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// devela::num::dom::int::wrapper::impl_prime
//
//! Implements prime-related methods for [`Int`].
//
// TOC
// - signed|unsigned:
// - is_prime
// - prime_nth
// - prime_pi
// - totient
use super::super::_docs::*;
use crate::{Int, IntError::Overflow, IntResult as Result, is, isize_up, paste, usize_up};
/// Implements prime-related methods for [`Int`].
///
/// # Args
/// $t: the input/output type
/// $up: the upcasted type to do the operations on (for prime_pi)
///
/// $d: the doclink suffix for the method name
macro_rules! __impl_int_prime {
() => {
__impl_int_prime![signed
i8 |i16 | "",
i16 |i32 | "-1",
i32 |i64 | "-2",
i64 |i128 | "-3",
i128 |i128 | "-4",
isize |isize_up | "-5",
];
__impl_int_prime![unsigned
u8 |u16 | "-6",
u16 |u32 | "-7",
u32 |u64 | "-8",
u64 |u128 | "-9",
u128 |u128 | "-10",
usize |usize_up | "-11",
];
};
(signed $( $t:ty | $up:ty | $d:literal ),+ $(,)?) => {
$( __impl_int_prime![@signed $t|$up | $d]; )+
};
(unsigned $( $t:ty | $up:ty | $d:literal ),+ $(,)?) => {
$( __impl_int_prime![@unsigned $t|$up | $d]; )+
};
(
// implements signed ops
@signed $t:ty | $up:ty | $d:literal) => { paste! {
///
#[doc = "# Integer prime-related methods for `" $t "`\n\n"]
#[doc = "- [is_prime](#method.is_prime" $d ")"]
#[doc = "- [prime_nth](#method.prime_nth" $d ")"]
#[doc = "- [prime_pi](#method.prime_pi" $d ")"]
#[doc = "- [totient](#method.totient" $d ")"]
///
impl Int<$t> {
/// Returns `true` if `n` is prime.
///
/// This approach uses optimized trial division, which means it checks
/// only odd numbers starting from 3 and up to the square root of the
/// given number. This is based on the fact that if a number is
/// divisible by a number larger than its square root, the result of the
/// division will be smaller than the square root, and it would have
/// already been checked in previous iterations.
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert![Int(127_" $t ").is_prime()];"]
#[doc = "assert![Int(2_" $t ").is_prime()];"]
#[doc = "assert![!Int(1_" $t ").is_prime()];"]
#[doc = "assert![!Int(-2_" $t ").is_prime()];"]
/// ```
#[must_use]
pub const fn is_prime(self) -> bool {
match self.0 {
..=1 => false,
2..=3 => true,
_ => {
is![self.0 % 2 == 0, return false];
let limit = is![let Ok(s) = self.sqrt_floor(), s.0, unreachable!()];
let mut i = 3;
while i <= limit { is![self.0 % i == 0, return false]; i += 2; }
true
}
}
}
/// Finds the 0-indexed `nth` prime number.
///
/// Note: If `nth` is negative, this function treats it as its absolute
/// value. For example, a value of `-3` will be treated as `3`, and the
/// function will return the 3rd prime number.
/// # Errors
/// Returns [`Overflow`] if the result can't fit the type.
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert_eq![Ok(Int(2)), Int(0_" $t ").prime_nth()];"]
#[doc = "assert_eq![Ok(Int(3)), Int(1_" $t ").prime_nth()];"]
#[doc = "assert_eq![Ok(Int(127)), Int(30_" $t ").prime_nth()];"]
#[doc = "assert_eq![Ok(Int(127)), Int(-30_" $t ").prime_nth()];"]
/// # #[cfg(feature = "_int_i8")]
/// assert![Int(31_i8).prime_nth().is_err()];
/// ```
pub const fn prime_nth(self) -> Result<Int<$t>> {
let [nth, mut count, mut i] = [self.0.abs(), 1, 2];
loop {
if Int(i).is_prime() {
is![count - 1 == nth, return Ok(Int(i))];
count += 1;
}
i = is![let Some(i) = i.checked_add(1), i, return Err(Overflow(None))];
}
}
/// Counts the number of primes upto and including `n`.
///
#[doc = _DOC_INT_NOTATION_PI!()]
///
#[doc = "It upcasts internally to [`" $up "`] for the inner operations."]
/// # Panics
/// It can panic if `n == i128|u128`, at the last iteration of a loop
/// that would take an unfeasable amount of time.
///
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert_eq![1, Int(2_" $t ").prime_pi()];"]
#[doc = "assert_eq![2, Int(3_" $t ").prime_pi()];"]
#[doc = "assert_eq![31, Int(127_" $t ").prime_pi()];"]
#[doc = "assert_eq![0, Int(-5_" $t ").prime_pi()];"]
/// ```
/// # Links
/// - <https://mathworld.wolfram.com/PrimeCountingFunction.html>.
/// - <https://en.wikipedia.org/wiki/Prime-counting_function>.
pub const fn prime_pi(self) -> usize {
let (mut prime_count, mut i) = (0_usize, 0 as $up);
while i <= self.0 as $up {
is![Int(i as $t).is_prime(), prime_count += 1];
i += 1;
}
prime_count
}
/// Counts the number of integers $<|n|$ that are relatively prime to `n`.
///
/// Note: If `n` is negative, this function treats it as its absolute
/// value. For example, a value of `-3` will be treated as `3`.
///
/// # Formulation
/// ## Algorithm
#[doc = _DOC_INT_ALGORITHM_TOTIENT!()]
///
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert_eq![Int(2), Int(4_" $t ").totient()];"]
#[doc = "assert_eq![Int(6), Int(9_" $t ").totient()];"]
#[doc = "assert_eq![Int(12), Int(13_" $t ").totient()];"]
#[doc = "assert_eq![Int(22), Int(-23_" $t ").totient()];"]
#[doc = "assert_eq![Int(2), Int(-3_" $t ").totient()];"]
/// ```
/// # Links
/// - <https://en.wikipedia.org/wiki/Euler%27s_totient_function>.
pub const fn totient(self) -> Int<$t> {
let (mut n, mut result, mut i) = (self.0.abs(), self.0.abs(), 2);
while i * i <= n {
if n % i == 0 {
while n % i == 0 { n /= i; }
result -= result / i;
}
i += 1;
}
is![n > 1, result -= result / n];
Int(result)
}
}
}};
(
// implements unsigned ops
@unsigned $t:ty | $up:ty | $d:literal) => { paste! {
///
#[doc = "# Integer prime-related methods for `" $t "`\n\n"]
#[doc = "- [is_prime](#method.is_prime" $d ")"]
#[doc = "- [prime_nth](#method.prime_nth" $d ")"]
#[doc = "- [prime_pi](#method.prime_pi" $d ")"]
#[doc = "- [totient](#method.totient" $d ")"]
///
impl Int<$t> {
/// Returns `true` if `n` is prime.
///
/// This approach uses optimized trial division, which means it checks
/// only odd numbers starting from 3 and up to the square root of the
/// given number. This is based on the fact that if a number is
/// divisible by a number larger than its square root, the result of the
/// division will be smaller than the square root, and it would have
/// already been checked in previous iterations.
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert![Int(127_" $t ").is_prime()];"]
#[doc = "assert![Int(2_" $t ").is_prime()];"]
#[doc = "assert![!Int(1_" $t ").is_prime()];"]
/// ```
#[must_use]
pub const fn is_prime(self) -> bool {
match self.0 {
..=1 => false,
2..=3 => true,
_ => {
is![self.0 % 2 == 0, return false];
let limit = self.sqrt_floor().0;
let mut i = 3;
while i <= limit { is![self.0 % i == 0, return false]; i += 2; }
true
}
}
}
/// Finds the 0-indexed `nth` prime number.
/// # Errors
/// Returns [`Overflow`] if the result can't fit the type.
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert_eq![Ok(Int(2)), Int(0_" $t ").prime_nth()];"]
#[doc = "assert_eq![Ok(Int(3)), Int(1_" $t ").prime_nth()];"]
#[doc = "assert_eq![Ok(Int(251)), Int(53_" $t ").prime_nth()];"]
/// // assert![Int(54_u8).prime_nth().is_err()];
/// ```
pub const fn prime_nth(self) -> Result<Int<$t>> {
let [nth, mut count, mut i] = [self.0, 1, 2];
loop {
if Int(i).is_prime() {
is![count - 1 == nth, return Ok(Int(i))];
count += 1;
}
i = is![let Some(i) = i.checked_add(1), i, return Err(Overflow(None))];
}
}
/// Counts the number of primes upto and including `n`.
///
#[doc = _DOC_INT_NOTATION_PI!()]
///
#[doc = "It upcasts internally to [`" $up "`] for the inner operations."]
/// # Panics
/// It can panic if `n == i128|u128`, at the last iteration of a loop
/// that would take an unfeasable amount of time.
///
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert_eq![1, Int(2_" $t ").prime_pi()];"]
#[doc = "assert_eq![2, Int(3_" $t ").prime_pi()];"]
#[doc = "assert_eq![31, Int(127_" $t ").prime_pi()];"]
/// ```
/// # Links
/// - <https://mathworld.wolfram.com/PrimeCountingFunction.html>.
/// - <https://en.wikipedia.org/wiki/Prime-counting_function>.
pub const fn prime_pi(self) -> usize {
let (mut prime_count, mut i) = (0_usize, 0 as $up);
while i <= self.0 as $up {
is![Int(i as $t).is_prime(), prime_count += 1];
i += 1;
}
prime_count
}
/// Counts the number of integers $<n$ that are relatively prime to `n`.
///
/// # Formulation
/// ## Algorithm
#[doc = _DOC_INT_ALGORITHM_TOTIENT!()]
///
/// # Examples
/// ```
/// # use devela::Int;
#[doc = "assert_eq![Int(2), Int(4_" $t ").totient()];"]
#[doc = "assert_eq![Int(6), Int(9_" $t ").totient()];"]
#[doc = "assert_eq![Int(12), Int(13_" $t ").totient()];"]
/// ```
/// # Links
/// - <https://en.wikipedia.org/wiki/Euler%27s_totient_function>.
pub const fn totient(self) -> Int<$t> {
let (mut n, mut result, mut i) = (self.0, self.0, 2);
while i * i <= n {
if n % i == 0 {
while n % i == 0 { n /= i; }
result -= result / i;
}
i += 1;
}
is![n > 1, result -= result / n];
Int(result)
}
}
}};
}
__impl_int_prime!();