/rust/registry/src/index.crates.io-1949cf8c6b5b557f/num-integer-0.1.45/src/roots.rs

Source
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);

Coverage Report

Created: 2025-12-31 06:43

Line	Count	Source
1		use core;
2		use core::mem;
3		use traits::checked_pow;
4		use traits::PrimInt;
5		use Integer;
6
7		/// Provides methods to compute an integer's square root, cube root,
8		/// and arbitrary `n`th root.
9		pub trait Roots: Integer {
10		/// Returns the truncated principal `n`th root of an integer
11		/// -- `if x >= 0 { ⌊ⁿ√x⌋ } else { ⌈ⁿ√x⌉ }`
12		///
13		/// This is solving for `r` in `rⁿ = x`, rounding toward zero.
14		/// If `x` is positive, the result will satisfy `rⁿ ≤ x < (r+1)ⁿ`.
15		/// If `x` is negative and `n` is odd, then `(r-1)ⁿ < x ≤ rⁿ`.
16		///
17		/// # Panics
18		///
19		/// Panics if `n` is zero:
20		///
21		/// ```should_panic
22		/// # use num_integer::Roots;
23		/// println!("can't compute ⁰√x : {}", 123.nth_root(0));
24		/// ```
25		///
26		/// or if `n` is even and `self` is negative:
27		///
28		/// ```should_panic
29		/// # use num_integer::Roots;
30		/// println!("no imaginary numbers... {}", (-1).nth_root(10));
31		/// ```
32		///
33		/// # Examples
34		///
35		/// ```
36		/// use num_integer::Roots;
37		///
38		/// let x: i32 = 12345;
39		/// assert_eq!(x.nth_root(1), x);
40		/// assert_eq!(x.nth_root(2), x.sqrt());
41		/// assert_eq!(x.nth_root(3), x.cbrt());
42		/// assert_eq!(x.nth_root(4), 10);
43		/// assert_eq!(x.nth_root(13), 2);
44		/// assert_eq!(x.nth_root(14), 1);
45		/// assert_eq!(x.nth_root(std::u32::MAX), 1);
46		///
47		/// assert_eq!(std::i32::MAX.nth_root(30), 2);
48		/// assert_eq!(std::i32::MAX.nth_root(31), 1);
49		/// assert_eq!(std::i32::MIN.nth_root(31), -2);
50		/// assert_eq!((std::i32::MIN + 1).nth_root(31), -1);
51		///
52		/// assert_eq!(std::u32::MAX.nth_root(31), 2);
53		/// assert_eq!(std::u32::MAX.nth_root(32), 1);
54		/// ```
55		fn nth_root(&self, n: u32) -> Self;
56
57		/// Returns the truncated principal square root of an integer -- `⌊√x⌋`
58		///
59		/// This is solving for `r` in `r² = x`, rounding toward zero.
60		/// The result will satisfy `r² ≤ x < (r+1)²`.
61		///
62		/// # Panics
63		///
64		/// Panics if `self` is less than zero:
65		///
66		/// ```should_panic
67		/// # use num_integer::Roots;
68		/// println!("no imaginary numbers... {}", (-1).sqrt());
69		/// ```
70		///
71		/// # Examples
72		///
73		/// ```
74		/// use num_integer::Roots;
75		///
76		/// let x: i32 = 12345;
77		/// assert_eq!((x * x).sqrt(), x);
78		/// assert_eq!((x * x + 1).sqrt(), x);
79		/// assert_eq!((x * x - 1).sqrt(), x - 1);
80		/// ```
81		#[inline]
82	0	fn sqrt(&self) -> Self {
83	0	self.nth_root(2)
84	0	}
85
86		/// Returns the truncated principal cube root of an integer --
87		/// `if x >= 0 { ⌊∛x⌋ } else { ⌈∛x⌉ }`
88		///
89		/// This is solving for `r` in `r³ = x`, rounding toward zero.
90		/// If `x` is positive, the result will satisfy `r³ ≤ x < (r+1)³`.
91		/// If `x` is negative, then `(r-1)³ < x ≤ r³`.
92		///
93		/// # Examples
94		///
95		/// ```
96		/// use num_integer::Roots;
97		///
98		/// let x: i32 = 1234;
99		/// assert_eq!((x * x * x).cbrt(), x);
100		/// assert_eq!((x * x * x + 1).cbrt(), x);
101		/// assert_eq!((x * x * x - 1).cbrt(), x - 1);
102		///
103		/// assert_eq!((-(x * x * x)).cbrt(), -x);
104		/// assert_eq!((-(x * x * x + 1)).cbrt(), -x);
105		/// assert_eq!((-(x * x * x - 1)).cbrt(), -(x - 1));
106		/// ```
107		#[inline]
108	0	fn cbrt(&self) -> Self {
109	0	self.nth_root(3)
110	0	}
111		}
112
113		/// Returns the truncated principal square root of an integer --
114		/// see [Roots::sqrt](trait.Roots.html#method.sqrt).
115		#[inline]
116	0	pub fn sqrt<T: Roots>(x: T) -> T {
117	0	x.sqrt()
118	0	}
119
120		/// Returns the truncated principal cube root of an integer --
121		/// see [Roots::cbrt](trait.Roots.html#method.cbrt).
122		#[inline]
123	0	pub fn cbrt<T: Roots>(x: T) -> T {
124	0	x.cbrt()
125	0	}
126
127		/// Returns the truncated principal `n`th root of an integer --
128		/// see [Roots::nth_root](trait.Roots.html#tymethod.nth_root).
129		#[inline]
130	0	pub fn nth_root<T: Roots>(x: T, n: u32) -> T {
131	0	x.nth_root(n)
132	0	}
133
134		macro_rules! signed_roots {
135		($T:ty, $U:ty) => {
136		impl Roots for $T {
137		#[inline]
138	0	fn nth_root(&self, n: u32) -> Self {
139	0	if *self >= 0 {
140	0	(*self as $U).nth_root(n) as Self
141		} else {
142	0	assert!(n.is_odd(), "even roots of a negative are imaginary");
143	0	-((self.wrapping_neg() as $U).nth_root(n) as Self)
144		}
145	0	} Unexecuted instantiation: <i8 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 Unexecuted instantiation: <i16 as num_integer::roots::Roots>::nth_root
146
147		#[inline]
148	0	fn sqrt(&self) -> Self {
149	0	assert!(*self >= 0, "the square root of a negative is imaginary");
150	0	(*self as $U).sqrt() as Self
151	0	} Unexecuted instantiation: <i8 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 Unexecuted instantiation: <i16 as num_integer::roots::Roots>::sqrt
152
153		#[inline]
154	0	fn cbrt(&self) -> Self {
155	0	if *self >= 0 {
156	0	(*self as $U).cbrt() as Self
157		} else {
158	0	-((self.wrapping_neg() as $U).cbrt() as Self)
159		}
160	0	} Unexecuted instantiation: <i8 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 Unexecuted instantiation: <i16 as num_integer::roots::Roots>::cbrt
161		}
162		};
163		}
164
165		signed_roots!(i8, u8);
166		signed_roots!(i16, u16);
167		signed_roots!(i32, u32);
168		signed_roots!(i64, u64);
169		#[cfg(has_i128)]
170		signed_roots!(i128, u128);
171		signed_roots!(isize, usize);
172
173		#[inline]
174	0	fn fixpoint<T, F>(mut x: T, f: F) -> T
175	0	where
176	0	T: Integer + Copy,
177	0	F: Fn(T) -> T,
178		{
179	0	let mut xn = f(x);
180	0	while x < xn {
181	0	x = xn;
182	0	xn = f(x);
183	0	}
184	0	while x > xn {
185	0	x = xn;
186	0	xn = f(x);
187	0	}
188	0	x
189	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>::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::<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}>
190
191		#[inline]
192	0	fn bits<T>() -> u32 {
193	0	8 * mem::size_of::<T>() as u32
194	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>
195
196		#[inline]
197	0	fn log2<T: PrimInt>(x: T) -> u32 {
198	0	debug_assert!(x > T::zero());
199	0	bits::<T>() - 1 - x.leading_zeros()
200	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::<u16> Unexecuted instantiation: num_integer::roots::log2::<u64>
201
202		macro_rules! unsigned_roots {
203		($T:ident) => {
204		impl Roots for $T {
205		#[inline]
206	0	fn nth_root(&self, n: u32) -> Self {
207	0	fn go(a: $T, n: u32) -> $T {
208		// Specialize small roots
209	0	match n {
210	0	0 => panic!("can't find a root of degree 0!"),
211	0	1 => return a,
212	0	2 => return a.sqrt(),
213	0	3 => return a.cbrt(),
214	0	_ => (),
215		}
216
217		// The root of values less than 2ⁿ can only be 0 or 1.
218	0	if bits::<$T>() <= n \|\| a < (1 << n) {
219	0	return (a > 0) as $T;
220	0	}
221
222	0	if bits::<$T>() > 64 {
223		// 128-bit division is slow, so do a bitwise `nth_root` until it's small enough.
224	0	return if a <= core::u64::MAX as $T {
225	0	(a as u64).nth_root(n) as $T
226		} else {
227	0	let lo = (a >> n).nth_root(n) << 1;
228	0	let hi = lo + 1;
229		// 128-bit `checked_mul` also involves division, but we can't always
230		// compute `hiⁿ` without risking overflow. Try to avoid it though...
231	0	if hi.next_power_of_two().trailing_zeros() * n >= bits::<$T>() {
232	0	match checked_pow(hi, n as usize) {
233	0	Some(x) if x <= a => hi,
234	0	_ => lo,
235		}
236		} else {
237	0	if hi.pow(n) <= a {
238	0	hi
239		} else {
240	0	lo
241		}
242		}
243		};
244	0	}
245
246		#[cfg(feature = "std")]
247		#[inline]
248	0	fn guess(x: $T, n: u32) -> $T {
249		// for smaller inputs, `f64` doesn't justify its cost.
250	0	if bits::<$T>() <= 32 \|\| x <= core::u32::MAX as $T {
251	0	1 << ((log2(x) + n - 1) / n)
252		} else {
253	0	((x as f64).ln() / f64::from(n)).exp() as $T
254		}
255	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: <usize as num_integer::roots::Roots>::nth_root::go::guess Unexecuted instantiation: <u128 as num_integer::roots::Roots>::nth_root::go::guess
256
257		#[cfg(not(feature = "std"))]
258		#[inline]
259		fn guess(x: $T, n: u32) -> $T {
260		1 << ((log2(x) + n - 1) / n)
261		}
262
263		// https://en.wikipedia.org/wiki/Nth_root_algorithm
264	0	let n1 = n - 1;
265	0	let next = \|x: $T\| {
266	0	let y = match checked_pow(x, n1 as usize) {
267	0	Some(ax) => a / ax,
268	0	None => 0,
269		};
270	0	(y + x * n1 as $T) / n as $T
271	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: <usize as num_integer::roots::Roots>::nth_root::go::{closure#0} Unexecuted instantiation: <u128 as num_integer::roots::Roots>::nth_root::go::{closure#0}
272	0	fixpoint(guess(a, n), next)
273	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
274	0	go(*self, n)
275	0	} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::nth_root Unexecuted instantiation: <u128 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: <usize as num_integer::roots::Roots>::nth_root
276
277		#[inline]
278	0	fn sqrt(&self) -> Self {
279	0	fn go(a: $T) -> $T {
280	0	if bits::<$T>() > 64 {
281		// 128-bit division is slow, so do a bitwise `sqrt` until it's small enough.
282	0	return if a <= core::u64::MAX as $T {
283	0	(a as u64).sqrt() as $T
284		} else {
285	0	let lo = (a >> 2u32).sqrt() << 1;
286	0	let hi = lo + 1;
287	0	if hi * hi <= a {
288	0	hi
289		} else {
290	0	lo
291		}
292		};
293	0	}
294
295	0	if a < 4 {
296	0	return (a > 0) as $T;
297	0	}
298
299		#[cfg(feature = "std")]
300		#[inline]
301	0	fn guess(x: $T) -> $T {
302	0	(x as f64).sqrt() as $T
303	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: <usize as num_integer::roots::Roots>::sqrt::go::guess Unexecuted instantiation: <u128 as num_integer::roots::Roots>::sqrt::go::guess
304
305		#[cfg(not(feature = "std"))]
306		#[inline]
307		fn guess(x: $T) -> $T {
308		1 << ((log2(x) + 1) / 2)
309		}
310
311		// https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
312	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: <usize as num_integer::roots::Roots>::sqrt::go::{closure#0} Unexecuted instantiation: <u128 as num_integer::roots::Roots>::sqrt::go::{closure#0}
313	0	fixpoint(guess(a), next)
314	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
315	0	go(*self)
316	0	} 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
317
318		#[inline]
319	0	fn cbrt(&self) -> Self {
320	0	fn go(a: $T) -> $T {
321	0	if bits::<$T>() > 64 {
322		// 128-bit division is slow, so do a bitwise `cbrt` until it's small enough.
323	0	return if a <= core::u64::MAX as $T {
324	0	(a as u64).cbrt() as $T
325		} else {
326	0	let lo = (a >> 3u32).cbrt() << 1;
327	0	let hi = lo + 1;
328	0	if hi * hi * hi <= a {
329	0	hi
330		} else {
331	0	lo
332		}
333		};
334	0	}
335
336	0	if bits::<$T>() <= 32 {
337		// Implementation based on Hacker's Delight `icbrt2`
338	0	let mut x = a;
339	0	let mut y2 = 0;
340	0	let mut y = 0;
341	0	let smax = bits::<$T>() / 3;
342	0	for s in (0..smax + 1).rev() {
343	0	let s = s * 3;
344	0	y2 *= 4;
345	0	y *= 2;
346	0	let b = 3 * (y2 + y) + 1;
347	0	if x >> s >= b {
348	0	x -= b << s;
349	0	y2 += 2 * y + 1;
350	0	y += 1;
351	0	}
352		}
353	0	return y;
354	0	}
355
356	0	if a < 8 {
357	0	return (a > 0) as $T;
358	0	}
359	0	if a <= core::u32::MAX as $T {
360	0	return (a as u32).cbrt() as $T;
361	0	}
362
363		#[cfg(feature = "std")]
364		#[inline]
365	0	fn guess(x: $T) -> $T {
366	0	(x as f64).cbrt() as $T
367	0	} Unexecuted instantiation: <u64 as num_integer::roots::Roots>::cbrt::go::guess Unexecuted instantiation: <usize as num_integer::roots::Roots>::cbrt::go::guess 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: <u128 as num_integer::roots::Roots>::cbrt::go::guess
368
369		#[cfg(not(feature = "std"))]
370		#[inline]
371		fn guess(x: $T) -> $T {
372		1 << ((log2(x) + 2) / 3)
373		}
374
375		// https://en.wikipedia.org/wiki/Cube_root#Numerical_methods
376	0	let next = \|x: $T\| (a / (x * x) + x * 2) / 3; Unexecuted instantiation: <u64 as num_integer::roots::Roots>::cbrt::go::{closure#0} Unexecuted instantiation: <usize as num_integer::roots::Roots>::cbrt::go::{closure#0} 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: <u128 as num_integer::roots::Roots>::cbrt::go::{closure#0}
377	0	fixpoint(guess(a), next)
378	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
379	0	go(*self)
380	0	} 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
381		}
382		};
383		}
384
385		unsigned_roots!(u8);
386		unsigned_roots!(u16);
387		unsigned_roots!(u32);
388		unsigned_roots!(u64);
389		#[cfg(has_i128)]
390		unsigned_roots!(u128);
391		unsigned_roots!(usize);