/rust/registry/src/index.crates.io-6f17d22bba15001f/num-integer-0.1.46/src/roots.rs

Source (jump to first uncovered line)
use crate::Integer;
use core::mem;
use num_traits::{checked_pow, PrimInt};

/// 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);
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);
unsigned_roots!(u128);
unsigned_roots!(usize);

Coverage Report

Created: 2025-07-12 07:18

Line	Count	Source (jump to first uncovered line)
1		use crate::Integer;
2		use core::mem;
3		use num_traits::{checked_pow, PrimInt};
4
5		/// Provides methods to compute an integer's square root, cube root,
6		/// and arbitrary `n`th root.
7		pub trait Roots: Integer {
8		/// Returns the truncated principal `n`th root of an integer
9		/// -- `if x >= 0 { ⌊ⁿ√x⌋ } else { ⌈ⁿ√x⌉ }`
10		///
11		/// This is solving for `r` in `rⁿ = x`, rounding toward zero.
12		/// If `x` is positive, the result will satisfy `rⁿ ≤ x < (r+1)ⁿ`.
13		/// If `x` is negative and `n` is odd, then `(r-1)ⁿ < x ≤ rⁿ`.
14		///
15		/// # Panics
16		///
17		/// Panics if `n` is zero:
18		///
19		/// ```should_panic
20		/// # use num_integer::Roots;
21		/// println!("can't compute ⁰√x : {}", 123.nth_root(0));
22		/// ```
23		///
24		/// or if `n` is even and `self` is negative:
25		///
26		/// ```should_panic
27		/// # use num_integer::Roots;
28		/// println!("no imaginary numbers... {}", (-1).nth_root(10));
29		/// ```
30		///
31		/// # Examples
32		///
33		/// ```
34		/// use num_integer::Roots;
35		///
36		/// let x: i32 = 12345;
37		/// assert_eq!(x.nth_root(1), x);
38		/// assert_eq!(x.nth_root(2), x.sqrt());
39		/// assert_eq!(x.nth_root(3), x.cbrt());
40		/// assert_eq!(x.nth_root(4), 10);
41		/// assert_eq!(x.nth_root(13), 2);
42		/// assert_eq!(x.nth_root(14), 1);
43		/// assert_eq!(x.nth_root(std::u32::MAX), 1);
44		///
45		/// assert_eq!(std::i32::MAX.nth_root(30), 2);
46		/// assert_eq!(std::i32::MAX.nth_root(31), 1);
47		/// assert_eq!(std::i32::MIN.nth_root(31), -2);
48		/// assert_eq!((std::i32::MIN + 1).nth_root(31), -1);
49		///
50		/// assert_eq!(std::u32::MAX.nth_root(31), 2);
51		/// assert_eq!(std::u32::MAX.nth_root(32), 1);
52		/// ```
53		fn nth_root(&self, n: u32) -> Self;
54
55		/// Returns the truncated principal square root of an integer -- `⌊√x⌋`
56		///
57		/// This is solving for `r` in `r² = x`, rounding toward zero.
58		/// The result will satisfy `r² ≤ x < (r+1)²`.
59		///
60		/// # Panics
61		///
62		/// Panics if `self` is less than zero:
63		///
64		/// ```should_panic
65		/// # use num_integer::Roots;
66		/// println!("no imaginary numbers... {}", (-1).sqrt());
67		/// ```
68		///
69		/// # Examples
70		///
71		/// ```
72		/// use num_integer::Roots;
73		///
74		/// let x: i32 = 12345;
75		/// assert_eq!((x * x).sqrt(), x);
76		/// assert_eq!((x * x + 1).sqrt(), x);
77		/// assert_eq!((x * x - 1).sqrt(), x - 1);
78		/// ```
79		#[inline]
80	0	fn sqrt(&self) -> Self {
81	0	self.nth_root(2)
82	0	}
83
84		/// Returns the truncated principal cube root of an integer --
85		/// `if x >= 0 { ⌊∛x⌋ } else { ⌈∛x⌉ }`
86		///
87		/// This is solving for `r` in `r³ = x`, rounding toward zero.
88		/// If `x` is positive, the result will satisfy `r³ ≤ x < (r+1)³`.
89		/// If `x` is negative, then `(r-1)³ < x ≤ r³`.
90		///
91		/// # Examples
92		///
93		/// ```
94		/// use num_integer::Roots;
95		///
96		/// let x: i32 = 1234;
97		/// assert_eq!((x * x * x).cbrt(), x);
98		/// assert_eq!((x * x * x + 1).cbrt(), x);
99		/// assert_eq!((x * x * x - 1).cbrt(), x - 1);
100		///
101		/// assert_eq!((-(x * x * x)).cbrt(), -x);
102		/// assert_eq!((-(x * x * x + 1)).cbrt(), -x);
103		/// assert_eq!((-(x * x * x - 1)).cbrt(), -(x - 1));
104		/// ```
105		#[inline]
106	0	fn cbrt(&self) -> Self {
107	0	self.nth_root(3)
108	0	}
109		}
110
111		/// Returns the truncated principal square root of an integer --
112		/// see [Roots::sqrt](trait.Roots.html#method.sqrt).
113		#[inline]
114	0	pub fn sqrt<T: Roots>(x: T) -> T {
115	0	x.sqrt()
116	0	}
117
118		/// Returns the truncated principal cube root of an integer --
119		/// see [Roots::cbrt](trait.Roots.html#method.cbrt).
120		#[inline]
121	0	pub fn cbrt<T: Roots>(x: T) -> T {
122	0	x.cbrt()
123	0	}
124
125		/// Returns the truncated principal `n`th root of an integer --
126		/// see [Roots::nth_root](trait.Roots.html#tymethod.nth_root).
127		#[inline]
128	0	pub fn nth_root<T: Roots>(x: T, n: u32) -> T {
129	0	x.nth_root(n)
130	0	}
131
132		macro_rules! signed_roots {
133		($T:ty, $U:ty) => {
134		impl Roots for $T {
135		#[inline]
136	0	fn nth_root(&self, n: u32) -> Self {
137	0	if *self >= 0 {
138	0	(*self as $U).nth_root(n) as Self
139		} else {
140	0	assert!(n.is_odd(), "even roots of a negative are imaginary");
141	0	-((self.wrapping_neg() as $U).nth_root(n) as Self)
142		}
143	0	} Unexecuted instantiation: <i8 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <i16 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <i32 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <i64 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <i128 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <isize as num_integer::roots::Roots>::nth_root
144
145		#[inline]
146	0	fn sqrt(&self) -> Self {
147	0	assert!(*self >= 0, "the square root of a negative is imaginary");
148	0	(*self as $U).sqrt() as Self
149	0	} Unexecuted instantiation: <i8 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <i16 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <i32 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <i64 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <i128 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <isize as num_integer::roots::Roots>::sqrt
150
151		#[inline]
152	0	fn cbrt(&self) -> Self {
153	0	if *self >= 0 {
154	0	(*self as $U).cbrt() as Self
155		} else {
156	0	-((self.wrapping_neg() as $U).cbrt() as Self)
157		}
158	0	} Unexecuted instantiation: <i8 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <i16 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <i32 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <i64 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <i128 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <isize as num_integer::roots::Roots>::cbrt
159		}
160		};
161		}
162
163		signed_roots!(i8, u8);
164		signed_roots!(i16, u16);
165		signed_roots!(i32, u32);
166		signed_roots!(i64, u64);
167		signed_roots!(i128, u128);
168		signed_roots!(isize, usize);
169
170		#[inline]
171	0	fn fixpoint<T, F>(mut x: T, f: F) -> T
172	0	where
173	0	T: Integer + Copy,
174	0	F: Fn(T) -> T,
175	0	{
176	0	let mut xn = f(x);
177	0	while x < xn {
178	0	x = xn;
179	0	xn = f(x);
180	0	}
181	0	while x > xn {
182	0	x = xn;
183	0	xn = f(x);
184	0	}
185	0	x
186	0	} Unexecuted instantiation: num_integer::roots::fixpoint::<u8, <u8 as num_integer::roots::Roots>::cbrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u8, <u8 as num_integer::roots::Roots>::sqrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u8, <u8 as num_integer::roots::Roots>::nth_root::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<usize, <usize as num_integer::roots::Roots>::cbrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<usize, <usize as num_integer::roots::Roots>::sqrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<usize, <usize as num_integer::roots::Roots>::nth_root::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u32, <u32 as num_integer::roots::Roots>::cbrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u32, <u32 as num_integer::roots::Roots>::sqrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u32, <u32 as num_integer::roots::Roots>::nth_root::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u128, <u128 as num_integer::roots::Roots>::cbrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u128, <u128 as num_integer::roots::Roots>::sqrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u128, <u128 as num_integer::roots::Roots>::nth_root::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u16, <u16 as num_integer::roots::Roots>::cbrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u16, <u16 as num_integer::roots::Roots>::sqrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u16, <u16 as num_integer::roots::Roots>::nth_root::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u64, <u64 as num_integer::roots::Roots>::cbrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u64, <u64 as num_integer::roots::Roots>::sqrt::go::{closure#0}> Unexecuted instantiation: num_integer::roots::fixpoint::<u64, <u64 as num_integer::roots::Roots>::nth_root::go::{closure#0}>
187
188		#[inline]
189	0	fn bits<T>() -> u32 {
190	0	8 * mem::size_of::<T>() as u32
191	0	} Unexecuted instantiation: num_integer::roots::bits::<u8> Unexecuted instantiation: num_integer::roots::bits::<usize> Unexecuted instantiation: num_integer::roots::bits::<u32> Unexecuted instantiation: num_integer::roots::bits::<u128> Unexecuted instantiation: num_integer::roots::bits::<u16> Unexecuted instantiation: num_integer::roots::bits::<u64>
192
193		#[inline]
194	0	fn log2<T: PrimInt>(x: T) -> u32 {
195	0	debug_assert!(x > T::zero());
196	0	bits::<T>() - 1 - x.leading_zeros()
197	0	} Unexecuted instantiation: num_integer::roots::log2::<u8> Unexecuted instantiation: num_integer::roots::log2::<usize> Unexecuted instantiation: num_integer::roots::log2::<u32> Unexecuted instantiation: num_integer::roots::log2::<u128> Unexecuted instantiation: num_integer::roots::log2::<u16> Unexecuted instantiation: num_integer::roots::log2::<u64>
198
199		macro_rules! unsigned_roots {
200		($T:ident) => {
201		impl Roots for $T {
202		#[inline]
203	0	fn nth_root(&self, n: u32) -> Self {
204	0	fn go(a: $T, n: u32) -> $T {
205	0	// Specialize small roots
206	0	match n {
207	0	0 => panic!("can't find a root of degree 0!"),
208	0	1 => return a,
209	0	2 => return a.sqrt(),
210	0	3 => return a.cbrt(),
211	0	_ => (),
212	0	}
213	0
214	0	// The root of values less than 2ⁿ can only be 0 or 1.
215	0	if bits::<$T>() <= n \|\| a < (1 << n) {
216	0	return (a > 0) as $T;
217	0	}
218	0
219	0	if bits::<$T>() > 64 {
220		// 128-bit division is slow, so do a bitwise `nth_root` until it's small enough.
221	0	return if a <= core::u64::MAX as $T {
222	0	(a as u64).nth_root(n) as $T
223		} else {
224	0	let lo = (a >> n).nth_root(n) << 1;
225	0	let hi = lo + 1;
226	0	// 128-bit `checked_mul` also involves division, but we can't always
227	0	// compute `hiⁿ` without risking overflow. Try to avoid it though...
228	0	if hi.next_power_of_two().trailing_zeros() * n >= bits::<$T>() {
229	0	match checked_pow(hi, n as usize) {
230	0	Some(x) if x <= a => hi,
231	0	_ => lo,
232		}
233		} else {
234	0	if hi.pow(n) <= a {
235	0	hi
236		} else {
237	0	lo
238		}
239		}
240		};
241	0	}
242
243		#[cfg(feature = "std")]
244		#[inline]
245	0	fn guess(x: $T, n: u32) -> $T {
246	0	// for smaller inputs, `f64` doesn't justify its cost.
247	0	if bits::<$T>() <= 32 \|\| x <= core::u32::MAX as $T {
248	0	1 << ((log2(x) + n - 1) / n)
249		} else {
250	0	((x as f64).ln() / f64::from(n)).exp() as $T
251		}
252	0	} Unexecuted instantiation: <u8 as num_integer::roots::Roots>::nth_root::go::guess Unexecuted instantiation: <u16 as num_integer::roots::Roots>::nth_root::go::guess Unexecuted instantiation: <u32 as num_integer::roots::Roots>::nth_root::go::guess Unexecuted instantiation: <u64 as num_integer::roots::Roots>::nth_root::go::guess Unexecuted instantiation: <u128 as num_integer::roots::Roots>::nth_root::go::guess Unexecuted instantiation: <usize as num_integer::roots::Roots>::nth_root::go::guess
253
254		#[cfg(not(feature = "std"))]
255		#[inline]
256		fn guess(x: $T, n: u32) -> $T {
257		1 << ((log2(x) + n - 1) / n)
258		}
259
260		// https://en.wikipedia.org/wiki/Nth_root_algorithm
261	0	let n1 = n - 1;
262	0	let next = \|x: $T\| {
263	0	let y = match checked_pow(x, n1 as usize) {
264	0	Some(ax) => a / ax,
265	0	None => 0,
266		};
267	0	(y + x * n1 as $T) / n as $T
268	0	}; Unexecuted instantiation: <u8 as num_integer::roots::Roots>::nth_root::go::{closure#0} Unexecuted instantiation: <u16 as num_integer::roots::Roots>::nth_root::go::{closure#0} Unexecuted instantiation: <u32 as num_integer::roots::Roots>::nth_root::go::{closure#0} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::nth_root::go::{closure#0} Unexecuted instantiation: <u128 as num_integer::roots::Roots>::nth_root::go::{closure#0} Unexecuted instantiation: <usize as num_integer::roots::Roots>::nth_root::go::{closure#0}
269	0	fixpoint(guess(a, n), next)
270	0	} Unexecuted instantiation: <u8 as num_integer::roots::Roots>::nth_root::go Unexecuted instantiation: <u16 as num_integer::roots::Roots>::nth_root::go Unexecuted instantiation: <u32 as num_integer::roots::Roots>::nth_root::go Unexecuted instantiation: <u64 as num_integer::roots::Roots>::nth_root::go Unexecuted instantiation: <u128 as num_integer::roots::Roots>::nth_root::go Unexecuted instantiation: <usize as num_integer::roots::Roots>::nth_root::go
271	0	go(*self, n)
272	0	} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <u8 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <u16 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <u32 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <u64 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <u128 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <usize as num_integer::roots::Roots>::nth_root
273
274		#[inline]
275	0	fn sqrt(&self) -> Self {
276	0	fn go(a: $T) -> $T {
277	0	if bits::<$T>() > 64 {
278		// 128-bit division is slow, so do a bitwise `sqrt` until it's small enough.
279	0	return if a <= core::u64::MAX as $T {
280	0	(a as u64).sqrt() as $T
281		} else {
282	0	let lo = (a >> 2u32).sqrt() << 1;
283	0	let hi = lo + 1;
284	0	if hi * hi <= a {
285	0	hi
286		} else {
287	0	lo
288		}
289		};
290	0	}
291	0
292	0	if a < 4 {
293	0	return (a > 0) as $T;
294	0	}
295
296		#[cfg(feature = "std")]
297		#[inline]
298	0	fn guess(x: $T) -> $T {
299	0	(x as f64).sqrt() as $T
300	0	} Unexecuted instantiation: <u8 as num_integer::roots::Roots>::sqrt::go::guess Unexecuted instantiation: <u16 as num_integer::roots::Roots>::sqrt::go::guess Unexecuted instantiation: <u32 as num_integer::roots::Roots>::sqrt::go::guess Unexecuted instantiation: <u64 as num_integer::roots::Roots>::sqrt::go::guess Unexecuted instantiation: <u128 as num_integer::roots::Roots>::sqrt::go::guess Unexecuted instantiation: <usize as num_integer::roots::Roots>::sqrt::go::guess
301
302		#[cfg(not(feature = "std"))]
303		#[inline]
304		fn guess(x: $T) -> $T {
305		1 << ((log2(x) + 1) / 2)
306		}
307
308		// https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
309	0	let next = \|x: $T\| (a / x + x) >> 1; Unexecuted instantiation: <u8 as num_integer::roots::Roots>::sqrt::go::{closure#0} Unexecuted instantiation: <u16 as num_integer::roots::Roots>::sqrt::go::{closure#0} Unexecuted instantiation: <u32 as num_integer::roots::Roots>::sqrt::go::{closure#0} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::sqrt::go::{closure#0} Unexecuted instantiation: <u128 as num_integer::roots::Roots>::sqrt::go::{closure#0} Unexecuted instantiation: <usize as num_integer::roots::Roots>::sqrt::go::{closure#0}
310	0	fixpoint(guess(a), next)
311	0	} Unexecuted instantiation: <u8 as num_integer::roots::Roots>::sqrt::go Unexecuted instantiation: <u16 as num_integer::roots::Roots>::sqrt::go Unexecuted instantiation: <u32 as num_integer::roots::Roots>::sqrt::go Unexecuted instantiation: <u64 as num_integer::roots::Roots>::sqrt::go Unexecuted instantiation: <u128 as num_integer::roots::Roots>::sqrt::go Unexecuted instantiation: <usize as num_integer::roots::Roots>::sqrt::go
312	0	go(*self)
313	0	} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <usize as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <u8 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <u16 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <u32 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <u64 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <u128 as num_integer::roots::Roots>::sqrt Unexecuted instantiation: <usize as num_integer::roots::Roots>::sqrt
314
315		#[inline]
316	0	fn cbrt(&self) -> Self {
317	0	fn go(a: $T) -> $T {
318	0	if bits::<$T>() > 64 {
319		// 128-bit division is slow, so do a bitwise `cbrt` until it's small enough.
320	0	return if a <= core::u64::MAX as $T {
321	0	(a as u64).cbrt() as $T
322		} else {
323	0	let lo = (a >> 3u32).cbrt() << 1;
324	0	let hi = lo + 1;
325	0	if hi * hi * hi <= a {
326	0	hi
327		} else {
328	0	lo
329		}
330		};
331	0	}
332	0
333	0	if bits::<$T>() <= 32 {
334		// Implementation based on Hacker's Delight `icbrt2`
335	0	let mut x = a;
336	0	let mut y2 = 0;
337	0	let mut y = 0;
338	0	let smax = bits::<$T>() / 3;
339	0	for s in (0..smax + 1).rev() {
340	0	let s = s * 3;
341	0	y2 *= 4;
342	0	y *= 2;
343	0	let b = 3 * (y2 + y) + 1;
344	0	if x >> s >= b {
345	0	x -= b << s;
346	0	y2 += 2 * y + 1;
347	0	y += 1;
348	0	}
349		}
350	0	return y;
351	0	}
352	0
353	0	if a < 8 {
354	0	return (a > 0) as $T;
355	0	}
356	0	if a <= core::u32::MAX as $T {
357	0	return (a as u32).cbrt() as $T;
358	0	}
359
360		#[cfg(feature = "std")]
361		#[inline]
362	0	fn guess(x: $T) -> $T {
363	0	(x as f64).cbrt() as $T
364	0	} Unexecuted instantiation: <u8 as num_integer::roots::Roots>::cbrt::go::guess Unexecuted instantiation: <u16 as num_integer::roots::Roots>::cbrt::go::guess Unexecuted instantiation: <u32 as num_integer::roots::Roots>::cbrt::go::guess Unexecuted instantiation: <u64 as num_integer::roots::Roots>::cbrt::go::guess Unexecuted instantiation: <u128 as num_integer::roots::Roots>::cbrt::go::guess Unexecuted instantiation: <usize as num_integer::roots::Roots>::cbrt::go::guess
365
366		#[cfg(not(feature = "std"))]
367		#[inline]
368		fn guess(x: $T) -> $T {
369		1 << ((log2(x) + 2) / 3)
370		}
371
372		// https://en.wikipedia.org/wiki/Cube_root#Numerical_methods
373	0	let next = \|x: $T\| (a / (x * x) + x * 2) / 3; Unexecuted instantiation: <u8 as num_integer::roots::Roots>::cbrt::go::{closure#0} Unexecuted instantiation: <u16 as num_integer::roots::Roots>::cbrt::go::{closure#0} Unexecuted instantiation: <u32 as num_integer::roots::Roots>::cbrt::go::{closure#0} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::cbrt::go::{closure#0} Unexecuted instantiation: <u128 as num_integer::roots::Roots>::cbrt::go::{closure#0} Unexecuted instantiation: <usize as num_integer::roots::Roots>::cbrt::go::{closure#0}
374	0	fixpoint(guess(a), next)
375	0	} Unexecuted instantiation: <u8 as num_integer::roots::Roots>::cbrt::go Unexecuted instantiation: <u16 as num_integer::roots::Roots>::cbrt::go Unexecuted instantiation: <u32 as num_integer::roots::Roots>::cbrt::go Unexecuted instantiation: <u64 as num_integer::roots::Roots>::cbrt::go Unexecuted instantiation: <u128 as num_integer::roots::Roots>::cbrt::go Unexecuted instantiation: <usize as num_integer::roots::Roots>::cbrt::go
376	0	go(*self)
377	0	} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <u8 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <u16 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <u32 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <u64 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <u128 as num_integer::roots::Roots>::cbrt Unexecuted instantiation: <usize as num_integer::roots::Roots>::cbrt
378		}
379		};
380		}
381
382		unsigned_roots!(u8);
383		unsigned_roots!(u16);
384		unsigned_roots!(u32);
385		unsigned_roots!(u64);
386		unsigned_roots!(u128);
387		unsigned_roots!(usize);