Module exact 

Exact numbers: arbitrary-precision integers and rationals.

These are the two element types that never round. An Ext is J’s “extended” integer, a num-bigint BigInt; a Rat is a pair of them in lowest terms with a positive denominator.

Functions here are the arithmetic only, plus the exactness tests the languages’ type rules turn on — “is this square exact”, “is this power a whole number”. Where an operation has no exact answer the function says so (None) and the caller widens to float; the type rules and the diagnostics live in verb.rs.

Structs

Rat

A rational number in lowest terms; the denominator is always positive and never zero.

Constants

MAX_BITS

The largest magnitude a power is allowed to reach, in bits. A bignum power grows without warning — 2 ^ 10000000x is a gigabyte — so the arithmetic refuses rather than exhausts the machine.

Functions

exact_root

The exact n-th root, when the argument has one. n must be positive.

exact_sqrt

The exact square root, when the argument has one.

ext_binomial

x ! y on whole numbers: the number of ways to choose x things from y. Follows J for the degenerate cases — a negative or oversized x gives 0, and 0 ! y is 1.

ext_factorial

! y on a whole number: the exact factorial. None for a negative argument (a pole) or one large enough to exhaust the machine.

ext_pow

base ^ exp for a whole nonnegative exponent, refusing a result too big to hold.

ext_residue

x | y: y reduced modulo x, the residue taking x’s sign; 0 | y is y.

ext_to_f64

ext_to_i64

f64_exact

A double as the rational it exactly is: mantissa over a power of two.

f64_to_ext

A double as an exact integer, when it is one.

f64_to_rat

A double as the simplest rational within J’s comparison tolerance of it: the first continued-fraction convergent that is close enough. An integral value is exact, so x: 1e30 keeps every digit the double really holds.

rat_gcd

The greatest common divisor of two rationals: gcd(numerators) over lcm(denominators), which is the largest rational dividing both a whole number of times.

rat_lcm

The least common multiple of two rationals, lcm(numerators) over gcd(denominators); zero when either is zero.

rat_residue

x | y on rationals: y - x * <. y % x, as on the reals.

ratio_to_f64

num / den as the nearest double. Dividing the two to_f64s overflows as soon as either half leaves the double range, so the ratio is scaled by its own bit lengths first.

Type Aliases

Ext

An extended-precision integer (J x).