Add Fractional, Real, and RealExt traits and update Round trait. Rename Natural trait to Integer.

[brendanzab]

As part of the numeric trait reform (see issue #4819), I have added the following traits to core::num and implemented them for floating point types:

pub trait Round {
    fn floor(&self) -> Self;
    fn ceil(&self) -> Self;
    fn round(&self) -> Self;
    fn trunc(&self) -> Self;
    fn fract(&self) -> Self;
}

pub trait Fractional: Num
                    + Ord
                    + Round
                    + Quot<Self,Self> {
    fn recip(&self) -> Self;
}

pub trait Real: Signed
              + Fractional {
    // Common Constants
    fn pi() -> Self;
    fn two_pi() -> Self;
    fn frac_pi_2() -> Self;
    fn frac_pi_3() -> Self;
    fn frac_pi_4() -> Self;
    fn frac_pi_6() -> Self;
    fn frac_pi_8() -> Self;
    fn frac_1_pi() -> Self;
    fn frac_2_pi() -> Self;
    fn frac_2_sqrtpi() -> Self;
    fn sqrt2() -> Self;
    fn frac_1_sqrt2() -> Self;
    fn e() -> Self;
    fn log2_e() -> Self;
    fn log10_e() -> Self;
    fn log_2() -> Self;
    fn log_10() -> Self;

    // Exponential functions
    fn pow(&self, n: Self) -> Self;
    fn exp(&self) -> Self;
    fn exp2(&self) -> Self;
    fn expm1(&self) -> Self;
    fn ldexp(&self, n: int) -> Self;
    fn log(&self) -> Self;
    fn log2(&self) -> Self;
    fn log10(&self) -> Self;
    fn log_radix(&self) -> Self;
    fn ilog_radix(&self) -> int;
    fn sqrt(&self) -> Self;
    fn rsqrt(&self) -> Self;
    fn cbrt(&self) -> Self;

    // Angular conversions
    fn to_degrees(&self) -> Self;
    fn to_radians(&self) -> Self;

    // Triganomic functions
    fn hypot(&self, other: Self) -> Self;
    fn sin(&self) -> Self;
    fn cos(&self) -> Self;
    fn tan(&self) -> Self;

    // Inverse triganomic functions
    fn asin(&self) -> Self;
    fn acos(&self) -> Self;
    fn atan(&self) -> Self;
    fn atan2(&self, other: Self) -> Self;

    // Hyperbolic triganomic functions
    fn sinh(&self) -> Self;
    fn cosh(&self) -> Self;
    fn tanh(&self) -> Self;
}

/// Methods that are harder to implement and not commonly used.
pub trait RealExt: Real {
    // Gamma functions
    fn lgamma(&self) -> (int, Self);
    fn tgamma(&self) -> Self;

    // Bessel functions
    fn j0(&self) -> Self;
    fn j1(&self) -> Self;
    fn jn(&self, n: int) -> Self;
    fn y0(&self) -> Self;
    fn y1(&self) -> Self;
    fn yn(&self, n: int) -> Self;
} 

The constants in Real could be associated items in the future (see issue #5527). At the moment I have left the constants in {float|f32|f64}::consts in case folks need to access these at compile time. There are also instances of int in Real and RealExt. In the future these could be replaced with an associated INTEGER type on Real.

Natural has also been renamed to Integer. This is because Natural normally means 'positive integer' in mathematics. It is therefore strange to implement it on signed integer types. Integer is probably a better choice.

I have also switched some of the Integer methods to take borrowed pointers as arguments. This brings them in line with the Quot and Rem traits, and is be better for large Integer types like BigInt and BigUint because they don't need to be copied unnecessarily.

There has also been considerable discussion on the mailing list and IRC about the renaming of the Div and Modulo traits to Quot and Rem. Depending on the outcome of these discussions they might be renamed again.

[brendanzab]

Updated to fix a build error with std::rational::Ratio.

[brson]

👍 Nice work.

[brendanzab]

@brson Cheers! Is it just me or is @bors taking a while on this one?