use core;
use core::mem;
use traits::checked_pow;
use traits::PrimInt;
use Integer;
/// Provides methods to compute an integer's square root, cube root,
/// and arbitrary `n`th root.
pub trait Roots: Integer {
/// Returns the truncated principal `n`th root of an integer
/// -- `if x >= 0 { ⌊ⁿ√x⌋ } else { ⌈ⁿ√x⌉ }`
///
/// This is solving for `r` in `rⁿ = x`, rounding toward zero.
/// If `x` is positive, the result will satisfy `rⁿ ≤ x < (r+1)ⁿ`.
/// If `x` is negative and `n` is odd, then `(r-1)ⁿ < x ≤ rⁿ`.
///
/// # Panics
///
/// Panics if `n` is zero:
///
/// ```should_panic
/// # use num_integer::Roots;
/// println!("can't compute ⁰√x : {}", 123.nth_root(0));
/// ```
///
/// or if `n` is even and `self` is negative:
///
/// ```should_panic
/// # use num_integer::Roots;
/// println!("no imaginary numbers... {}", (-1).nth_root(10));
/// ```
///
/// # Examples
///
/// ```
/// use num_integer::Roots;
///
/// let x: i32 = 12345;
/// assert_eq!(x.nth_root(1), x);
/// assert_eq!(x.nth_root(2), x.sqrt());
/// assert_eq!(x.nth_root(3), x.cbrt());
/// assert_eq!(x.nth_root(4), 10);
/// assert_eq!(x.nth_root(13), 2);
/// assert_eq!(x.nth_root(14), 1);
/// assert_eq!(x.nth_root(std::u32::MAX), 1);
///
/// assert_eq!(std::i32::MAX.nth_root(30), 2);
/// assert_eq!(std::i32::MAX.nth_root(31), 1);
/// assert_eq!(std::i32::MIN.nth_root(31), -2);
/// assert_eq!((std::i32::MIN + 1).nth_root(31), -1);
///
/// assert_eq!(std::u32::MAX.nth_root(31), 2);
/// assert_eq!(std::u32::MAX.nth_root(32), 1);
/// ```
fn nth_root(&self, n: u32) -> Self;
/// Returns the truncated principal square root of an integer -- `⌊√x⌋`
///
/// This is solving for `r` in `r² = x`, rounding toward zero.
/// The result will satisfy `r² ≤ x < (r+1)²`.
///
/// # Panics
///
/// Panics if `self` is less than zero:
///
/// ```should_panic
/// # use num_integer::Roots;
/// println!("no imaginary numbers... {}", (-1).sqrt());
/// ```
///
/// # Examples
///
/// ```
/// use num_integer::Roots;
///
/// let x: i32 = 12345;
/// assert_eq!((x * x).sqrt(), x);
/// assert_eq!((x * x + 1).sqrt(), x);
/// assert_eq!((x * x - 1).sqrt(), x - 1);
/// ```
#[inline]
fn sqrt(&self) -> Self {
self.nth_root(2)
}
/// Returns the truncated principal cube root of an integer --
/// `if x >= 0 { ⌊∛x⌋ } else { ⌈∛x⌉ }`
///
/// This is solving for `r` in `r³ = x`, rounding toward zero.
/// If `x` is positive, the result will satisfy `r³ ≤ x < (r+1)³`.
/// If `x` is negative, then `(r-1)³ < x ≤ r³`.
///
/// # Examples
///
/// ```
/// use num_integer::Roots;
///
/// let x: i32 = 1234;
/// assert_eq!((x * x * x).cbrt(), x);
/// assert_eq!((x * x * x + 1).cbrt(), x);
/// assert_eq!((x * x * x - 1).cbrt(), x - 1);
///
/// assert_eq!((-(x * x * x)).cbrt(), -x);
/// assert_eq!((-(x * x * x + 1)).cbrt(), -x);
/// assert_eq!((-(x * x * x - 1)).cbrt(), -(x - 1));
/// ```
#[inline]
fn cbrt(&self) -> Self {
self.nth_root(3)
}
}
/// Returns the truncated principal square root of an integer --
/// see [Roots::sqrt](trait.Roots.html#method.sqrt).
#[inline]
pub fn sqrt<T: Roots>(x: T) -> T {
x.sqrt()
}
/// Returns the truncated principal cube root of an integer --
/// see [Roots::cbrt](trait.Roots.html#method.cbrt).
#[inline]
pub fn cbrt<T: Roots>(x: T) -> T {
x.cbrt()
}
/// Returns the truncated principal `n`th root of an integer --
/// see [Roots::nth_root](trait.Roots.html#tymethod.nth_root).
#[inline]
pub fn nth_root<T: Roots>(x: T, n: u32) -> T {
x.nth_root(n)
}
macro_rules! signed_roots {
($T:ty, $U:ty) => {
impl Roots for $T {
#[inline]
fn nth_root(&self, n: u32) -> Self {
if *self >= 0 {
(*self as $U).nth_root(n) as Self
} else {
assert!(n.is_odd(), "even roots of a negative are imaginary");
-((self.wrapping_neg() as $U).nth_root(n) as Self)
}
}
#[inline]
fn sqrt(&self) -> Self {
assert!(*self >= 0, "the square root of a negative is imaginary");
(*self as $U).sqrt() as Self
}
#[inline]
fn cbrt(&self) -> Self {
if *self >= 0 {
(*self as $U).cbrt() as Self
} else {
-((self.wrapping_neg() as $U).cbrt() as Self)
}
}
}
};
}
signed_roots!(i8, u8);
signed_roots!(i16, u16);
signed_roots!(i32, u32);
signed_roots!(i64, u64);
#[cfg(has_i128)]
signed_roots!(i128, u128);
signed_roots!(isize, usize);
#[inline]
fn fixpoint<T, F>(mut x: T, f: F) -> T
where
T: Integer + Copy,
F: Fn(T) -> T,
{
let mut xn = f(x);
while x < xn {
x = xn;
xn = f(x);
}
while x > xn {
x = xn;
xn = f(x);
}
x
}
#[inline]
fn bits<T>() -> u32 {
8 * mem::size_of::<T>() as u32
}
#[inline]
fn log2<T: PrimInt>(x: T) -> u32 {
debug_assert!(x > T::zero());
bits::<T>() - 1 - x.leading_zeros()
}
macro_rules! unsigned_roots {
($T:ident) => {
impl Roots for $T {
#[inline]
fn nth_root(&self, n: u32) -> Self {
fn go(a: $T, n: u32) -> $T {
// Specialize small roots
match n {
0 => panic!("can't find a root of degree 0!"),
1 => return a,
2 => return a.sqrt(),
3 => return a.cbrt(),
_ => (),
}
// The root of values less than 2ⁿ can only be 0 or 1.
if bits::<$T>() <= n || a < (1 << n) {
return (a > 0) as $T;
}
if bits::<$T>() > 64 {
// 128-bit division is slow, so do a bitwise `nth_root` until it's small enough.
return if a <= core::u64::MAX as $T {
(a as u64).nth_root(n) as $T
} else {
let lo = (a >> n).nth_root(n) << 1;
let hi = lo + 1;
// 128-bit `checked_mul` also involves division, but we can't always
// compute `hiⁿ` without risking overflow. Try to avoid it though...
if hi.next_power_of_two().trailing_zeros() * n >= bits::<$T>() {
match checked_pow(hi, n as usize) {
Some(x) if x <= a => hi,
_ => lo,
}
} else {
if hi.pow(n) <= a {
hi
} else {
lo
}
}
};
}
#[cfg(feature = "std")]
#[inline]
fn guess(x: $T, n: u32) -> $T {
// for smaller inputs, `f64` doesn't justify its cost.
if bits::<$T>() <= 32 || x <= core::u32::MAX as $T {
1 << ((log2(x) + n - 1) / n)
} else {
((x as f64).ln() / f64::from(n)).exp() as $T
}
}
#[cfg(not(feature = "std"))]
#[inline]
fn guess(x: $T, n: u32) -> $T {
1 << ((log2(x) + n - 1) / n)
}
//
https://en.wikipedia.org/wiki/Nth_root_algorithm
let n1 = n - 1;
let next = |x: $T| {
let y = match checked_pow(x, n1 as usize) {
Some(ax) => a / ax,
None => 0,
};
(y + x * n1 as $T) / n as $T
};
fixpoint(guess(a, n), next)
}
go(*self, n)
}
#[inline]
fn sqrt(&self) -> Self {
fn go(a: $T) -> $T {
if bits::<$T>() > 64 {
// 128-bit division is slow, so do a bitwise `sqrt` until it's small enough.
return if a <= core::u64::MAX as $T {
(a as u64).sqrt() as $T
} else {
let lo = (a >> 2u32).sqrt() << 1;
let hi = lo + 1;
if hi * hi <= a {
hi
} else {
lo
}
};
}
if a < 4 {
return (a > 0) as $T;
}
#[cfg(feature = "std")]
#[inline]
fn guess(x: $T) -> $T {
(x as f64).sqrt() as $T
}
#[cfg(not(feature = "std"))]
#[inline]
fn guess(x: $T) -> $T {
1 << ((log2(x) + 1) / 2)
}
//
https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
let next = |x: $T| (a / x + x) >> 1;
fixpoint(guess(a), next)
}
go(*self)
}
#[inline]
fn cbrt(&self) -> Self {
fn go(a: $T) -> $T {
if bits::<$T>() > 64 {
// 128-bit division is slow, so do a bitwise `cbrt` until it's small enough.
return if a <= core::u64::MAX as $T {
(a as u64).cbrt() as $T
} else {
let lo = (a >> 3u32).cbrt() << 1;
let hi = lo + 1;
if hi * hi * hi <= a {
hi
} else {
lo
}
};
}
if bits::<$T>() <= 32 {
// Implementation based on Hacker's Delight `icbrt2`
let mut x = a;
let mut y2 = 0;
let mut y = 0;
let smax = bits::<$T>() / 3;
for s in (0..smax + 1).rev() {
let s = s * 3;
y2 *= 4;
y *= 2;
let b = 3 * (y2 + y) + 1;
if x >> s >= b {
x -= b << s;
y2 += 2 * y + 1;
y += 1;
}
}
return y;
}
if a < 8 {
return (a > 0) as $T;
}
if a <= core::u32::MAX as $T {
return (a as u32).cbrt() as $T;
}
#[cfg(feature = "std")]
#[inline]
fn guess(x: $T) -> $T {
(x as f64).cbrt() as $T
}
#[cfg(not(feature = "std"))]
#[inline]
fn guess(x: $T) -> $T {
1 << ((log2(x) + 2) / 3)
}
//
https://en.wikipedia.org/wiki/Cube_root#Numerical_methods
let next = |x: $T| (a / (x * x) + x * 2) / 3;
fixpoint(guess(a), next)
}
go(*self)
}
}
};
}
unsigned_roots!(u8);
unsigned_roots!(u16);
unsigned_roots!(u32);
unsigned_roots!(u64);
#[cfg(has_i128)]
unsigned_roots!(u128);
unsigned_roots!(usize);
