3) }
static four     { BigInt.small_( 4) }
static five     { BigInt.small_( 5) }
static six      { BigInt.small_( 6) }
static seven    { BigInt.small_( 7) }
static eight    { BigInt.small_( 8) }
static nine     { BigInt.small_( 9) }
static ten      { BigInt.small_(10) }

// Private method to initialize static fields.
static init_() {
// All possible digits for bases 2 to 36.
__alphabet = "0123456789abcdefghijklmnopqrstuvwxyz"

// Maximum 'small' integer = 2^53 - 1
__maxSmall = 9007199254740991

// Random number generator.
__rand = Random.new()

// Threshold between small and big integers in 'list' format.
__threshold = smallToList_(9007199254740992)

// List of powers of 2 up to 1e7.
__powersOfTwo = powersOfTwo_

// Length of __powersOfTwo (pre-computed).
__powers2Length = 24

// Last element of __powersOfTwo (pre-computed).
__highestPower2 = 8388608
}

// Returns the maximum 'small' BigInt = 2^53-1.
static maxSmall { BigInt.small_(__maxSmall) }

// Private property which returns powers of 2 up to 1e7.
static powersOfTwo_ {
var pot = [1]
while (2 * pot[-1] <= 1e7) pot.add(2 * pot[-1])
return pot
}

// Private method which returns the lowest one bit of a BigInt (rough).
static roughLOB_(n) {
var lobmask_i = 1 << 30
var v = n.value_
var x
if (n.isSmall) {
} else {
var base = 1e7
var lobmask_bi = (base & -base) * (base & -base) | lobmask_i
x = v[0] + v[1]*base | lobmask_bi
}
return x & -x
}

// Private method to return the integer logarithm of 'value' with respect to base 'base'
// i.e.the number of times 'base' can be multiplied by itself without exceeding 'value'.
static integerLogarithm_(value, base) {
if (base <= value) {
var tmp = integerLogarithm_(value, base.square)
var p = tmp[0]
var e = tmp[1]
var t = p * base
return (t <= value) ? [t, e*2 + 1] : [p, e * 2]
}
return [one, 0]
}

// Private method to determine whether a number is a small integer or not.
static isSmall_(n) { (n is Num) && n.isInteger && n.abs <= __maxSmall }

// Private method to convert a small integer to list format.
static smallToList_(n) {
if (n < 1e7) return [n]
if (n < 1e14) return [n % 1e7, (n/1e7).floor]
return [n % 1e7, ((n/1e7).floor) % 1e7, (n/1e14).floor]
}

// Private method to convert a list to a small integer where possible.
static listToSmall_(a) {
trim_(a)
var length = a.count
var base = 1e7
if (length < 4 && compareAbs_(a, __threshold) < 0) {
if (length == 0) return 0
if (length == 1) return a[0]
if (length == 2) return a[0] + a[1]*base
return a[0] + (a[1] + a[2]*base) * base
}
return a
}

// Private method to check whether the magnitude of a number n <= 1e7.
static shiftIsSmall_(n) { n.abs <= 1e7 }

// Private method to remove any trailing zero elements from a list.
static trim_(a) {
var i = a.count - 1
while (i >= 0 && a[i] == 0) {
a.removeAt(i)
i = i - 1
}
}

// Private method to compare two lists, first by length and then by contents.
static compareAbs_(a, b) {
if (a.count != b.count) return (a.count > b.count) ? 1 : -1
var i = a.count - 1
while (i >= 0) {
if (a[i] != b[i]) return (a[i] > b[i]) ? 1 : -1
i = i - 1
}
return 0
}

// Private method to add two lists where a.count >= b.count.
var la = a.count
var lb = b.count
var r = List.filled(la, 0)
var carry = 0
var base = 1e7
var sum = 0
var i = 0
while (i < lb) {
sum = a[i] + b[i] + carry
carry = (sum >= base) ? 1 : 0
r[i] = sum - carry*base
i = i + 1
}
while (i < la) {
sum = a[i] + carry
carry = (sum == base) ? 1 : 0
r[i] = sum - carry*base
i = i + 1
}
return r
}

// Private method to add two lists regardless of length.

// Private method to add 'carry' (0 <= carry <= maxSmall) to a list 'a'.
var l = a.count
var r = List.filled(l, 0)
var base = 1e7
var sum = 0
for (i in 0...l) {
sum = a[i] - base + carry
carry = (sum/base).floor
r[i] = sum - carry*base
carry = carry + 1
}
while (carry > 0) {
carry = (carry/base).floor
}
return r
}

// Private method to subtract two lists where a.count >= b.count.
static subtract_(a, b) {
var la = a.count
var lb = b.count
var r = List.filled(la, 0)
var borrow = 0
var base = 1e7
var diff = 0
var i = 0
while (i < lb) {
diff = a[i] - borrow - b[i]
if (diff < 0) {
diff = diff + base
borrow = 1
} else borrow = 0
r[i] = diff
i = i + 1
}
while (i < la) {
diff = a[i] - borrow
if (diff < 0) {
diff = diff + base
} else {
r[i] = diff
i = i + 1
break
}
r[i] = diff
i = i + 1
}
while (i < la) {
r[i] = a[i]
i = i + 1
}
trim_(r)
return r
}

// Private method to subtract lists regardless of length.
static subtractAny_(a, b, signed) {
var value
if (compareAbs_(a, b) >= 0) {
value = subtract_(a, b)
} else {
value = subtract_(b, a)
signed = !signed
}
value = listToSmall_(value)
if (value is Num) {
if (signed) value = -value
return BigInt.small_(value)
}
return BigInt.big_(value, signed)
}

// Private method to subtract 'b' (0 <= b <= maxSmall) from a list 'a'.
static subtractSmall_(a, b, signed) {
var l = a.count
var r = List.filled(l, 0)
var carry = -b
var base = 1e7
for (i in 0...l) {
var diff = a[i] + carry
carry = (diff/base).floor
diff = diff % base
r[i] = (diff < 0) ? diff + base : diff
}
r = listToSmall_(r)
if (r is Num) {
if (signed) r = -r
return BigInt.small_(r)
}
return BigInt.big_(r, signed)
}

// Private method to multiply two lists.
static multiplyLong_(a, b) {
var la = a.count
var lb = b.count
var l  = la + lb
var r  = List.filled(l, 0)
var base = 1e7
for (i in 0...la) {
var ai = a[i]
for (j in 0...lb) {
var bj = b[j]
var prod = ai*bj + r[i + j]
var carry = (prod/base).floor
r[i + j] = prod - carry*base
r[i + j + 1] = r[i + j + 1] + carry
}
}
trim_(r)
return r
}

/// Private method to multiply a list 'a' by 'b' (|b| < 1e7).
static multiplySmall_(a, b) {
var l = a.count
var r = List.filled(l, 0)
var base = 1e7
var carry = 0
for (i in 0...l) {
var prod = a[i]*b + carry
carry = (prod/base).floor
r[i] = prod - carry*base
}
while (carry > 0) {
carry = (carry/base).floor
}
return r
}

// Private helper method for multiplyKaratsuba_ method.
static shiftLeft_(x, n) {
var r = []
while (n > 0) {
n = n - 1
}
return r + x
}

// Private method to multiply two lists 'x' and 'y' using the Karatsuba algorithm.
static multiplyKaratsuba_(x, y) {
var n = (x.count > y.count) ? y.count : x.count
if (n <= 30) return multiplyLong_(x, y)
n = (n/2).floor
var a = x[0...n]
var b = x[n..-1]
var c = y[0...n]
var d = y[n..-1]
var ac = multiplyKaratsuba_(a, c)
var bd = multiplyKaratsuba_(b, d)
var s = subtract_(subtract_(abcd, ac), bd)
trim_(prod)
return prod
}

// Private method to determine whether Karatsuba multiplication may be beneficial.
static useKaratsuba_(l1, l2) { -0.012*l1 - 0.012*l2 + 0.000015*l1*l2 > 0 }

// Private method to multiply a small integer 'a' (a >= 0) by a list 'b'.
static multiplySmallAndList_(a, b, signed) {
if (a < 1e7) return BigInt.big_(multiplySmall_(b, a), signed)
return BigInt.big_(multiplyLong_(b, smallToList_(a)), signed)
}

// Private method to square a list.
static square_(a) {
var l = a.count
var r = List.filled(l + l, 0)
var base = 1e7
for (i in 0...l) {
var ai = a[i]
var carry = 0 - ai*ai
var j = i
while (j < l) {
var aj = a[j]
var prod = 2*ai*aj + r[i + j] + carry
carry = (prod/base).floor
r[i + j] = prod - carry*base
j = j + 1
}
r[i + l] = carry
}
trim_(r)
return r
}

// Private method to 'div/mod' two lists, better for smaller sizes.
static divMod1_(a, b) {
var la = a.count
var lb = b.count
var base = 1e7
var result = List.filled(la - lb + 1, 0)
var divMostSigDigit = b[-1]
// normalization
var lambda = (base/(2*divMostSigDigit)).ceil
var remainder = multiplySmall_(a, lambda)
var divisor = multiplySmall_(b, lambda)
divMostSigDigit = divisor[lb-1]
var shift = la - lb
while (shift >= 0) {
var quotDigit = base - 1
if (remainder[shift+lb] != divMostSigDigit) {
quotDigit = ((remainder[shift+lb]*base + remainder[shift+lb-1])/divMostSigDigit).floor
}
var carry = 0
var borrow = 0
var l = divisor.count
for (i in 0...l) {
carry = carry + quotDigit*divisor[i]
var q = (carry/base).floor
borrow = borrow + remainder[shift+i] - (carry - q*base)
carry = q
if (borrow < 0) {
remainder[shift+i] = borrow + base
borrow = -1
} else {
remainder[shift+i] = borrow
borrow = 0
}
}
while (borrow != 0) {
quotDigit = quotDigit - 1
carry = 0
for (i in 0...l) {
carry = carry + remainder[shift+i] - base + divisor[i]
if (carry < 0) {
remainder[shift+i] = carry + base
carry = 0
} else {
remainder[shift+i] = carry
carry = 1
}
}
borrow = borrow + carry
}
result[shift] = quotDigit
shift = shift - 1
}
// denormalization
remainder = divModSmall_(remainder, lambda)[0]
return [listToSmall_(result), listToSmall_(remainder)]
}

// Private method to 'div/mod' two lists, better for larger sizes.
static divMod2_(a, b) {
var la = a.count
var lb = b.count
var result = []
var part = []
var base = 1e7
while (la != 0) {
la = la - 1
part = a[la..la] + part
trim_(part)
if (compareAbs_(part, b) < 0) {
} else {
var xlen = part.count
var highx = part[xlen-1]*base + part[xlen-2]
var highy = b[lb-1]*base + b[lb-2]
if (xlen > lb) highx = (highx + 1) * base
var guess = (highx/highy).ceil
var check
while (true) {
check = multiplySmall_(b, guess)
if (compareAbs_(check, part) <= 0) break
guess = guess - 1
if (guess == 0) break
}
part = subtract_(part, check)
}
}
result = result[-1..0]
return [listToSmall_(result), listToSmall_(part)]
}

// Private method to 'div/mod' a list 'value' with an integer 'lambda'.
static divModSmall_(value, lambda) {
var length = value.count
var quot = List.filled(length, 0)
var base = 1e7
var remainder = 0
var i = length - 1
while (i >= 0) {
var divisor = remainder*base + value[i]
var q = (divisor/lambda).truncate
remainder = divisor - q*lambda
quot[i] = q | 0
i = i - 1
}
return [quot, remainder | 0]
}

// Private method to 'div/mod' any two BigInts.
static divModAny_(self, n) {
var a = self.value_
if (!(n is BigInt)) n = BigInt.new(n)
var b = n.value_
if (b == 0) Fiber.abort("Cannot divide by zero.")
if (self.isSmall) {
if (n.isSmall) {
var c = (a/b).truncate
return [BigInt.small_(c), BigInt.small_(a % b)]
}
return [zero, self]
}
var quotient
if (n.isSmall) {
if (b == 1) return [self, zero]
if (b == -1) return [-self, zero]
var ab = b.abs
if (ab < 1e7) {
var value = divModSmall_(a, ab)
quotient = listToSmall_(value[0])
var remainder = value[1]
if (self.signed_) remainder = -remainder
if (quotient is Num) {
if (self.signed_ != n.signed_) quotient = -quotient
return [BigInt.small_(quotient), BigInt.small_(remainder)]
}
return [BigInt.big_(quotient, self.signed_ != n.signed_), BigInt.small_(remainder)]
}
b = smallToList_(ab)
}
var comparison = compareAbs_(a, b)
if (comparison == -1) return [BigInt.zero, self]
if (comparison == 0) return [(self.signed_ == n.signed_) ? one : minusOne, zero]

// divMod1 is faster on smaller input sizes
var value = (a.count + b.count <= 200) ? divMod1_(a, b) : divMod2_(a, b)
quotient = value[0]
var qSign = self.signed_ != n.signed_
var mod = value[1]
var mSign = self.signed_
if (quotient is Num) {
if (qSign) quotient = -quotient
quotient = BigInt.small_(quotient)
} else quotient = BigInt.big_(quotient, qSign)
if (mod is Num) {
if (mSign) mod = -mod
mod = BigInt.small_(mod)
} else mod = BigInt.big_(mod, mSign)
return [quotient, mod]
}

// Private method to determine if a BigInt is a basic prime or not.
static isBasicPrime_(n) {
if (!(n is BigInt)) n = BigInt.new(n)
if (n.isUnit) return false
if (n == two || n == three || n == five) return true
if (n.isEven || n.isDivisibleBy(three) || n.isDivisibleBy(five)) return false
if (n < 49) return true
return null // unknown if prime or not
}

// Private method to apply the Miller-Rabin test.
static millerRabinTest_(n, a) {
var nPrev = n.dec
var b = nPrev
var r = 0
while (b.isEven) {
b = b / two
r = r + 1
}
for (i in 0...a.count) {
if (n >= a[i]) {
var x = (a[i] is BigInt) ? a[i] : BigInt.new(a[i])
x = x.modPow(b, n)
if (!x.isUnit && x != nPrev) {
var d = r - 1
var next = false
while (d != 0) {
x = x.square % n
if (x.isUnit) return false
if (x == nPrev) {
next = true
break
}
d = d - 1
}
if (!next) return false
}
}
}
return true
}

// Private method to perform bitwise operations depending on the 'fn' passed in.
static bitwise_(x, y, fn) {
if (!(y is BigInt)) y = BigInt.new(y)
var xSign = x.isNegative
var ySign = y.isNegative
var xRem = xSign ? ~x : x
var yRem = ySign ? ~y : y
var result = []
while (!xRem.isZero || !yRem.isZero) {
var xDivMod = divModAny_(xRem, __highestPower2)
var xDigit = xDivMod[1].toNum
if (xSign) {
xDigit = __highestPower2 - 1 - xDigit // two's complement for negative numbers
}
var yDivMod = divModAny_(yRem, __highestPower2)
var yDigit = yDivMod[1].toNum
if (ySign) {
yDigit = __highestPower2 - 1 - yDigit // two's complement for negative numbers
}
xRem = xDivMod[0]
yRem = yDivMod[0]
}
var sum = fn.call(xSign ? 1 : 0, ySign ? 1 : 0) != 0 ? minusOne : zero
var i = result.count - 1
var hp2 = BigInt.small_(__highestPower2)
while (i >= 0) {
sum = sum * hp2 + BigInt.new(result[i])
i = i - 1
}
return sum
}

// Returns the greater of two BigInts.
static max(a, b) {
if (!(a is BigInt)) a = BigInt.new(a)
if (!(b is BigInt)) b = BigInt.new(b)
return (a > b) ? a : b
}

// Returns the lesser of two BigInts.
static min(a, b) {
if (!(a is BigInt)) a = BigInt.new(a)
if (!(b is BigInt)) b = BigInt.new(b)
return (a < b) ? a : b
}

// Returns the positive difference of two BigInts.
static dim(a, b) {
if (!(a is BigInt)) a = BigInt.new(a)
if (!(b is BigInt)) b = BigInt.new(b)
if (a >= b) return a - b
return zero
}

// Returns the greatest common denominator of a and b.
static gcd(a, b) {
if (!(a is BigInt)) a = BigInt.new(a)
if (!(b is BigInt)) b = BigInt.new(b)
a = a.abs
b = b.abs
if (a == b) return a
if (a.isZero) return b
if (b.isZero) return a
var c = one
while (a.isEven && b.isEven) {
var d = min(roughLOB_(a), roughLOB_(b))
a = a / d
b = b / d
c = c * d
}
while (a.isEven) a = a / roughLOB_(a)
while (true) {
while (b.isEven) b = b / roughLOB_(b)
if (a > b) {
var t = b
b = a
a = t
}
b = b - a
if (b.isZero) break
}
return c.isUnit ? a : a * c
}

// Returns the least common multiple of a and b.
static lcm(a, b) {
if (!(a is BigInt)) a = BigInt.new(a)
if (!(b is BigInt)) b = BigInt.new(b)
if (a.isZero && b.isZero) return BigInt.zero
return a / gcd(a, b) * b
}

// Returns the factorial of 'n'.
static factorial(n) {
if (!(n is Num && n >= 0)) Fiber.abort("Argument must be a non-negative integer")
if (n < 2) return BigInt.one
var fact = BigInt.one
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)) Fiber.abort("First argument must be a non-negative integer")
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 = BigInt.one
for (e in f) {
if (e < 0) Fiber.abort("The elements of the list must be non-negative integers.")
if (e > 1) prod = prod * BigInt.factorial(e)
}
return BigInt.factorial(n)/prod
}

// Returns the binomial coefficent of n over k.
static binomial(n, k) { multinomial(n, [k, n-k]) }

// Returns whether or not 'n' is an instance of BigInt.
static isInstance(n) { n is BigInt }

// Private helper method for 'randBetween'.
static fromList_(digits, base, isNegative) {
var digits2 = digits.map { |d| BigInt.new(d) }.toList
return parseBaseFromList_(digits2, BigInt.new(base || 10), isNegative)
}

// Returns a random number between 'a' (inclusive) and 'b' (exclusive).
static randBetween(a, b) {
if (!(a is BigInt)) a = BigInt.new(a)
if (!(b is BigInt)) b = BigInt.new(b)
var low  = min(a, b)
var high = max(a, b)
var range = high - low + one
if (range.isSmall) return low + (range.value_ * __rand.float()).floor
var digits = toBase_(range, 1e7)[0]
var result = []
var restricted = true
for (i in 0...digits.count) {
var top = restricted ? digits[i] : 1e7
var digit = (top * __rand.float()).truncate
if (digit < top) restricted = false
}
return low + fromList_(result, 1e7, false)
}

// Private method to provide the components for converting a BigInt to a different base.
static toBase_(n, base) {
if (!(base is BigInt)) base = BigInt.new(base)
if (base < two) Fiber.abort("Bases less than 2 are not supported.")
var neg = false
if (n.isNegative) {
neg = true
n = n.abs
}
var out = []
var left = n
while (left.isNegative || left.compareAbs(base) >= 0) {
var divmod = left.divMod(base)
left = divmod[0]
var digit = divmod[1]
if (digit.isNegative) {
digit = (base - digit).abs
left = left.inc
}
}
return [out[-1..0], neg]
}

// Private method to parse a list, in a given base, to a BigInt.
static parseBaseFromList_(digits, base, isNegative) {
var val = zero
var pow = one
var i = digits.count - 1
while (i >= 0) {
val = val + digits[i]*pow
pow = pow * base
i = i - 1
}
return isNegative ? -val : val
}

// Private method to parse a numeric string, in a given base (2 to 36), to a BigInt.
static parseBase_(text, base) {
text = text.trim()
if (text.count == 0) Fiber.abort("Invalid base string.")
if (base > 10) text = lower_(text)
var alphabet = __alphabet[0...base]
var isNegative = text[0] == "-"
if (isNegative || text[0] == "+") {
text = text[1..-1]
if (text.count == 0) Fiber.abort("Invalid base string.")
}
text = text.trimStart("0")
if (text == "") text = "0"
base = BigInt.small_(base)
var digits = []
for (c in text) {
var ix = alphabet.indexOf(c)
if (ix == - 1) Fiber.abort("%(c) is not a valid digit in base %(base).")
}
return parseBaseFromList_(digits, base, isNegative)
}

// Private helper function to convert a string to lower case.
static lower_(s) { s.codePoints.map { |c|
return String.fromCodePoint((c >= 65 && c <= 90) ? c + 32 : c)
}.join() }

// Private method to parse a base 10 numeric string 'v' to the components for a BigInt.
static parseString_(v) {
v = v.trim()
if (v.count == 0) Fiber.abort("Invalid integer.")
var signed = v[0] == "-"
if (signed || v[0] == "+") {
v = v[1..-1]
if (v.count == 0) Fiber.abort("Invalid integer.")
}
v = v.trimStart("0")
if (v == "") v = "0"
v = lower_(v)
var split = v.split("e")
if (split.count > 2) Fiber.abort("Invalid integer.")
if (split.count == 2) {
var exp = split[1]
if (exp[0] == "+") exp = exp[1..-1]
exp = Num.fromString(exp)
if (!isSmall_(exp)) Fiber.abort("Exponent is not valid.")
var text = split[0]
var dp = text.indexOf(".")
if (dp >= 0) {
exp = exp - (text.count - dp - 1)
text = text[0...dp] + text[dp+1..-1]
}
if (exp < 0) Fiber.abort("Exponent cannot be negative.")
text = text + ("0" * exp)
v = text
}
var isValid = v.count > 0 && v.all { |d| "0123456789".contains(d) }
if (!isValid) Fiber.abort("Invalid integer.")
if (v.count <= 16) {
var n = Num.fromString(v)
if (isSmall_(n)) return (signed) ? [-n, true] : [n, false]
}
var r = []
var max = v.count
var lb = 7
var min = max - lb
while (max > 0) {
min = min - lb
if (min < 0) min = 0
max = max - lb
}
trim_(r)
return [r, signed]
}

// Constructs a BigInt object from either a numeric base 10 string or a 'safe' integer.
construct new(value) {
if (!(value is String) && !BigInt.isSmall_(value)) {
Fiber.abort("Value must be a base 10 numeric string or a safe integer.")
}
if (value is String) {
var res = BigInt.parseString_(value)
_value  = res[0]
_signed = res[1]
} else {
_value = value
_signed = value < 0
}
}

// Creates a BigInt object from an (unprefixed) numeric string in a given base (2 to 36).
static fromBaseString(s, base) {
if (!(s is String)) Fiber.abort("Value must be a numeric string in the given base.")
if (!((base is Num) && base.isInteger && base >= 2 && base <= 36)) {
Fiber.abort("Base must be an integer between 2 and 36.")
}
if (base == 10) return BigInt.new(s)
return parseBase_(s, base)
}

// Private constructor which creates a BigInt object from a list of integers and a bool.
construct big_(a, signed) {
_value = a
_signed = signed
}

// Private constructor which creates a BigInt object from a 'safe' integer and a bool.
construct small_(a) {
_value = a
_signed = a < 0
}

// Private properties for internal use.
value_   { _value  }
signed_  { _signed }

// Public self-evident properties.
isSmall    { !(_value is List) }
isEven     { (this.isSmall) ? (_value & 1) == 0 : (_value[0] & 1) == 0 }
isOdd      { (this.isSmall) ? (_value & 1) == 1 : (_value[0] & 1) == 1 }
isPositive { (this.isSmall) ? (_value > 0) : !_signed }
isNegative { (this.isSmall) ? (_value < 0) :  _signed }
isUnit     { (this.isSmall) ? _value.abs == 1 : false }
isZero     { (this.isSmall) ? _value == 0 : false }

isDivisibleBy(n) {
if (!(n is BigInt)) n = BigInt.new(n)
if (n.isZero) return false
if (n.isUnit) return true
if (n.compareAbs(BigInt.two) == 0) return this.isEven
return (this % n).isZero
}

// Returns true if the current instance is prime, false otherwise.
// Setting 'strict' to true enforces the GRH-supported lower bound of 2*log(N)^2.
isPrime(strict) {
if (!(strict is Bool)) Fiber.abort("Argument must be a boolean.")
var isbp = BigInt.isBasicPrime_(this)
if (isbp != null) return isbp
var n = this.abs
var bits = n.bitLength
if (bits <= 64) {
return BigInt.millerRabinTest_(n, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37])
}
var logN = 2.log * bits.toNum
var t = ((strict) ? (2 * logN.pow(2)) : logN).ceil
var a = []
return BigInt.millerRabinTest_(n, a)
}

// Returns true if the current instance is very likely to be prime, false otherwise.
// The larger the number of 'iterations', the lower the chance of a false positive.
isProbablePrime(iterations) {
if (!((iterations is Num) && iterations.isInteger && iterations > 0)) {
Fiber.abort("Iterations must be a positive integer.")
}
var isbp = BigInt.isBasicPrime_(this)
if (isbp != null) return isbp
var n = this.abs
var a = []
for (i in 0...iterations) a.add(BigInt.randBetween(BigInt.two, n - BigInt.two))
return BigInt.millerRabinTest_(n, a)
}

// Convenience versions of the above methods which use default parameter values.
isPrime         { isPrime(false) }
isProbablePrime { isProbablePrime(5) }

// Returns the next probable prime number greater than 'this'.
nextPrime(iterations) {
if (this < BigInt.two) return BigInt.two
var n = isEven ? this + 1 : this + 2
while (true) {
if (n.isProbablePrime(iterations)) return n
n = n + 2
}
}

// Returns the previous probable prime number less than 'this' or null if there isn't one.
prevPrime(iterations) {
if (this < BigInt.three) return null
if (this == BigInt.three) return BigInt.two
var n = isEven ? this - 1 : this - 2
while (true) {
if (n.isProbablePrime(iterations)) return n
n = n - 2
}
}

// Convenience versions of the above methods which use 5 iterations.
nextPrime { nextPrime(5) }
prevPrime { prevPrime(5) }

// Negates a BigInt.
- { (this.isSmall) ? BigInt.small_(-_value) : BigInt.big_(_value, !_signed) }

// Private method which adds a BigInt to a 'small' current instance.
var a = _value
if (_signed != n.signed_) return this - (-n)
var b = n.value_
if (n.isSmall) {
var c = a + b
if (BigInt.isSmall_(c)) return BigInt.small_(c)
b = BigInt.smallToList_(b.abs)
}
}

// Adds a BigInt to the current instance.
+ (n) {
if (!(n is BigInt)) n = BigInt.new(n)
if (n.isZero) return this.copy()
if (_signed != n.signed_) return this - (-n)
var a = _value
var b = n.value_
if (n.isSmall) {
}
}

// Private method which subtracts a BigInt from a 'small' current instance.
smallSubtract_(n) {
var a = _value
if (_signed != n.signed_) return this + (-n)
var b = n.value_
if (n.isSmall) return BigInt.small_(a - b)
return BigInt.subtractSmall_(b, a.abs, !_signed)
}

// Subtracts a BigInt from the current instance.
- (n) {
if (!(n is BigInt)) n = BigInt.new(n)
if (n.isZero) return this.copy()
if (this.isSmall) return smallSubtract_(n)
if (_signed != n.signed_) return this + (-n)
var a = _value
var b = n.value_
if (n.isSmall) return BigInt.subtractSmall_(a, b.abs, _signed)
return BigInt.subtractAny_(a, b, _signed)
}

// Private method which multiplies the current instance by a 'small' BigInt.
multiplyBySmall_(a) {
if (this.isSmall) {
var c = a.value_ * _value
if (BigInt.isSmall_(c)) return BigInt.small_(c)
c = a.value_.abs
return BigInt.multiplySmallAndList_(c, BigInt.smallToList_(_value.abs), _signed != a.signed_)
}
if (a.value_ == 0) return BigInt.zero
if (a.value_ == 1) return this.copy()
if (a.value_ == -1) return -this
return BigInt.multiplySmallAndList_(a.value_.abs, _value, _signed != a.signed_)
}

// Multiplies the current instance by a BigInt.
* (n) {
if (!(n is BigInt)) n = BigInt.new(n)
if (this.isSmall) return n.multiplyBySmall_(this)
var a = _value
var b = n.value_
var signed = _signed != n.signed_
if (n.isSmall) {
if (b == 0) return BigInt.zero
if (b == 1) return this.copy()
if (b == -1) return -this
var ab = b.abs
if (ab < 1e7) return BigInt.big_(BigInt.multiplySmall_(a, ab), signed)
b = BigInt.smallToList_(ab)
}
if (BigInt.useKaratsuba_(a.count, b.count)) {
return BigInt.big_(BigInt.multiplyKaratsuba_(a, b), signed)
}
return BigInt.big_(BigInt.multiplyLong_(a, b), signed)
}

// Square the current instance.
square {
if (this.isSmall) {
var value = _value * _value
if (BigInt.isSmall_(value)) return BigInt.small_(value)
return BigInt.big_(BigInt.square_(BigInt.smallToList_(_value.abs)), false)
}
return BigInt.big_(BigInt.square_(_value), false)
}

// Cube the current instance.
cube { square * this }

// Returns true if the current instance is a perfect square, false otherwise.
isSquare {
var root = isqrt
return root.square == this
}

// Returns true if the current instance is a perfect cube, false otherwise.
isCube {
var root = icbrt
return root.square * root == this
}

// Returns the integer n'th root of the current instance
// i.e. the precise n'th root truncated towards zero if not an integer.
// Throws an error if the current instance is negative and n is even.
iroot(n) {
if (!((n is Num) && n.isInteger && n > 0)) {
Fiber.abort("Argument must be a positive integer.")
}
if (n == 1) return this.copy()
var t = copy()
var neg = t < 0
if (neg) {
if (n % 2 == 0) {
Fiber.abort("Cannot take the %(n)th root of a negative number.")
} else {
t = -t
}
}
n = n - 1
var s = t + 1
var u = t
while (u < s) {
s = u
u = ((u * n) + t / u.pow(n)) / (n + 1)
}
return (neg) ? -s : s
}

// Returns the cube root of the current instance i.e. the largest integer 'r' such that
// r.cube <= this.
icbrt {
if (isSmall) return BigInt.small_(toSmall.cbrt.floor)
return iroot(3)
}

// Returns the integer square root of the current instance i.e. the largest integer 'r' such that
// r.square <= this. Throws an error if the current instance is negative.
isqrt {
if (isNegative) Fiber.abort("Cannot take the square root of a negative number.")
if (isSmall) return BigInt.small_(toSmall.sqrt.floor)
var one = BigInt.one
var a = one << ((bitLength + one) >> one)
while (true) {
var b = (this/a + a) >> one
if (b >= a) return a
a = b
}
}

// Returns a list containing the quotient and the remainder after dividing the current instance
// by a BigInt. The sign of the remainder will match the sign of the dividend.
divMod(n) {
if (!(n is BigInt)) n = BigInt.new(n)
return BigInt.divModAny_(this, n)
}

// Divides the current instance by a BigInt.
/(n) {
if (!(n is BigInt)) n = BigInt.new(n)
return BigInt.divModAny_(this, n)[0]
}

// Returns the remainder after dividing the current instance by a BigInt.
// The sign of the remainder will match the sign of the dividend.
%(n) {
if (!(n is BigInt)) n = BigInt.new(n)
return BigInt.divModAny_(this, n)[1]
}

// Returns the current instance raised to the power of a 'small' BigInt.
// If the exponent is less than 0, returns 0. O.pow(0) returns one.
pow(n) {
if (!(n is BigInt)) n = BigInt.new(n)
var a = _value
var b = n.value_
if (b == 0) return BigInt.one
if (a == 0) return BigInt.zero
if (a == 1) return BigInt.one
if (a == -1) return n.isEven ? BigInt.one : BigInt.minusOne
if (n.signed_) return BigInt.zero
if (!n.isSmall) Fiber.abort("The exponent %(n) is too large.")
if (this.isSmall) {
var value = a.pow(b)
if (BigInt.isSmall_(value)) return BigInt.small_(value.truncate)
}
var x = this
var y = BigInt.one
while (true) {
if ((b & 1) == 1) {
y = y * x
b = b - 1
}
if (b == 0) break
b = b / 2
x = x.square
}
return y
}

// Returns the current instance to the power 'exp' modulo 'mod'.
modPow(exp, mod) {
if (!(exp is BigInt)) exp = BigInt.new(exp)
if (!(mod is BigInt)) mod = BigInt.new(mod)
if (mod.isZero) Fiber.abort("Cannot take modPow with modulus 0.")
var r = BigInt.one
var base = this % mod
if (exp.isNegative) {
exp = exp * BigInt.minusOne
base = base.modInv(mod)
}
while (exp.isPositive) {
if (base.isZero) return BigInt.zero
if (exp.isOdd) r = (r * base) % mod
exp = exp / BigInt.two
base = base.square % mod
}
return r
}

// Returns the multiplicative inverse of the current instance modulo 'r'.
modInv(n) {
if (!(n is BigInt)) n = BigInt.new(n)
var r = n
var newR = this.abs
var t = BigInt.zero
var newT = BigInt.one
while (!newR.isZero) {
var q = r / newR
var lastT = t
var lastR = r
t = newT
r = newR
newT = lastT - q*newT
newR = lastR - q*newR
}
if (!r.isUnit) Fiber.abort("%(this) and %(n) are not co-prime.")
if (t.compare(BigInt.zero) == -1) t = t + n
if (this.isNegative) return -t
return t
}

// Returns the sign of the current instance: 0 if zero, 1 if positive and -1 otherwise.
sign { (this.isZero) ? 0 : (this.isPositive) ? 1 : -1 }

// Increments the current instance by one.
inc {
var value = _value
if (this.isSmall) {
if (value + 1 <= __maxSmall) return BigInt.small_(value + 1)
return BigInt.big_(__threshold, false)
}
if (_signed) return BigInt.subtractSmall_(value, 1, _signed)
}

// Decrements the current instance by one.
dec {
var value = _value
if (this.isSmall) {
if (value - 1 >= -__maxSmall) return BigInt.small_(value - 1)
return BigInt.big_(__threshold, true)
}
if (_signed) return BigInt.big_(BigInt.addSmall_(value, 1), true)
return BigInt.subtractSmall_(value, 1, _signed)
}

// Bitwise operators. The operands are treated as if they were represented
// using two's complement representation.
~     { (-this).dec }
&(n)  { BigInt.bitwise_(this, n, Fn.new { |a, b| a & b }) }
|(n)  { BigInt.bitwise_(this, n, Fn.new { |a, b| a | b }) }
^(n)  { BigInt.bitwise_(this, n, Fn.new { |a, b| a ^ b }) }

<<(n) {
if (!(n is BigInt)) n = BigInt.new(n)
n = n.toNum
if (!BigInt.shiftIsSmall_(n)) Fiber.abort("%(n) is too large for shifting.")
if (n < 0) return this >> (-n)
var result = this
if (result.isZero) return result
var hp2 = BigInt.small_(__highestPower2)
while (n >= __powers2Length) {
result = result * hp2
n = n - (__powers2Length - 1)
}
return result * __powersOfTwo[n]
}

>>(n) {
if (!(n is BigInt)) n = BigInt.new(n)
n = n.toNum
if (!BigInt.shiftIsSmall_(n)) Fiber.abort("%(n) is too large for shifting.")
if (n < 0) return this << (-n)
var result = this
var remQuo
var hp2 = BigInt.small_(__highestPower2)
while (n >= __powers2Length) {
if (result.isZero || (result.isNegative && result.isUnit)) return result
remQuo = BigInt.divModAny_(result, hp2)
result = remQuo[1].isNegative ? remQuo[0].dec : remQuo[0]
n = n - (__powers2Length - 1)
}
remQuo = BigInt.divModAny_(result, __powersOfTwo[n])
return remQuo[1].isNegative ? remQuo[0].dec : remQuo[0]
}

// Returns the absolute value of the current instance.
abs { (this.isSmall) ? BigInt.small_(_value.abs) : BigInt.big_(_value, false) }

// Compares the current instance with a BigInt. If they are equal returns 0.
// If 'this' is greater, returns 1. Otherwise returns -1.
// Also allows a comparison with an infinite number.
compare(n) {
if ((n is Num) && n.isInfinity) return -n.sign
if (!(n is BigInt)) n = BigInt.new(n)
var a = _value
var b = n.value_
if (this.isSmall) {
if (n.isSmall) return (a == b) ? 0 : a > b ? 1 : -1
if (_signed != n.signed_) return (_signed) ? -1 : 1
return _signed ? 1 : -1
}
if (_signed != n.signed_) return n.signed_ ? 1 : -1
if (n.isSmall) return _signed ? -1 : 1
return BigInt.compareAbs_(a, b) * (_signed ? -1 : 1)
}

// As 'compare' but compares absolute values.
compareAbs(n) {
if ((n is Num) && n.isInfinity) return -n.sign
if (!(n is BigInt)) n = BigInt.new(n)
if (this.isSmall) {
var a = _value.abs
var b = n.value_
if (n.isSmall) {
b = b.abs
return (a == b) ? 0 : (a > b) ? 1 : -1
}
return -1
}
if (n.isSmall) return 1
return BigInt.compareAbs_(_value, n.value_)
}

// Returns the number of digits required to represent the current instance in binary.
bitLength {
var n = this
if (n.isNegative) n = (-n) - BigInt.one
if (n.isZero) return BigInt.zero
return BigInt.new(BigInt.integerLogarithm_(n, BigInt.two)[1]) + BigInt.one
}

// Returns true if the 'n'th bit of the current instance is set or false otherwise.
testBit(n) {
if (n.type != Num || !n.isInteger || n < 0) Fiber.abort("Argument must be a non-negative integer.")
return (this >> n) & BigInt.one != BigInt.zero
}

// The inherited 'clone' method just returns 'this' as BigInt objects are immutable.
// If you need an actual copy use this method instead.
copy() { (this.isSmall) ? BigInt.small_(_value) : BigInt.big_(_value, _signed) }

// Converts the current instance to a 'small' integer where possible.
// Otherwise returns null.
toSmall { (this.isSmall) ? _value : null }

// Converts the current instance to a Num where possible.
// Will probably lose accuracy if the current instance is not 'small'.
toNum { Num.fromString(this.toString) }

// Returns the string representation of the current instance in a given base (2 to 36).
toBaseString(base) {
if (!((base is Num) && base.isInteger && base >= 2 && base <= 36)) {
Fiber.abort("Base must be an integer between 2 and 36.")
}
if (base == 10) return this.toString
var lst = BigInt.toBase_(this, base)
return (lst[1] ? "-" : "") + lst[0].map { |d| __alphabet[d] }.join("")
}

// Returns the string representation of the current instance in base 10.
toString {
var v = (this.isSmall) ? BigInt.smallToList_(_value.abs) : _value
var l = v.count - 1
var str = v[l].toString
var zeros = "0000000"
var digit = ""
l = l - 1
while (l >= 0) {
digit = v[l].toString
str = str + zeros[digit.count..-1] + digit
l = l - 1
}
var sign = _signed ? "-" : ""
return sign + str
}

// Creates and returns a range of BigInts from 'this' to 'n' with a step of 1.
// 'this' and 'n' must both be 'small' integers. 'n' can be a Num or a BigInt.
..(n)  { BigInts.range(this, n, 1, true)  }  // inclusive of 'n'
...(n) { BigInts.range(this, n, 1, false) }  // exclusive of 'n'

/* Prime factorization methods. */

// Private worker method for Pollard's Rho algorithm.
static pollardRho_(m, seed, c) {
var g = Fn.new { |x| (x*x + c) % m }
var x = BigInt.new(seed)
var y = BigInt.new(seed)
var z = BigInt.one
var d = BigInt.one
var count = 0
while (true) {
x = g.call(x)
y = g.call(g.call(y))
d = (x - y).abs % m
z = z * d
count = count + 1
if (count == 100) {
d = BigInt.gcd(z, m)
if (d != BigInt.one) break
z = BigInt.one
count = 0
}
}
if (d == m) return BigInt.zero
return d
}

// Returns a factor (BigInt) of 'm' (a BigInt or an integral Num) using the
// Pollard's Rho algorithm. Both the 'seed' and 'c' can be set to integers.
// Returns BigInt.zero in the event of failure.
static pollardRho(m, seed, c) {
if (m < 2) return BigInt.zero
if (m is Num) m = BigInt.new(m)
if (m.isProbablePrime(10)) return m.copy()
if (m.isSquare) return m.isqrt
for (p in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]) {
if (m.isDivisibleBy(p)) return BigInt.new(p)
}
return pollardRho_(m, seed, c)
}

// Convenience version of the above method which uses a seed of 2 and a value for c of 1.
static pollardRho(m) { pollardRho(m, 2, 1) }

// Private method for factorizing smaller numbers (BigInt) using a wheel with basis [2, 3, 5].
static primeFactorsWheel_(m) {
var n = m.copy()
var inc = [4, 2, 4, 2, 4, 6, 2, 6]
var factors = []
var k = BigInt.new(37)
var i = 0
while (k * k <= n) {
if (n.isDivisibleBy(k)) {
n = n / k
} else {
k = k + inc[i]
i = (i + 1) % 8
}
}
return factors
}

// Private worker method (recursive) to obtain the prime factors of a number (BigInt).
static primeFactors_(m, trialDivs) {
if (m.isProbablePrime(10)) return [m.copy()]
var n = m.copy()
var factors = []
var seed = 2
var c = 1
var checkPrime = true
var threshold = 1e11 // from which using PR may be advantageous
if (trialDivs) {
for (p in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]) {
while (n.isDivisibleBy(p)) {
n = n / p
}
}
}
while (n > BigInt.one) {
if (checkPrime && n.isProbablePrime(10)) {
break
}
if (n >= threshold) {
var d = pollardRho_(n, seed, c)
if (d != BigInt.zero) {
n = n / d
checkPrime = true
} else if (c == 1) {
if (n.isSquare) {
n = n.isqrt
var pf = primeFactors_(n, false)
break
} else {
c = 2
checkPrime = false
}
} else if (c < 101) {
c = c + 1
} else if (seed < 101) {
seed = seed + 1
} else {
break
}
} else {
break
}
}
factors.sort()
return factors
}

// Returns a list of the primes factors (BigInt) of 'm' (a Bigint or an integral Num)
// using the wheel based factorization and/or Pollard's Rho algorithm as appropriate.
static primeFactors(m) {
if (m < 2) return []
if (m is Num) m = BigInt.new(m)
return primeFactors_(m, true)
}

// Returns all the divisors of 'n' including 1 and 'n' itself.
static divisors(n) {
if (n < 1) return []
if (n is Num) n = BigInt.new(n)
var divs = []
var divs2 = []
var i = one
var k = (n.isEven) ? one : two
var sqrt = n.isqrt
while (i <= sqrt) {
if (n.isDivisibleBy(i)) {
var j = n / i
}
i = i + k
}
if (!divs2.isEmpty) divs = divs + divs2[-1..0]
return divs
}

// 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 larger numbers.
static divisors2(n) {
if (n is Num) n = BigInt.new(n)
var pf = primeFactors(n)
if (pf.count == 0) return (n == 1) ? [one] : pf
var arr = []
if (pf.count == 1) {
} else {
var prevPrime = pf[0]
var count = 1
for (i in 1...pf.count) {
if (pf[i] == prevPrime) {
count = count + 1
} else {
prevPrime = pf[i]
count = 1
}
}
}
var divisors = []
var generateDivs
generateDivs = Fn.new { |currIndex, currDivisor|
if (currIndex == arr.count) {
return
}
for (i in 0..arr[currIndex][1]) {
generateDivs.call(currIndex+1, currDivisor)
currDivisor = currDivisor * arr[currIndex][0]
}
}
generateDivs.call(0, one)
return divisors.sort()
}

// As 'properDivisors' but uses 'divisors2' method.
static properDivisors2(n) {
var d = divisors2(n)
var c = d.count
return (c <= 1) ? [] : d[0..-2]
}

// Returns the sum of all the divisors of 'n' including 1 and 'n' itself.
static divisorSum(n) {
if (n < 1) return zero
if (n is Num) n = BigInt.new(n)
var total = one
var power = two
while (n % 2 == 0) {
total = total + power
power = power * 2
n = n / 2
}
var i = three
while (i * i <= n) {
var sum = one
power = i
while (n % i == 0) {
sum = sum + power
power = power * i
n = n / i
}
total = total * sum
i = i + 2
}
if (n > 1) total = total * (n + 1)
}

// Returns the number of divisors of 'n' including 1 and 'n' itself.
static divisorCount(n) {
if (n < 1) return 0
if (n is Num) n = BigInt.new(n)
var count = 0
var prod = 1
while (n % 2 == 0) {
count = count + 1
n = n / 2
}
prod = prod * (1 + count)
var i = three
while (i * i <= n) {
count = 0
while (n % i == 0) {
count = count + 1
n = n / i
}
prod = prod * (1 + count)
i = i + 2
}
if (n > 2) prod = prod * 2
return prod
}
}

/*  BigInts contains various routines applicable to lists of big integers
and for creating and iterating though ranges of such numbers. */
class BigInts is Sequence {
static sum(a)  { a.reduce(BigInt.zero) { |acc, x| acc + x } }
static prod(a) { a.reduce(BigInt.one)  { |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 } }

// Private helper method for creating ranges.
static checkValue_(v, name) {
if (v is BigInt && v.isSmall) {
return v.toSmall
} else if (v is Num && v.isInteger && v.abs <= Num.maxSafeInteger) {
return v
} else {
Fiber.abort("Invalid value for '%(name)'.")
}
}

// Creates a range of 'small' BigInts analogous to the Range class but allowing for steps > 1.
// Use the 'ascending' parameter to check that the range's direction is as intended.
construct range(from, to, step, inclusive, ascending) {
from = BigInts.checkValue_(from, "from")
to   = BigInts.checkValue_(to, "to")
step = BigInts.checkValue_(step, "step")
if (step < 1) Fiber.abort("'step' must be a positive integer.")
if (ascending && from > to) Fiber.abort("'from' cannot exceed 'to'.")
if (!ascending && from < to) Fiber.abort("'to' cannot exceed 'from'.")
_rng = inclusive ? from..to : from...to
_step = step
}

// Convenience versions of 'range' which use default values for some parameters.
static range(from, to, step, inclusive) { range(from, to, step, inclusive, from <= to) }
static range(from, to, step)            { range(from, to, step, true, from <= to) }
static range(from, to)                  { range(from, to, 1, true, from <= to) }

from        {  _rng.from        }
to          {  _rng.to          }
step        {  _step            }
min         {  from.min(to)     }
max         {  from.max(to)     }
isInclusive {  _rng.isInclusive }
isAscending {  from <= to       }

// Iterator protocol methods.
iterate(iterator) {
if (!iterator || _step == 1) {
return _rng.iterate(iterator)
} else {
var count = _step
while (count > 0 && iterator) {
iterator = _rng.iterate(iterator)
count = count - 1
}
return iterator
}
}

iteratorValue(iterate) { BigInt.small_(_rng.iteratorValue(iterate)) }
}

/* BigRat represents a rational number as a BigInt numerator and (non-zero) denominator
expressed in their lowest terms. BigRat objects are immutable.
*/
class BigRat is Comparable {
// Private helper function to check that 'o' is a suitable type and throw an error otherwise.
// Rational numbers, numbers and numeric strings are returned as BigRats.
static check_(o) {
if (o is BigRat) return o
if (o is BigInt) return BigRat.new(o, BigInt.one)
if (o.type.toString == "Rat") return BigRat.fromRat(o)
if (o is Num) return BigRat.fromFloat(o)
if (o is String) return (o.contains("_") && o.contains("/")) ? fromMixedString(o) :
o.contains("/") ? fromRationalString(o) : fromDecimal(o)
Fiber.abort("Argument must either be a rational number, a number or a numeric string.")
}

// Constants.
static minusOne { BigRat.new(BigInt.minusOne, BigInt.one)   }
static zero     { BigRat.new(BigInt.zero,     BigInt.one)   }
static one      { BigRat.new(BigInt.one,      BigInt.one)   }
static two      { BigRat.new(BigInt.two,      BigInt.one)   }
static three    { BigRat.new(BigInt.three,    BigInt.one)   }
static four     { BigRat.new(BigInt.four,     BigInt.one)   }
static five     { BigRat.new(BigInt.five,     BigInt.one)   }
static six      { BigRat.new(BigInt.six,      BigInt.one)   }
static seven    { BigRat.new(BigInt.seven,    BigInt.one)   }
static eight    { BigRat.new(BigInt.eight,    BigInt.one)   }
static nine     { BigRat.new(BigInt.nine,     BigInt.one)   }
static ten      { BigRat.new(BigInt.ten,      BigInt.one)   }
static half     { BigRat.new(BigInt.one,      BigInt.two)   }
static third    { BigRat.new(BigInt.one,      BigInt.three) }
static quarter  { BigRat.new(BigInt.one,      BigInt.four)  }
static fifth    { BigRat.new(BigInt.one,      BigInt.five)  }
static sixth    { BigRat.new(BigInt.one,      BigInt.six)   }
static seventh  { BigRat.new(BigInt.one,      BigInt.seven) }
static eighth   { BigRat.new(BigInt.one,      BigInt.eight) }
static ninth    { BigRat.new(BigInt.one,      BigInt.nine)  }
static tenth    { BigRat.new(BigInt.one,      BigInt.ten)   }

// Constructs a new BigRat object by passing it a numerator and a denominator.
// These must either be BigInts or Nums/Strings which are capable of creating one
// when passed to the BigInt.new constructor.
construct new(n, d) {
n = (n is BigInt) ? n.copy() : BigInt.new(n)
d = (d is BigInt) ? d.copy() : BigInt.new(d)
if (d == BigInt.zero) Fiber.abort("Denominator must be a non-zero integer.")
if (n == BigInt.zero) {
d = BigInt.one
} else if (d < BigInt.zero) {
n = -n
d = -d
}
var g = BigInt.gcd(n, d).abs
if (g > BigInt.one) {
n = n / g
d = d / g
}
_n = n
_d = d
}

// Convenience method which constructs a new BigRat object by passing it just a numerator.
static new(n) { BigRat.new(n, BigInt.one) }

// Constructs a BigRat object from a Rat object. To use this method the Rat class needs
// to be imported from the Wren-rat module as, to minimize dependencies,
// this module does not do so.
static fromRat(r) {
if (r.type.toString != "Rat") Fiber.abort("Argument must be a rational number.")
return BigRat.new(r.num, r.den)
}

// Constructs a BigRat object from a string of the form "n/d".
// Improper fractions are allowed.
static fromRationalString(s) {
var nd = s.split("/")
if (nd.count != 2) Fiber.abort("Argument is not a suitable string.")
var n = BigInt.new(nd[0])
var d = BigInt.new(nd[1])
return BigRat.new(n, d)
}

// Constructs a BigRat object from a string of the form "i_n/d" where 'i' is an integer.
// Improper and negative fractional parts are allowed.
static fromMixedString(s) {
var ind = s.split("_")
if (ind.count != 2) Fiber.abort("Argument is not a suitable string.")
var nd = fromRationalString(ind[1])
var i = BigRat.new(ind[0])
var neg = i.isNegative || (i.isZero && ind[0][0] == "-")
return neg ? i - nd : i + nd
}

// Constructs a BigRat object from a decimal numeric string or value.
static fromDecimal(s) {
if (!(s is String)) s = s.toString
if (s == "") Fiber.abort("Argument cannot be an empty string.")
s = BigInt.lower_(s)
var parts = s.split("e")
if (parts.count > 2) Fiber.abort("Argument is invalid scientific notation.")
if (parts.count == 2) {
var isPositive = true
if (parts[1][0] == "-") {
parts[1] = parts[1][1..-1]
isPositive = false
}
if (parts[1][0] == "+") parts[1] = parts[1][1..-1]
var significand = fromDecimal(parts[0])
var p = BigInt.new(parts[1])
var exponent = BigRat.new(BigInt.ten.pow(p), BigInt.one)
return (isPositive) ? significand * exponent : significand / exponent
}
s = s.trim().trimStart("0")
if (s == "") return BigRat.zero
if (s.startsWith(".")) s = "0" + s
if (!s.contains(".")) {
return BigRat.new(s)
} else {
s = s.trimEnd("0")
}
if (s.endsWith(".")) return BigRat.new(s[0..-2])
var splits = s.split(".")
if (splits.count != 2) Fiber.abort("Argument is not a decimal.")
var num = splits[0]
var isNegative = false
if (num.startsWith("-")) {
isNegative = true
num = num[1..-1]
} else if (num.startsWith("+")) {
num = num[1..-1]
}
var den = splits[1]
if (!num.all { |c| "0123456789".contains(c) } || !den.all { |c| "0123456789".contains(c) }) {
Fiber.abort("Argument is not a decimal.")
}
num = BigInt.new(num + den)
if (isNegative) num = -num
den = BigInt.ten.pow(den.count)
return BigRat.new(num, den)
}

// Constructs a rational number from a floating point number provided the latter is an integer
// or has a decimal string representation.
static fromFloat(n) {
if (!(n is Num)) Fiber.abort("Argument must be a number.")
if (n.isInteger) return BigRat.new(n, BigInt.one)
return fromDecimal(n)
}

// Returns the greater of two BigRat objects.
static max(r1, r2) { (r1 < r2) ? r2 : r1 }

// Returns the smaller of two BigRat objects.
static min(r1, r2) { (r1 < r2) ? r1 : r2 }

// Determines whether a BigRat object is always shown as such or, if integral, as an integer.
static showAsInt     { __showAsInt }
static showAsInt=(b) { __showAsInt = b }

// Basic properties.
num        { _n }                   // numerator
den        { _d }                   // denominator
ratio      { [_n, _d] }             // a two element list of the above
isInteger  { _d == 1 }              // checks if integral or not
isPositive { _n > BigInt.zero }     // checks if positive
isNegative { _n < BigInt.zero }     // checks if negative
isUnit     { _n.abs == BigInt.one } // checks if plus or minus one
isZero     { _n == BigInt.zero }    // checks if zero

// Rounding methods (similar to those in Num class).
ceil { // higher integer, towards zero
if (isInteger) return this.copy()
var div = _n/_d
if (!this.isNegative) div = div.inc
return BigRat.new(div, BigInt.one)
}

floor { // lower integer
if (isInteger) return this.copy()
var div = _n/_d
if (this.isNegative) div = div.dec
return BigRat.new(div, BigInt.one)
}

truncate { this.isNegative ? ceil : floor } // lower integer, towards zero

round { // nearer integer
if (isInteger) return this.copy()
var div = _n / _d
if (_d == 2) {
div = isNegative ? div.dec : div.inc // round 1/2 away from zero
return BigRat.new(div, BigInt.one)
}
return (this + BigRat.half).floor
}

roundUp  { this >= 0 ? ceil : floor   } // rounds to higher integer, away from zero

round(digits) { // rounds to 'digits' decimal places
if (digits == 0) return round
return BigRat.fromDecimal(toDecimal(digits))
}

fraction { this - truncate } // fractional part (same sign as this.num)

// Reciprocal
inverse  { BigRat.new(_d, _n) }

// Integer division.
idiv(o)  { (this/o).truncate }

// Negation.
-{ BigRat.new(-_n, _d) }

// Arithmetic operators (work with numbers and numeric strings as well as other rationals).
+(o) { (o = BigRat.check_(o)) && BigRat.new(_n * o.den + _d * o.num, _d * o.den) }
-(o) { (o = BigRat.check_(o)) && (this + (-o)) }
*(o) { (o = BigRat.check_(o)) && BigRat.new(_n * o.num, _d * o.den) }
/(o) { (o = BigRat.check_(o)) && BigRat.new(_n * o.den, _d * o.num) }
%(o) { (o = BigRat.check_(o)) && (this - idiv(o) * o) }

// Computes integral powers.
pow(i) {
if (!((i is Num) && i.isInteger)) Fiber.abort("Argument must be an integer.")
if (i == 0) return BigRat.one
if (i == 1) return this.copy()
if (i == -1) return this.inverse
var np = _n.pow(i.abs)
var dp = _d.pow(i.abs)
return (i > 0) ? BigRat.new(np, dp) : BigRat.new(dp, np)
}

// Returns the square of the current instance.
square { BigRat.new(_n * _n , _d *_d) }

// Returns the cube of the current instance.
cube { square * this }

// Returns the square root of the current instance to 'digits' decimal places.
// Five more decimals is used to try to ensure accuracy though this is not guaranteed.
sqrt(digits) {
if (!((digits is Num) && digits.isInteger && digits >= 0)) {
Fiber.abort("Digits must be a non-negative integer.")
}
digits = digits + 5
var powd = BigInt.ten.pow(digits)
var sqtd = (powd.square * _n / _d).iroot(2)
return BigRat.new(sqtd, powd)
}

// Convenience version of the above method which uses 14 decimal places.
sqrt { sqrt(14) }

// Returns the cube root of the current instance to 'digits' decimal places.
// Five more decimals is used to try to ensure accuracy though this is not guaranteed.
cbrt(digits) {
if (!((digits is Num) && digits.isInteger && digits >= 0)) {
Fiber.abort("Digits must be a non-negative integer.")
}
digits = digits + 5
var powd = BigInt.ten.pow(digits)
var cbtd = (powd.cube * _n / _d).iroot(3)
return BigRat.new(cbtd, powd)
}

// Convenience version of the above method which uses 14 decimal places.
cbrt { cbrt(14) }

// Other methods.
inc  { this + BigRat.one }                     // increment
dec  { this - BigRat.one }                     // decrement
abs  { (_n >= BigInt.zero) ? copy() : -this }  // absolute value
sign { _n.sign }                               // sign

// The inherited 'clone' method just returns 'this' as BigRat objects are immutable.
// If you need an actual copy use this method instead.
copy() { BigRat.new(_n, _d) }

// Compares this BigRat with another one to enable comparison operators via Comparable trait.
compare(other) {
if ((other is Num) && other.isInfinity) return -other.sign
other = BigRat.check_(other)
if (_d == other.den) return _n.compare(other.num)
return (_n * other.den).compare(other.num * _d)
}

// As above but compares the absolute values of the BigRats.
compareAbs(other) { this.abs.compare(other.abs) }

// Returns this BigRat expressed as a BigInt with any fractional part truncated.
toBigInt { _n/_d }

// Converts the current instance to a Num where possible.
// Will probably lose accuracy if the numerator and/or denominator are not 'small'.
toFloat { _n.toNum / _d.toNum }

// Converts the current instance to an integer where possible with any fractional part truncated.
// Will probably lose accuracy if the numerator and/or denominator are not 'small'.
toInt { this.toFloat.truncate }

// Returns the decimal representation of this BigRat object to 'digits' decimal places.
// If 'rounded' is true, the value is rounded to that number of places with halves
// being rounded away from zero. Otherwise the value is truncated to that number of places.
// If 'zfill' is true, any unfilled decimal places are filled with zeros.
toDecimal(digits, rounded, zfill) {
if (!(digits is Num && digits.isInteger && digits >= 0)) {
Fiber.abort("Digits must be a non-negative integer")
}
if (rounded.type != Bool) Fiber.abort("Rounded must be true or false.")
if (zfill.type != Bool) Fiber.abort("Zfill must be true or false.")
var qr = _n.divMod(_d)
var rem = BigRat.new(qr[1].abs, _d)
// need to allow an extra digit if 'rounding' is true
var digits2 = (rounded) ? digits + 1 : digits
var shiftedRem = rem * BigRat.new("1e" + digits2.toString, BigInt.one)
var decPart = (shiftedRem.num / shiftedRem.den).toString
if (rounded) {
var finalByte = decPart[-1].bytes[0]
if (finalByte >= 53) {  // last character >= 5
decPart = (BigInt.new(decPart) + 5).toString
if (digits == 0) {
qr[0] = isNegative ? qr[0].dec : qr[0].inc
} else if (decPart.count == digits2 + 1) {
qr[0] = isNegative ? qr[0].dec : qr[0].inc
decPart = decPart[1..-1]
}
}
decPart = decPart[0...-1]  // remove last digit
}
if (decPart.count < digits) {
decPart = ("0" * (digits - decPart.count)) + decPart
}
if ((shiftedRem.num % shiftedRem.den) == BigInt.zero)  {
decPart = decPart.trimEnd("0")
}
if (digits < 1 || digits == 1 && decPart == "0") decPart = ""
var intPart = qr[0].toString
if (this.isNegative && qr[0] == 0) intPart = "-" + intPart
if (decPart == "") {
if (digits > 0) return intPart + (zfill ? "." + ("0" * digits) : "")
return intPart
}
return intPart + "." + decPart + (zfill ? ("0" * (digits - decPart.count)) : "")
}

// Convenience versions of the above which use default values for some or all parameters.
toDecimal(digits, rounded) { toDecimal(digits, rounded, false) } // never trailing zeros
toDecimal(digits)          { toDecimal(digits, true, false)    } // always rounded, never trailing zeros
toDecimal                  { toDecimal(14, true, false)        } // 14 digits, always rounded, never trailing zeros

// Returns a string represenation of this instance in the form "i_n/d" where 'i' is an integer.
toMixedString {
var sign = _n.isNegative ? "-" : ""
var nn = _n.abs
var q = nn / _d
var r = nn % _d
return sign + q.toString + "_" + r.toString + "/" + _d.toString
}

// Returns the string representation of this BigRat object depending on 'showAsInt'.
toString { (BigRat.showAsInt && _d == BigInt.one) ? "%(_n)" : "%(_n)/%(_d)" }
}

/*  BigRats contains various routines applicable to lists of big rational numbers */
class BigRats {
static sum(a)  { a.reduce(BigRat.zero) { |acc, x| acc + x } }
static mean(a) { sum(a)/a.count }
static prod(a) { a.reduce(BigRat.one) { |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 } }
}

/* BigDec represents an arbitrary length decimal number. Internally, it is simply a BigRat object
but with the number of decimal places in its expansion limited to a given 'precision'.
The minimum and default precision is 16 decimal places. BigDec objects are immutable.
*/
class BigDec is Comparable {
// Private helper function to check that 'o' is a suitable type and throw an error otherwise.
// Numbers, rational numbers and numeric strings are returned as BigDecs.
static check_(o) {
if (o is BigDec) return o
if (o is Num || o is String) return new(o)
if (o.type.toString == "Rat") return new_(BigRat.fromRat(o))
if (o is BigRat) return fromBigRat(o)
if (o is BigInt) return fromBigInt(o)
Fiber.abort("Argument must either be a number, a rational number or a numeric string.")
}

// Numeric constants (all have default precision)
static minusOne { fromInt(-1) }
static zero     { fromInt(0)  }
static one      { fromInt(1)  }
static two      { fromInt(2)  }
static three    { fromInt(3)  }
static four     { fromInt(4)  }
static five     { fromInt(5)  }
static six      { fromInt(6)  }
static seven    { fromInt(7)  }
static eight    { fromInt(8)  }
static nine     { fromInt(9)  }
static ten      { fromInt(10) }
static half     { new(0.5)    }
static quarter  { new(0.25)   }
static fifth    { new(0.2)    }
static eighth   { new(0.125)  }
static tenth    { new(0.1)    }

// Transcendental constants (all have precision 64)
static e     { new("2.7182818284590452353602874713526624977572470936999595749669676277", 64) }
static ln2   { new("0.6931471805599453094172321214581765680755001343602552541206800095", 64) }
static ln10  { new("2.3025850929940456840179914546843642076011014886287729760333279010", 64) }
static pi    { new("3.1415926535897932384626433832795028841971693993751058209749445923", 64) }
static tau   { new("6.2831853071795864769252867665590057683943387987502116419498891846", 64) }
static phi   { new("1.6180339887498948482045868343656381177203091798057628621354486227", 64) }
static sqrt2 { new("1.4142135623730950488016887242096980785696718753769480731766797380", 64) }

// Returns 'pi' to a given precision using the Almqvest Berndt AGM method.
// See https://rosettacode.org/wiki/Arithmetic-geometric_mean/Calculate_Pi for details.
static pi(prec) {
if (prec < 16) prec = 16
var an = BigRat.one
var bn = BigRat.half.sqrt(prec)
var tn = BigRat.half.square
var pn = BigRat.one
while (pn <= prec) {
var prevAn = an
an = (bn + an) * BigRat.half
bn = (bn * prevAn).sqrt(prec)
prevAn = prevAn - an
tn = tn - (prevAn.square * pn)
pn = pn + pn
}
var res = (an + bn).square / (tn * 4)
return BigDec.fromBigRat(res, prec)
}

// Returns the greater of two BigDec objects.
static max(d1, d2) { (d1 < d2) ? d2 : d1 }

// Returns the smaller of two BigDec objects.
static min(d1, d2) { (d1 < d2) ? d1 : d2 }

// Constructs a BigDec object from a number or decimal numeric string.
construct new(s, prec) {
_prec = prec.max(16)
_br = BigRat.fromDecimal(s)
if (s is String) _br = _br.round(_prec)
}

// Constructs a BigDec object from an integer.
construct fromInt(i, prec) {
if (!(i is Num && i.isInteger)) Fiber.abort ("Argument must be an integer.")
_br = BigRat.new(i, 1)
_prec = prec.max(16)
}

// Constructs a BigDec object from a big integer.
construct fromBigInt(bi, prec) {
if (!(bi is BigInt)) Fiber.abort ("Argument must be a big integer.")
_br = BigRat.new(bi, BigInt.one)
_prec = prec.max(16)
}

// Constructs a BigDec object from a rational number.
construct fromRat(r, prec) {
_br = BigRat.fromRat(r)
_prec = prec.max(16)
}

// Constructs a BigDec object from a big rational number.
construct fromBigRat(br, prec) {
if (!(br is BigRat)) Fiber.abort ("Argument must be a big rational number.")
_br = br.copy()
_prec = prec.max(16)
}

// Convenience versions of the above constructors which always use a precision of 16.
static new(s)         { new(s, 16) }
static fromInt(i)     { fromInt(i, 16) }
static fromBigInt(bi) { fromBigInt(bi, 16) }
static fromRat(r)     { fromRat(r, 16) }
static fromBigRat(br) { fromBigRat(br, 16) }

// Private constructor.
construct new_(br, prec) {
_br = br.round(prec)
_prec = prec
}

// Private method to return the internal BigRat without copying.
br_ { _br }

// Get or sets the precision for this object.
prec { _prec }
prec=(p) { _prec = p.floor.max(16) }

// Other properties.
isInteger  { _br.isInteger  }  // checks if integral or not
isPositive { _br.isPositive }  // checks if positive
isNegative { _br.isNegative }  // checks if negative
isUnit     { _br.isUnit     }  // checks if plus or minus one
isZero     { _br.isZero     }  // checks if zero

/* Note that the result of unary operations has the same precision as the operand
and the result of binary operations has the greater precision of the operands */

// Rounding methods (similar to those in Num class).
ceil     { BigDec.new_(_br.ceil, _prec) }      // higher integer, towards zero
floor    { BigDec.new_(_br.floor, _prec) }     // lower integer
truncate { BigDec.new_(_br.truncate, _prec) }  // lower integer, towards zero
round    { BigDec.new_(_br.round, _prec) }     // nearer integer
roundUp  { BigDec.new_(_br.roundUp, _prec) }   // higher integer, away from zero

round(digits) { BigDec.new(_br.round(digits), _prec) } // rounds to 'digits' decimal places
fraction { this - truncate }  // fractional part (same sign as this)

// Negation.
- { BigDec.new (-_br, _prec) }

// Arithmetic operators (work with numbers, rational numbers and numeric strings
// as well as other decimal numbers).
+(o) { (o = BigDec.check_(o)) && BigDec.new_(_br + o.br_, _prec.max(o.prec)) }
-(o) { (o = BigDec.check_(o)) && BigDec.new_(_br - o.br_, _prec.max(o.prec)) }
*(o) { (o = BigDec.check_(o)) && BigDec.new_(_br * o.br_, _prec.max(o.prec)) }
/(o) { (o = BigDec.check_(o)) && BigDec.new_(_br / o.br_, _prec.max(o.prec)) }
%(o) { (o = BigDec.check_(o)) && BigDec.new_(_br % o.br_, _prec.max(o.prec)) }

// Integral power methods.
pow(i) { BigDec.new_(_br.pow(i), _prec) }
square { BigDec.new_(_br.square, _prec) }
cube   { BigDec.new_(_br.cube,   _prec) }

// Roots to 'digits' places or 16 places by default.
sqrt(digits) { BigDec.new_(_br.sqrt(digits), _prec) }
cbrt(digits) { BigDec.new_(_br.cbrt(digits), _prec) }
sqrt         { sqrt(16) }
cbrt         { cbrt(16) }

// Other methods.
inverse { BigDec.new_(_br.inverse, _prec) }  // inverse
idiv(o) { (this/o).truncate }                // integer division
inc     { this + BigDec.one }                // increment
dec     { this - BigDec.one }                // decrement
abs     { BigDec.new_(_br.abs, _prec)}       // absolute value
sign    { _br.num.sign }                     // sign

// The inherited 'clone' method just returns 'this' as BigDec objects are immutable.
// If you need an actual copy use this method instead.
copy() { BigDec.new_(_br.copy(), _prec) }

// Compares this BigDec with 'other' to enable comparison operators via Comparable trait.
compare(other) { (other is BigDec) ? _br.compare(other.br_) : _br.compare(other) }

// As above but compares absolute values.
compareAbs(other) { (other is BigDec) ? _br.compareAbs(other.br_) : _br.compareAbs(other) }

// Returns this BigDec expressed as a BigInt with any fractional part truncated.
toBigInt { _br.num/_br.den }

// Returns this BigDec expressed as a BigRat.
toBigRat { _br.copy() }

// Converts the current instance to a Num where possible.
// Will probably lose accuracy for larger numbers.
toFloat { _br.toFloat }

// Converts the current instance to an integer where possible with any fractional part truncated.
// Will probably lose accuracy for integers >= 2^53.
toInt { this.toFloat.truncate }

// Returns the string representation of this BigDec object to 'digits' decimal places.
// If 'rounded' is true, the value is rounded to that number of places with halves
// being rounded away from zero. Otherwise the value is truncated to that number of places.
// If 'zfill' is true, any unfilled decimal places are filled with zeros.
toString(digits, rounded, zfill) { _br.toDecimal(digits, rounded, zfill) }

// Convenience versions of the above which use default values for some or all parameters
// and never add trailing zeros.
toString(digits, rounded) { toString(digits, rounded, false) }
toString(digits)          { toString(digits, true, false)    } // always rounded
toString                  { toString(_prec, true, false)     } // precision digits, always rounded
toDefaultString           { toString(16, true, false)        } // 16 digits, always rounded

// Private worker method for exponentiation using Taylor series.
exp_ {
var y = _br
var x = _br
var z = _br
var res = x + BigRat.one
var eps = BigRat.new(BigInt.one, BigInt.ten.pow(_prec))
var k = 2
var fact = BigRat.two
while (x.abs > eps) {
z = z * y
x = z / fact
res = res + x.round(_prec+2)
k = k + 1
fact = fact * k
}
return BigDec.fromBigRat(res, _prec)
}

// Returns the exponential of this instance (e ^ this) to the same precision.
exp {
if (_br.isInteger) return exp_
return (truncate/2).exp_.square * fraction.exp_
}

// Private worker method for natural logarithm using Taylor series.
log_ {
var y = _br
var x = (y - BigRat.one) / (y + BigRat.one)
var z = x * x
var res = BigRat.zero
var eps = BigRat.new(BigInt.one, BigInt.ten.pow(_prec))
var k = 1
while (x.abs > eps) {
res = res + (BigRat.two * x / k).round(_prec+2)
x = (x * z).round(_prec+2)
k = k + 2
}
return BigDec.new_(res, _prec)
}

// Returns the natural logarithm of this instance to the same precision.
log {
if (sign <= 0) Fiber.abort("Instance must be positive.")
if (this < BigDec.two) return log_
var d = this.copy()
var pow = 0
var two = BigDec.new(2, _prec)
while (d >= two) {
d = d / 2
pow = pow + 1
}
return d.log_ + two.log_ * pow
}

// Returns this ^ f where f may be fractional.
powf(f) {
if ((f is Num) && f.isInteger) return pow(f)
return (log * f).exp
}
}

/*  BigDecs contains various routines applicable to lists of big decimal numbers */
class BigDecs {
static sum(a)  { a.reduce(BigDec.zero) { |acc, x| acc + x } }
static mean(a) { sum(a)/a.count }
static prod(a) { a.reduce(BigDec.one) { |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 } }
}

// Initialize static fields.
BigInt.init_()
```