Module integer 

Functions of i64s: greatest common divisor, least common multiple, binomial coefficients, powers, square roots and residues. Each gives None where its result isn’t an i64. Their function objects are the ones that floats and complex numbers use: Gcd, Lcm and Residue from tolerant, whose integer forms ignore the tolerance they hold, and Binom and Pow. A slice form appends results to a vector and reports whether every item fitted. Divisor divides many integers by one, with multiplications in place of division instructions. Where every item has the same divisor, Residue’s slice form divides through FloatDivisor, which multiplies by the divisor’s reciprocal in f64 and vectorizes, for divisors and dividends below 2⁵¹ in magnitude. At types of 16 bits or fewer it multiplies 32-bit integers instead, and it takes Divisor for 64-bit dividends of 2⁵¹ or more.

Gcd, Lcm, Binom, Pow and Residue also take the narrower integer types of compact buffers, u8, i16 and i32, which they compute through these functions. A result that doesn’t fit its type fails. Pow floors a negative power.

Structs

Divisor

A fixed nonzero divisor for dividing many integers. Each division multiplies and shifts, where a division instruction would take several times as long. Setting one up costs more than one division.

Functions

binom

The binomial coefficient C(n, k), with the integer continuations C(n, k) = (-1)^k C(k - n - 1, k) for negative n, and C(n, k) = (-1)^(n - k) C(-k - 1, n - k) for negative k. None where the result is outside i64.

decode

The value of each cell of bases.len() digits in digits, in the mixed radix of bases, as APL’s ⊥ gives it, with each step wrapped to 64 bits, as BPL’s integer ⊥ wraps at i64. The first base is never used. Wrapping a value to a narrower width then gives what wrapping each step at that width gives.

encode

The digits of each of values in the mixed radix of bases, as APL’s ⊤ gives them: each value’s digits together, the most significant first. A base of 0 takes all of what remains as its digit. None where a digit isn’t an i64.

gcd

The greatest common divisor of x and y, which is never negative. gcd(0, 0) is 0. None only where it is 2^63, as for gcd(i64::MIN, 0).

lcm

The least common multiple of x and y, with the sign of x × y. It is 0 where either is 0.

mask

The mask whose bitwise and with an integer y gives y’s residue modulo x, where x is a positive power of two: x - 1. In two’s complement, y ∧ (x - 1) is y - x⌊y/x⌋ for every y of every width. None for any other x.

power

x^y. 0^0 is 1. A negative y floors the power, as [wrapped_power] does, with i64’s greatest value for an x of 0. None where the power doesn’t fit an i64.

residue

The residue of y modulo x, y - x⌊y/x⌋, which takes the sign of x. A zero x gives y. Every residue fits an i64.