auto merge of #6048 : bjz/rust/numeric-traits, r=pcwalton

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:

~~~rust
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](http://smallcultfollowing.com/babysteps/blog/2013/04/03/associated-items-continued/) 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.

12 files changed, 1148 insertions, 319 deletions

diff --git a/src/libcore/core.rc b/src/libcore/core.rc
index 61fbf98a7c6..71bbaf557ce 100644
--- a/src/libcore/core.rc
+++ b/src/libcore/core.rc
@@ -77,9 +77,7 @@ pub use kinds::{Const, Copy, Owned, Durable};
 pub use ops::{Drop};
 #[cfg(stage0)]
 pub use ops::{Add, Sub, Mul, Div, Modulo, Neg, Not};
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 pub use ops::{Add, Sub, Mul, Quot, Rem, Neg, Not};
 pub use ops::{BitAnd, BitOr, BitXor};
 pub use ops::{Shl, Shr, Index};
@@ -105,7 +103,9 @@ pub use iter::{BaseIter, ExtendedIter, EqIter, CopyableIter};
 pub use iter::{CopyableOrderedIter, CopyableNonstrictIter, Times};
 pub use iter::{ExtendedMutableIter};
 
-pub use num::{Num, Signed, Unsigned, Natural, NumCast};
+pub use num::{Num, NumCast};
+pub use num::{Signed, Unsigned, Integer};
+pub use num::{Round, Fractional, Real, RealExt};
 pub use ptr::Ptr;
 pub use to_str::ToStr;
 pub use clone::Clone;
diff --git a/src/libcore/num/f32.rs b/src/libcore/num/f32.rs
index 5d663844e5b..7d5807ba546 100644
--- a/src/libcore/num/f32.rs
+++ b/src/libcore/num/f32.rs
@@ -10,20 +10,10 @@
 
 //! Operations and constants for `f32`
 
-use num::strconv;
-use num::Signed;
-use num;
-use option::Option;
 use from_str;
-use to_str;
-
-#[cfg(notest)] use cmp::{Eq, Ord};
-#[cfg(stage0,notest)]
-use ops::{Add, Sub, Mul, Div, Modulo, Neg};
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
-use ops::{Add, Sub, Mul, Quot, Rem, Neg};
+use libc::c_int;
+use num::strconv;
+use prelude::*;
 
 pub use cmath::c_float_targ_consts::*;
 
@@ -233,6 +223,8 @@ pub fn logarithm(n: f32, b: f32) -> f32 {
     return log2(n) / log2(b);
 }
 
+impl Num for f32 {}
+
 #[cfg(notest)]
 impl Eq for f32 {
     #[inline(always)]
@@ -286,10 +278,7 @@ impl Div<f32,f32> for f32 {
     #[inline(always)]
     fn div(&self, other: &f32) -> f32 { *self / *other }
 }
-
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Quot<f32,f32> for f32 {
     #[inline(always)]
     fn quot(&self, other: &f32) -> f32 { *self / *other }
@@ -300,10 +289,7 @@ impl Modulo<f32,f32> for f32 {
     #[inline(always)]
     fn modulo(&self, other: &f32) -> f32 { *self % *other }
 }
-
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Rem<f32,f32> for f32 {
     #[inline(always)]
     fn rem(&self, other: &f32) -> f32 { *self % *other }
@@ -341,31 +327,188 @@ impl Signed for f32 {
     fn is_negative(&self) -> bool { *self < 0.0 || (1.0 / *self) == neg_infinity }
 }
 
-impl num::Round for f32 {
-    #[inline(always)]
-    fn round(&self, mode: num::RoundMode) -> f32 {
-        match mode {
-            num::RoundDown                           => floor(*self),
-            num::RoundUp                             => ceil(*self),
-            num::RoundToZero   if self.is_negative() => ceil(*self),
-            num::RoundToZero                         => floor(*self),
-            num::RoundFromZero if self.is_negative() => floor(*self),
-            num::RoundFromZero                       => ceil(*self)
-        }
-    }
-
+impl Round for f32 {
+    /// Round half-way cases toward `neg_infinity`
     #[inline(always)]
     fn floor(&self) -> f32 { floor(*self) }
+
+    /// Round half-way cases toward `infinity`
     #[inline(always)]
     fn ceil(&self) -> f32 { ceil(*self) }
+
+    /// Round half-way cases away from `0.0`
     #[inline(always)]
-    fn fract(&self) -> f32 {
-        if self.is_negative() {
-            (*self) - ceil(*self)
-        } else {
-            (*self) - floor(*self)
-        }
-    }
+    fn round(&self) -> f32 { round(*self) }
+
+    /// The integer part of the number (rounds towards `0.0`)
+    #[inline(always)]
+    fn trunc(&self) -> f32 { trunc(*self) }
+
+    ///
+    /// The fractional part of the number, satisfying:
+    ///
+    /// ~~~
+    /// assert!(x == trunc(x) + fract(x))
+    /// ~~~
+    ///
+    #[inline(always)]
+    fn fract(&self) -> f32 { *self - self.trunc() }
+}
+
+impl Fractional for f32 {
+    /// The reciprocal (multiplicative inverse) of the number
+    #[inline(always)]
+    fn recip(&self) -> f32 { 1.0 / *self }
+}
+
+impl Real for f32 {
+    /// Archimedes' constant
+    #[inline(always)]
+    fn pi() -> f32 { 3.14159265358979323846264338327950288 }
+
+    /// 2.0 * pi
+    #[inline(always)]
+    fn two_pi() -> f32 { 6.28318530717958647692528676655900576 }
+
+    /// pi / 2.0
+    #[inline(always)]
+    fn frac_pi_2() -> f32 { 1.57079632679489661923132169163975144 }
+
+    /// pi / 3.0
+    #[inline(always)]
+    fn frac_pi_3() -> f32 { 1.04719755119659774615421446109316763 }
+
+    /// pi / 4.0
+    #[inline(always)]
+    fn frac_pi_4() -> f32 { 0.785398163397448309615660845819875721 }
+
+    /// pi / 6.0
+    #[inline(always)]
+    fn frac_pi_6() -> f32 { 0.52359877559829887307710723054658381 }
+
+    /// pi / 8.0
+    #[inline(always)]
+    fn frac_pi_8() -> f32 { 0.39269908169872415480783042290993786 }
+
+    /// 1 .0/ pi
+    #[inline(always)]
+    fn frac_1_pi() -> f32 { 0.318309886183790671537767526745028724 }
+
+    /// 2.0 / pi
+    #[inline(always)]
+    fn frac_2_pi() -> f32 { 0.636619772367581343075535053490057448 }
+
+    /// 2.0 / sqrt(pi)
+    #[inline(always)]
+    fn frac_2_sqrtpi() -> f32 { 1.12837916709551257389615890312154517 }
+
+    /// sqrt(2.0)
+    #[inline(always)]
+    fn sqrt2() -> f32 { 1.41421356237309504880168872420969808 }
+
+    /// 1.0 / sqrt(2.0)
+    #[inline(always)]
+    fn frac_1_sqrt2() -> f32 { 0.707106781186547524400844362104849039 }
+
+    /// Euler's number
+    #[inline(always)]
+    fn e() -> f32 { 2.71828182845904523536028747135266250 }
+
+    /// log2(e)
+    #[inline(always)]
+    fn log2_e() -> f32 { 1.44269504088896340735992468100189214 }
+
+    /// log10(e)
+    #[inline(always)]
+    fn log10_e() -> f32 { 0.434294481903251827651128918916605082 }
+
+    /// log(2.0)
+    #[inline(always)]
+    fn log_2() -> f32 { 0.693147180559945309417232121458176568 }
+
+    /// log(10.0)
+    #[inline(always)]
+    fn log_10() -> f32 { 2.30258509299404568401799145468436421 }
+
+    #[inline(always)]
+    fn pow(&self, n: f32) -> f32 { pow(*self, n) }
+
+    #[inline(always)]
+    fn exp(&self) -> f32 { exp(*self) }
+
+    #[inline(always)]
+    fn exp2(&self) -> f32 { exp2(*self) }
+
+    #[inline(always)]
+    fn expm1(&self) -> f32 { expm1(*self) }
+
+    #[inline(always)]
+    fn ldexp(&self, n: int) -> f32 { ldexp(*self, n as c_int) }
+
+    #[inline(always)]
+    fn log(&self) -> f32 { ln(*self) }
+
+    #[inline(always)]
+    fn log2(&self) -> f32 { log2(*self) }
+
+    #[inline(always)]
+    fn log10(&self) -> f32 { log10(*self) }
+
+    #[inline(always)]
+    fn log_radix(&self) -> f32 { log_radix(*self) as f32 }
+
+    #[inline(always)]
+    fn ilog_radix(&self) -> int { ilog_radix(*self) as int }
+
+    #[inline(always)]
+    fn sqrt(&self) -> f32 { sqrt(*self) }
+
+    #[inline(always)]
+    fn rsqrt(&self) -> f32 { self.sqrt().recip() }
+
+    #[inline(always)]
+    fn cbrt(&self) -> f32 { cbrt(*self) }
+
+    /// Converts to degrees, assuming the number is in radians
+    #[inline(always)]
+    fn to_degrees(&self) -> f32 { *self * (180.0 / Real::pi::<f32>()) }
+
+    /// Converts to radians, assuming the number is in degrees
+    #[inline(always)]
+    fn to_radians(&self) -> f32 { *self * (Real::pi::<f32>() / 180.0) }
+
+    #[inline(always)]
+    fn hypot(&self, other: f32) -> f32 { hypot(*self, other) }
+
+    #[inline(always)]
+    fn sin(&self) -> f32 { sin(*self) }
+
+    #[inline(always)]
+    fn cos(&self) -> f32 { cos(*self) }
+
+    #[inline(always)]
+    fn tan(&self) -> f32 { tan(*self) }
+
+    #[inline(always)]
+    fn asin(&self) -> f32 { asin(*self) }
+
+    #[inline(always)]
+    fn acos(&self) -> f32 { acos(*self) }
+
+    #[inline(always)]
+    fn atan(&self) -> f32 { atan(*self) }
+
+    #[inline(always)]
+    fn atan2(&self, other: f32) -> f32 { atan2(*self, other) }
+
+    #[inline(always)]
+    fn sinh(&self) -> f32 { sinh(*self) }
+
+    #[inline(always)]
+    fn cosh(&self) -> f32 { cosh(*self) }
+
+    #[inline(always)]
+    fn tanh(&self) -> f32 { tanh(*self) }
 }
 
 /**
@@ -588,6 +731,111 @@ impl num::FromStrRadix for f32 {
 #[cfg(test)]
 mod tests {
     use f32::*;
+    use super::*;
+    use prelude::*;
+
+    macro_rules! assert_fuzzy_eq(
+        ($a:expr, $b:expr) => ({
+            let a = $a, b = $b;
+            if !((a - b).abs() < 1.0e-6) {
+                fail!(fmt!("The values were not approximately equal. Found: %? and %?", a, b));
+            }
+        })
+    )
+
+    #[test]
+    fn test_num() {
+        num::test_num(10f32, 2f32);
+    }
+
+    #[test]
+    fn test_floor() {
+        assert_fuzzy_eq!(1.0f32.floor(), 1.0f32);
+        assert_fuzzy_eq!(1.3f32.floor(), 1.0f32);
+        assert_fuzzy_eq!(1.5f32.floor(), 1.0f32);
+        assert_fuzzy_eq!(1.7f32.floor(), 1.0f32);
+        assert_fuzzy_eq!(0.0f32.floor(), 0.0f32);
+        assert_fuzzy_eq!((-0.0f32).floor(), -0.0f32);
+        assert_fuzzy_eq!((-1.0f32).floor(), -1.0f32);
+        assert_fuzzy_eq!((-1.3f32).floor(), -2.0f32);
+        assert_fuzzy_eq!((-1.5f32).floor(), -2.0f32);
+        assert_fuzzy_eq!((-1.7f32).floor(), -2.0f32);
+    }
+
+    #[test]
+    fn test_ceil() {
+        assert_fuzzy_eq!(1.0f32.ceil(), 1.0f32);
+        assert_fuzzy_eq!(1.3f32.ceil(), 2.0f32);
+        assert_fuzzy_eq!(1.5f32.ceil(), 2.0f32);
+        assert_fuzzy_eq!(1.7f32.ceil(), 2.0f32);
+        assert_fuzzy_eq!(0.0f32.ceil(), 0.0f32);
+        assert_fuzzy_eq!((-0.0f32).ceil(), -0.0f32);
+        assert_fuzzy_eq!((-1.0f32).ceil(), -1.0f32);
+        assert_fuzzy_eq!((-1.3f32).ceil(), -1.0f32);
+        assert_fuzzy_eq!((-1.5f32).ceil(), -1.0f32);
+        assert_fuzzy_eq!((-1.7f32).ceil(), -1.0f32);
+    }
+
+    #[test]
+    fn test_round() {
+        assert_fuzzy_eq!(1.0f32.round(), 1.0f32);
+        assert_fuzzy_eq!(1.3f32.round(), 1.0f32);
+        assert_fuzzy_eq!(1.5f32.round(), 2.0f32);
+        assert_fuzzy_eq!(1.7f32.round(), 2.0f32);
+        assert_fuzzy_eq!(0.0f32.round(), 0.0f32);
+        assert_fuzzy_eq!((-0.0f32).round(), -0.0f32);
+        assert_fuzzy_eq!((-1.0f32).round(), -1.0f32);
+        assert_fuzzy_eq!((-1.3f32).round(), -1.0f32);
+        assert_fuzzy_eq!((-1.5f32).round(), -2.0f32);
+        assert_fuzzy_eq!((-1.7f32).round(), -2.0f32);
+    }
+
+    #[test]
+    fn test_trunc() {
+        assert_fuzzy_eq!(1.0f32.trunc(), 1.0f32);
+        assert_fuzzy_eq!(1.3f32.trunc(), 1.0f32);
+        assert_fuzzy_eq!(1.5f32.trunc(), 1.0f32);
+        assert_fuzzy_eq!(1.7f32.trunc(), 1.0f32);
+        assert_fuzzy_eq!(0.0f32.trunc(), 0.0f32);
+        assert_fuzzy_eq!((-0.0f32).trunc(), -0.0f32);
+        assert_fuzzy_eq!((-1.0f32).trunc(), -1.0f32);
+        assert_fuzzy_eq!((-1.3f32).trunc(), -1.0f32);
+        assert_fuzzy_eq!((-1.5f32).trunc(), -1.0f32);
+        assert_fuzzy_eq!((-1.7f32).trunc(), -1.0f32);
+    }
+
+    #[test]
+    fn test_fract() {
+        assert_fuzzy_eq!(1.0f32.fract(), 0.0f32);
+        assert_fuzzy_eq!(1.3f32.fract(), 0.3f32);
+        assert_fuzzy_eq!(1.5f32.fract(), 0.5f32);
+        assert_fuzzy_eq!(1.7f32.fract(), 0.7f32);
+        assert_fuzzy_eq!(0.0f32.fract(), 0.0f32);
+        assert_fuzzy_eq!((-0.0f32).fract(), -0.0f32);
+        assert_fuzzy_eq!((-1.0f32).fract(), -0.0f32);
+        assert_fuzzy_eq!((-1.3f32).fract(), -0.3f32);
+        assert_fuzzy_eq!((-1.5f32).fract(), -0.5f32);
+        assert_fuzzy_eq!((-1.7f32).fract(), -0.7f32);
+    }
+
+    #[test]
+    fn test_real_consts() {
+        assert_fuzzy_eq!(Real::two_pi::<f32>(), 2f32 * Real::pi::<f32>());
+        assert_fuzzy_eq!(Real::frac_pi_2::<f32>(), Real::pi::<f32>() / 2f32);
+        assert_fuzzy_eq!(Real::frac_pi_3::<f32>(), Real::pi::<f32>() / 3f32);
+        assert_fuzzy_eq!(Real::frac_pi_4::<f32>(), Real::pi::<f32>() / 4f32);
+        assert_fuzzy_eq!(Real::frac_pi_6::<f32>(), Real::pi::<f32>() / 6f32);
+        assert_fuzzy_eq!(Real::frac_pi_8::<f32>(), Real::pi::<f32>() / 8f32);
+        assert_fuzzy_eq!(Real::frac_1_pi::<f32>(), 1f32 / Real::pi::<f32>());
+        assert_fuzzy_eq!(Real::frac_2_pi::<f32>(), 2f32 / Real::pi::<f32>());
+        assert_fuzzy_eq!(Real::frac_2_sqrtpi::<f32>(), 2f32 / Real::pi::<f32>().sqrt());
+        assert_fuzzy_eq!(Real::sqrt2::<f32>(), 2f32.sqrt());
+        assert_fuzzy_eq!(Real::frac_1_sqrt2::<f32>(), 1f32 / 2f32.sqrt());
+        assert_fuzzy_eq!(Real::log2_e::<f32>(), Real::e::<f32>().log2());
+        assert_fuzzy_eq!(Real::log10_e::<f32>(), Real::e::<f32>().log10());
+        assert_fuzzy_eq!(Real::log_2::<f32>(), 2f32.log());
+        assert_fuzzy_eq!(Real::log_10::<f32>(), 10f32.log());
+    }
 
     #[test]
     pub fn test_signed() {
diff --git a/src/libcore/num/f64.rs b/src/libcore/num/f64.rs
index 48f23fe8ba9..3b6198bfc47 100644
--- a/src/libcore/num/f64.rs
+++ b/src/libcore/num/f64.rs
@@ -10,20 +10,10 @@
 
 //! Operations and constants for `f64`
 
-use num::strconv;
-use num::Signed;
-use num;
-use option::Option;
-use to_str;
 use from_str;
-
-#[cfg(notest)] use cmp::{Eq, Ord};
-#[cfg(stage0,notest)]
-use ops::{Add, Sub, Mul, Div, Modulo, Neg};
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
-use ops::{Add, Sub, Mul, Quot, Rem, Neg};
+use libc::c_int;
+use num::strconv;
+use prelude::*;
 
 pub use cmath::c_double_targ_consts::*;
 pub use cmp::{min, max};
@@ -254,6 +244,8 @@ pub fn logarithm(n: f64, b: f64) -> f64 {
     return log2(n) / log2(b);
 }
 
+impl Num for f64 {}
+
 #[cfg(notest)]
 impl Eq for f64 {
     #[inline(always)]
@@ -300,9 +292,7 @@ impl Mul<f64,f64> for f64 {
 impl Div<f64,f64> for f64 {
     fn div(&self, other: &f64) -> f64 { *self / *other }
 }
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Quot<f64,f64> for f64 {
     #[inline(always)]
     fn quot(&self, other: &f64) -> f64 { *self / *other }
@@ -311,9 +301,7 @@ impl Quot<f64,f64> for f64 {
 impl Modulo<f64,f64> for f64 {
     fn modulo(&self, other: &f64) -> f64 { *self % *other }
 }
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Rem<f64,f64> for f64 {
     #[inline(always)]
     fn rem(&self, other: &f64) -> f64 { *self % *other }
@@ -349,31 +337,218 @@ impl Signed for f64 {
     fn is_negative(&self) -> bool { *self < 0.0 || (1.0 / *self) == neg_infinity }
 }
 
-impl num::Round for f64 {
-    #[inline(always)]
-    fn round(&self, mode: num::RoundMode) -> f64 {
-        match mode {
-            num::RoundDown                           => floor(*self),
-            num::RoundUp                             => ceil(*self),
-            num::RoundToZero   if self.is_negative() => ceil(*self),
-            num::RoundToZero                         => floor(*self),
-            num::RoundFromZero if self.is_negative() => floor(*self),
-            num::RoundFromZero                       => ceil(*self)
-        }
-    }
-
+impl Round for f64 {
+    /// Round half-way cases toward `neg_infinity`
     #[inline(always)]
     fn floor(&self) -> f64 { floor(*self) }
+
+    /// Round half-way cases toward `infinity`
     #[inline(always)]
     fn ceil(&self) -> f64 { ceil(*self) }
+
+    /// Round half-way cases away from `0.0`
     #[inline(always)]
-    fn fract(&self) -> f64 {
-        if self.is_negative() {
-            (*self) - ceil(*self)
-        } else {
-            (*self) - floor(*self)
-        }
+    fn round(&self) -> f64 { round(*self) }
+
+    /// The integer part of the number (rounds towards `0.0`)
+    #[inline(always)]
+    fn trunc(&self) -> f64 { trunc(*self) }
+
+    ///
+    /// The fractional part of the number, satisfying:
+    ///
+    /// ~~~
+    /// assert!(x == trunc(x) + fract(x))
+    /// ~~~
+    ///
+    #[inline(always)]
+    fn fract(&self) -> f64 { *self - self.trunc() }
+}
+
+impl Fractional for f64 {
+    /// The reciprocal (multiplicative inverse) of the number
+    #[inline(always)]
+    fn recip(&self) -> f64 { 1.0 / *self }
+}
+
+impl Real for f64 {
+    /// Archimedes' constant
+    #[inline(always)]
+    fn pi() -> f64 { 3.14159265358979323846264338327950288 }
+
+    /// 2.0 * pi
+    #[inline(always)]
+    fn two_pi() -> f64 { 6.28318530717958647692528676655900576 }
+
+    /// pi / 2.0
+    #[inline(always)]
+    fn frac_pi_2() -> f64 { 1.57079632679489661923132169163975144 }
+
+    /// pi / 3.0
+    #[inline(always)]
+    fn frac_pi_3() -> f64 { 1.04719755119659774615421446109316763 }
+
+    /// pi / 4.0
+    #[inline(always)]
+    fn frac_pi_4() -> f64 { 0.785398163397448309615660845819875721 }
+
+    /// pi / 6.0
+    #[inline(always)]
+    fn frac_pi_6() -> f64 { 0.52359877559829887307710723054658381 }
+
+    /// pi / 8.0
+    #[inline(always)]
+    fn frac_pi_8() -> f64 { 0.39269908169872415480783042290993786 }
+
+    /// 1.0 / pi
+    #[inline(always)]
+    fn frac_1_pi() -> f64 { 0.318309886183790671537767526745028724 }
+
+    /// 2.0 / pi
+    #[inline(always)]
+    fn frac_2_pi() -> f64 { 0.636619772367581343075535053490057448 }
+
+    /// 2.0 / sqrt(pi)
+    #[inline(always)]
+    fn frac_2_sqrtpi() -> f64 { 1.12837916709551257389615890312154517 }
+
+    /// sqrt(2.0)
+    #[inline(always)]
+    fn sqrt2() -> f64 { 1.41421356237309504880168872420969808 }
+
+    /// 1.0 / sqrt(2.0)
+    #[inline(always)]
+    fn frac_1_sqrt2() -> f64 { 0.707106781186547524400844362104849039 }
+
+    /// Euler's number
+    #[inline(always)]
+    fn e() -> f64 { 2.71828182845904523536028747135266250 }
+
+    /// log2(e)
+    #[inline(always)]
+    fn log2_e() -> f64 { 1.44269504088896340735992468100189214 }
+
+    /// log10(e)
+    #[inline(always)]
+    fn log10_e() -> f64 { 0.434294481903251827651128918916605082 }
+
+    /// log(2.0)
+    #[inline(always)]
+    fn log_2() -> f64 { 0.693147180559945309417232121458176568 }
+
+    /// log(10.0)
+    #[inline(always)]
+    fn log_10() -> f64 { 2.30258509299404568401799145468436421 }
+
+    #[inline(always)]
+    fn pow(&self, n: f64) -> f64 { pow(*self, n) }
+
+    #[inline(always)]
+    fn exp(&self) -> f64 { exp(*self) }
+
+    #[inline(always)]
+    fn exp2(&self) -> f64 { exp2(*self) }
+
+    #[inline(always)]
+    fn expm1(&self) -> f64 { expm1(*self) }
+
+    #[inline(always)]
+    fn ldexp(&self, n: int) -> f64 { ldexp(*self, n as c_int) }
+
+    #[inline(always)]
+    fn log(&self) -> f64 { ln(*self) }
+
+    #[inline(always)]
+    fn log2(&self) -> f64 { log2(*self) }
+
+    #[inline(always)]
+    fn log10(&self) -> f64 { log10(*self) }
+
+    #[inline(always)]
+    fn log_radix(&self) -> f64 { log_radix(*self) }
+
+    #[inline(always)]
+    fn ilog_radix(&self) -> int { ilog_radix(*self) as int }
+
+    #[inline(always)]
+    fn sqrt(&self) -> f64 { sqrt(*self) }
+
+    #[inline(always)]
+    fn rsqrt(&self) -> f64 { self.sqrt().recip() }
+
+    #[inline(always)]
+    fn cbrt(&self) -> f64 { cbrt(*self) }
+
+    /// Converts to degrees, assuming the number is in radians
+    #[inline(always)]
+    fn to_degrees(&self) -> f64 { *self * (180.0 / Real::pi::<f64>()) }
+
+    /// Converts to radians, assuming the number is in degrees
+    #[inline(always)]
+    fn to_radians(&self) -> f64 { *self * (Real::pi::<f64>() / 180.0) }
+
+    #[inline(always)]
+    fn hypot(&self, other: f64) -> f64 { hypot(*self, other) }
+
+    #[inline(always)]
+    fn sin(&self) -> f64 { sin(*self) }
+
+    #[inline(always)]
+    fn cos(&self) -> f64 { cos(*self) }
+
+    #[inline(always)]
+    fn tan(&self) -> f64 { tan(*self) }
+
+    #[inline(always)]
+    fn asin(&self) -> f64 { asin(*self) }
+
+    #[inline(always)]
+    fn acos(&self) -> f64 { acos(*self) }
+
+    #[inline(always)]
+    fn atan(&self) -> f64 { atan(*self) }
+
+    #[inline(always)]
+    fn atan2(&self, other: f64) -> f64 { atan2(*self, other) }
+
+    #[inline(always)]
+    fn sinh(&self) -> f64 { sinh(*self) }
+
+    #[inline(always)]
+    fn cosh(&self) -> f64 { cosh(*self) }
+
+    #[inline(always)]
+    fn tanh(&self) -> f64 { tanh(*self) }
+}
+
+impl RealExt for f64 {
+    #[inline(always)]
+    fn lgamma(&self) -> (int, f64) {
+        let mut sign = 0;
+        let result = lgamma(*self, &mut sign);
+        (sign as int, result)
     }
+
+    #[inline(always)]
+    fn tgamma(&self) -> f64 { tgamma(*self) }
+
+    #[inline(always)]
+    fn j0(&self) -> f64 { j0(*self) }
+
+    #[inline(always)]
+    fn j1(&self) -> f64 { j1(*self) }
+
+    #[inline(always)]
+    fn jn(&self, n: int) -> f64 { jn(n as c_int, *self) }
+
+    #[inline(always)]
+    fn y0(&self) -> f64 { y0(*self) }
+
+    #[inline(always)]
+    fn y1(&self) -> f64 { y1(*self) }
+
+    #[inline(always)]
+    fn yn(&self, n: int) -> f64 { yn(n as c_int, *self) }
 }
 
 /**
@@ -596,6 +771,112 @@ impl num::FromStrRadix for f64 {
 #[cfg(test)]
 mod tests {
     use f64::*;
+    use super::*;
+    use prelude::*;
+
+    macro_rules! assert_fuzzy_eq(
+        ($a:expr, $b:expr) => ({
+            let a = $a, b = $b;
+            if !((a - b).abs() < 1.0e-6) {
+                fail!(fmt!("The values were not approximately equal. \
+                            Found: %? and expected %?", a, b));
+            }
+        })
+    )
+
+    #[test]
+    fn test_num() {
+        num::test_num(10f64, 2f64);
+    }
+
+    #[test]
+    fn test_floor() {
+        assert_fuzzy_eq!(1.0f64.floor(), 1.0f64);
+        assert_fuzzy_eq!(1.3f64.floor(), 1.0f64);
+        assert_fuzzy_eq!(1.5f64.floor(), 1.0f64);
+        assert_fuzzy_eq!(1.7f64.floor(), 1.0f64);
+        assert_fuzzy_eq!(0.0f64.floor(), 0.0f64);
+        assert_fuzzy_eq!((-0.0f64).floor(), -0.0f64);
+        assert_fuzzy_eq!((-1.0f64).floor(), -1.0f64);
+        assert_fuzzy_eq!((-1.3f64).floor(), -2.0f64);
+        assert_fuzzy_eq!((-1.5f64).floor(), -2.0f64);
+        assert_fuzzy_eq!((-1.7f64).floor(), -2.0f64);
+    }
+
+    #[test]
+    fn test_ceil() {
+        assert_fuzzy_eq!(1.0f64.ceil(), 1.0f64);
+        assert_fuzzy_eq!(1.3f64.ceil(), 2.0f64);
+        assert_fuzzy_eq!(1.5f64.ceil(), 2.0f64);
+        assert_fuzzy_eq!(1.7f64.ceil(), 2.0f64);
+        assert_fuzzy_eq!(0.0f64.ceil(), 0.0f64);
+        assert_fuzzy_eq!((-0.0f64).ceil(), -0.0f64);
+        assert_fuzzy_eq!((-1.0f64).ceil(), -1.0f64);
+        assert_fuzzy_eq!((-1.3f64).ceil(), -1.0f64);
+        assert_fuzzy_eq!((-1.5f64).ceil(), -1.0f64);
+        assert_fuzzy_eq!((-1.7f64).ceil(), -1.0f64);
+    }
+
+    #[test]
+    fn test_round() {
+        assert_fuzzy_eq!(1.0f64.round(), 1.0f64);
+        assert_fuzzy_eq!(1.3f64.round(), 1.0f64);
+        assert_fuzzy_eq!(1.5f64.round(), 2.0f64);
+        assert_fuzzy_eq!(1.7f64.round(), 2.0f64);
+        assert_fuzzy_eq!(0.0f64.round(), 0.0f64);
+        assert_fuzzy_eq!((-0.0f64).round(), -0.0f64);
+        assert_fuzzy_eq!((-1.0f64).round(), -1.0f64);
+        assert_fuzzy_eq!((-1.3f64).round(), -1.0f64);
+        assert_fuzzy_eq!((-1.5f64).round(), -2.0f64);
+        assert_fuzzy_eq!((-1.7f64).round(), -2.0f64);
+    }
+
+    #[test]
+    fn test_trunc() {
+        assert_fuzzy_eq!(1.0f64.trunc(), 1.0f64);
+        assert_fuzzy_eq!(1.3f64.trunc(), 1.0f64);
+        assert_fuzzy_eq!(1.5f64.trunc(), 1.0f64);
+        assert_fuzzy_eq!(1.7f64.trunc(), 1.0f64);
+        assert_fuzzy_eq!(0.0f64.trunc(), 0.0f64);
+        assert_fuzzy_eq!((-0.0f64).trunc(), -0.0f64);
+        assert_fuzzy_eq!((-1.0f64).trunc(), -1.0f64);
+        assert_fuzzy_eq!((-1.3f64).trunc(), -1.0f64);
+        assert_fuzzy_eq!((-1.5f64).trunc(), -1.0f64);
+        assert_fuzzy_eq!((-1.7f64).trunc(), -1.0f64);
+    }
+
+    #[test]
+    fn test_fract() {
+        assert_fuzzy_eq!(1.0f64.fract(), 0.0f64);
+        assert_fuzzy_eq!(1.3f64.fract(), 0.3f64);
+        assert_fuzzy_eq!(1.5f64.fract(), 0.5f64);
+        assert_fuzzy_eq!(1.7f64.fract(), 0.7f64);
+        assert_fuzzy_eq!(0.0f64.fract(), 0.0f64);
+        assert_fuzzy_eq!((-0.0f64).fract(), -0.0f64);
+        assert_fuzzy_eq!((-1.0f64).fract(), -0.0f64);
+        assert_fuzzy_eq!((-1.3f64).fract(), -0.3f64);
+        assert_fuzzy_eq!((-1.5f64).fract(), -0.5f64);
+        assert_fuzzy_eq!((-1.7f64).fract(), -0.7f64);
+    }
+
+    #[test]
+    fn test_real_consts() {
+        assert_fuzzy_eq!(Real::two_pi::<f64>(), 2.0 * Real::pi::<f64>());
+        assert_fuzzy_eq!(Real::frac_pi_2::<f64>(), Real::pi::<f64>() / 2f64);
+        assert_fuzzy_eq!(Real::frac_pi_3::<f64>(), Real::pi::<f64>() / 3f64);
+        assert_fuzzy_eq!(Real::frac_pi_4::<f64>(), Real::pi::<f64>() / 4f64);
+        assert_fuzzy_eq!(Real::frac_pi_6::<f64>(), Real::pi::<f64>() / 6f64);
+        assert_fuzzy_eq!(Real::frac_pi_8::<f64>(), Real::pi::<f64>() / 8f64);
+        assert_fuzzy_eq!(Real::frac_1_pi::<f64>(), 1f64 / Real::pi::<f64>());
+        assert_fuzzy_eq!(Real::frac_2_pi::<f64>(), 2f64 / Real::pi::<f64>());
+        assert_fuzzy_eq!(Real::frac_2_sqrtpi::<f64>(), 2f64 / Real::pi::<f64>().sqrt());
+        assert_fuzzy_eq!(Real::sqrt2::<f64>(), 2f64.sqrt());
+        assert_fuzzy_eq!(Real::frac_1_sqrt2::<f64>(), 1f64 / 2f64.sqrt());
+        assert_fuzzy_eq!(Real::log2_e::<f64>(), Real::e::<f64>().log2());
+        assert_fuzzy_eq!(Real::log10_e::<f64>(), Real::e::<f64>().log10());
+        assert_fuzzy_eq!(Real::log_2::<f64>(), 2f64.log());
+        assert_fuzzy_eq!(Real::log_10::<f64>(), 10f64.log());
+    }
 
     #[test]
     pub fn test_signed() {
diff --git a/src/libcore/num/float.rs b/src/libcore/num/float.rs
index 036d295943c..9c0412b422f 100644
--- a/src/libcore/num/float.rs
+++ b/src/libcore/num/float.rs
@@ -20,21 +20,10 @@
 
 // PORT this must match in width according to architecture
 
-use f64;
-use num::strconv;
-use num::Signed;
-use num;
-use option::Option;
-use to_str;
 use from_str;
-
-#[cfg(notest)] use cmp::{Eq, Ord};
-#[cfg(stage0,notest)]
-use ops::{Add, Sub, Mul, Div, Modulo, Neg};
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
-use ops::{Add, Sub, Mul, Quot, Rem, Neg};
+use libc::c_int;
+use num::strconv;
+use prelude::*;
 
 pub use f64::{add, sub, mul, quot, rem, lt, le, eq, ne, ge, gt};
 pub use f64::logarithm;
@@ -382,6 +371,8 @@ pub fn tan(x: float) -> float {
     f64::tan(x as f64) as float
 }
 
+impl Num for float {}
+
 #[cfg(notest)]
 impl Eq for float {
     #[inline(always)]
@@ -412,37 +403,218 @@ impl num::One for float {
     fn one() -> float { 1.0 }
 }
 
-impl num::Round for float {
-    #[inline(always)]
-    fn round(&self, mode: num::RoundMode) -> float {
-        match mode {
-            num::RoundDown
-                => f64::floor(*self as f64) as float,
-            num::RoundUp
-                => f64::ceil(*self as f64) as float,
-            num::RoundToZero   if self.is_negative()
-                => f64::ceil(*self as f64) as float,
-            num::RoundToZero
-                => f64::floor(*self as f64) as float,
-            num::RoundFromZero if self.is_negative()
-                => f64::floor(*self as f64) as float,
-            num::RoundFromZero
-                => f64::ceil(*self as f64) as float
-        }
-    }
+impl Round for float {
+    /// Round half-way cases toward `neg_infinity`
+    #[inline(always)]
+    fn floor(&self) -> float { floor(*self as f64) as float }
 
+    /// Round half-way cases toward `infinity`
     #[inline(always)]
-    fn floor(&self) -> float { f64::floor(*self as f64) as float}
+    fn ceil(&self) -> float { ceil(*self as f64) as float }
+
+    /// Round half-way cases away from `0.0`
     #[inline(always)]
-    fn ceil(&self) -> float { f64::ceil(*self as f64) as float}
+    fn round(&self) -> float { round(*self as f64) as float }
+
+    /// The integer part of the number (rounds towards `0.0`)
     #[inline(always)]
-    fn fract(&self) -> float {
-        if self.is_negative() {
-            (*self) - (f64::ceil(*self as f64) as float)
-        } else {
-            (*self) - (f64::floor(*self as f64) as float)
-        }
+    fn trunc(&self) -> float { trunc(*self as f64) as float }
+
+    ///
+    /// The fractional part of the number, satisfying:
+    ///
+    /// ~~~
+    /// assert!(x == trunc(x) + fract(x))
+    /// ~~~
+    ///
+    #[inline(always)]
+    fn fract(&self) -> float { *self - self.trunc() }
+}
+
+impl Fractional for float {
+    /// The reciprocal (multiplicative inverse) of the number
+    #[inline(always)]
+    fn recip(&self) -> float { 1.0 / *self }
+}
+
+impl Real for float {
+    /// Archimedes' constant
+    #[inline(always)]
+    fn pi() -> float { 3.14159265358979323846264338327950288 }
+
+    /// 2.0 * pi
+    #[inline(always)]
+    fn two_pi() -> float { 6.28318530717958647692528676655900576 }
+
+    /// pi / 2.0
+    #[inline(always)]
+    fn frac_pi_2() -> float { 1.57079632679489661923132169163975144 }
+
+    /// pi / 3.0
+    #[inline(always)]
+    fn frac_pi_3() -> float { 1.04719755119659774615421446109316763 }
+
+    /// pi / 4.0
+    #[inline(always)]
+    fn frac_pi_4() -> float { 0.785398163397448309615660845819875721 }
+
+    /// pi / 6.0
+    #[inline(always)]
+    fn frac_pi_6() -> float { 0.52359877559829887307710723054658381 }
+
+    /// pi / 8.0
+    #[inline(always)]
+    fn frac_pi_8() -> float { 0.39269908169872415480783042290993786 }
+
+    /// 1.0 / pi
+    #[inline(always)]
+    fn frac_1_pi() -> float { 0.318309886183790671537767526745028724 }
+
+    /// 2.0 / pi
+    #[inline(always)]
+    fn frac_2_pi() -> float { 0.636619772367581343075535053490057448 }
+
+    /// 2 .0/ sqrt(pi)
+    #[inline(always)]
+    fn frac_2_sqrtpi() -> float { 1.12837916709551257389615890312154517 }
+
+    /// sqrt(2.0)
+    #[inline(always)]
+    fn sqrt2() -> float { 1.41421356237309504880168872420969808 }
+
+    /// 1.0 / sqrt(2.0)
+    #[inline(always)]
+    fn frac_1_sqrt2() -> float { 0.707106781186547524400844362104849039 }
+
+    /// Euler's number
+    #[inline(always)]
+    fn e() -> float { 2.71828182845904523536028747135266250 }
+
+    /// log2(e)
+    #[inline(always)]
+    fn log2_e() -> float { 1.44269504088896340735992468100189214 }
+
+    /// log10(e)
+    #[inline(always)]
+    fn log10_e() -> float { 0.434294481903251827651128918916605082 }
+
+    /// log(2.0)
+    #[inline(always)]
+    fn log_2() -> float { 0.693147180559945309417232121458176568 }
+
+    /// log(10.0)
+    #[inline(always)]
+    fn log_10() -> float { 2.30258509299404568401799145468436421 }
+
+    #[inline(always)]
+    fn pow(&self, n: float) -> float { pow(*self as f64, n as f64) as float }
+
+    #[inline(always)]
+    fn exp(&self) -> float { exp(*self as f64) as float }
+
+    #[inline(always)]
+    fn exp2(&self) -> float { exp2(*self as f64) as float }
+
+    #[inline(always)]
+    fn expm1(&self) -> float { expm1(*self as f64) as float }
+
+    #[inline(always)]
+    fn ldexp(&self, n: int) -> float { ldexp(*self as f64, n as c_int) as float }
+
+    #[inline(always)]
+    fn log(&self) -> float { ln(*self as f64) as float }
+
+    #[inline(always)]
+    fn log2(&self) -> float { log2(*self as f64) as float }
+
+    #[inline(always)]
+    fn log10(&self) -> float { log10(*self as f64) as float }
+
+    #[inline(always)]
+    fn log_radix(&self) -> float { log_radix(*self as f64) as float }
+
+    #[inline(always)]
+    fn ilog_radix(&self) -> int { ilog_radix(*self as f64) as int }
+
+    #[inline(always)]
+    fn sqrt(&self) -> float { sqrt(*self) }
+
+    #[inline(always)]
+    fn rsqrt(&self) -> float { self.sqrt().recip() }
+
+    #[inline(always)]
+    fn cbrt(&self) -> float { cbrt(*self as f64) as float }
+
+    /// Converts to degrees, assuming the number is in radians
+    #[inline(always)]
+    fn to_degrees(&self) -> float { *self * (180.0 / Real::pi::<float>()) }
+
+    /// Converts to radians, assuming the number is in degrees
+    #[inline(always)]
+    fn to_radians(&self) -> float { *self * (Real::pi::<float>() / 180.0) }
+
+    #[inline(always)]
+    fn hypot(&self, other: float) -> float { hypot(*self as f64, other as f64) as float }
+
+    #[inline(always)]
+    fn sin(&self) -> float { sin(*self) }
+
+    #[inline(always)]
+    fn cos(&self) -> float { cos(*self) }
+
+    #[inline(always)]
+    fn tan(&self) -> float { tan(*self) }
+
+    #[inline(always)]
+    fn asin(&self) -> float { asin(*self as f64) as float }
+
+    #[inline(always)]
+    fn acos(&self) -> float { acos(*self as f64) as float }
+
+    #[inline(always)]
+    fn atan(&self) -> float { atan(*self) }
+
+    #[inline(always)]
+    fn atan2(&self, other: float) -> float { atan2(*self as f64, other as f64) as float }
+
+    #[inline(always)]
+    fn sinh(&self) -> float { sinh(*self as f64) as float }
+
+    #[inline(always)]
+    fn cosh(&self) -> float { cosh(*self as f64) as float }
+
+    #[inline(always)]
+    fn tanh(&self) -> float { tanh(*self as f64) as float }
+}
+
+impl RealExt for float {
+    #[inline(always)]
+    fn lgamma(&self) -> (int, float) {
+        let mut sign = 0;
+        let result = lgamma(*self as f64, &mut sign);
+        (sign as int, result as float)
     }
+
+    #[inline(always)]
+    fn tgamma(&self) -> float { tgamma(*self as f64) as float }
+
+    #[inline(always)]
+    fn j0(&self) -> float { j0(*self as f64) as float }
+
+    #[inline(always)]
+    fn j1(&self) -> float { j1(*self as f64) as float }
+
+    #[inline(always)]
+    fn jn(&self, n: int) -> float { jn(n as c_int, *self as f64) as float }
+
+    #[inline(always)]
+    fn y0(&self) -> float { y0(*self as f64) as float }
+
+    #[inline(always)]
+    fn y1(&self) -> float { y1(*self as f64) as float }
+
+    #[inline(always)]
+    fn yn(&self, n: int) -> float { yn(n as c_int, *self as f64) as float }
 }
 
 #[cfg(notest)]
@@ -468,9 +640,7 @@ impl Div<float,float> for float {
     #[inline(always)]
     fn div(&self, other: &float) -> float { *self / *other }
 }
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Quot<float,float> for float {
     #[inline(always)]
     fn quot(&self, other: &float) -> float { *self / *other }
@@ -480,9 +650,7 @@ impl Modulo<float,float> for float {
     #[inline(always)]
     fn modulo(&self, other: &float) -> float { *self % *other }
 }
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Rem<float,float> for float {
     #[inline(always)]
     fn rem(&self, other: &float) -> float { *self % *other }
@@ -524,6 +692,109 @@ mod tests {
     use super::*;
     use prelude::*;
 
+    macro_rules! assert_fuzzy_eq(
+        ($a:expr, $b:expr) => ({
+            let a = $a, b = $b;
+            if !((a - b).abs() < 1.0e-6) {
+                fail!(fmt!("The values were not approximately equal. Found: %? and %?", a, b));
+            }
+        })
+    )
+
+    #[test]
+    fn test_num() {
+        num::test_num(10f, 2f);
+    }
+
+    #[test]
+    fn test_floor() {
+        assert_fuzzy_eq!(1.0f.floor(), 1.0f);
+        assert_fuzzy_eq!(1.3f.floor(), 1.0f);
+        assert_fuzzy_eq!(1.5f.floor(), 1.0f);
+        assert_fuzzy_eq!(1.7f.floor(), 1.0f);
+        assert_fuzzy_eq!(0.0f.floor(), 0.0f);
+        assert_fuzzy_eq!((-0.0f).floor(), -0.0f);
+        assert_fuzzy_eq!((-1.0f).floor(), -1.0f);
+        assert_fuzzy_eq!((-1.3f).floor(), -2.0f);
+        assert_fuzzy_eq!((-1.5f).floor(), -2.0f);
+        assert_fuzzy_eq!((-1.7f).floor(), -2.0f);
+    }
+
+    #[test]
+    fn test_ceil() {
+        assert_fuzzy_eq!(1.0f.ceil(), 1.0f);
+        assert_fuzzy_eq!(1.3f.ceil(), 2.0f);
+        assert_fuzzy_eq!(1.5f.ceil(), 2.0f);
+        assert_fuzzy_eq!(1.7f.ceil(), 2.0f);
+        assert_fuzzy_eq!(0.0f.ceil(), 0.0f);
+        assert_fuzzy_eq!((-0.0f).ceil(), -0.0f);
+        assert_fuzzy_eq!((-1.0f).ceil(), -1.0f);
+        assert_fuzzy_eq!((-1.3f).ceil(), -1.0f);
+        assert_fuzzy_eq!((-1.5f).ceil(), -1.0f);
+        assert_fuzzy_eq!((-1.7f).ceil(), -1.0f);
+    }
+
+    #[test]
+    fn test_round() {
+        assert_fuzzy_eq!(1.0f.round(), 1.0f);
+        assert_fuzzy_eq!(1.3f.round(), 1.0f);
+        assert_fuzzy_eq!(1.5f.round(), 2.0f);
+        assert_fuzzy_eq!(1.7f.round(), 2.0f);
+        assert_fuzzy_eq!(0.0f.round(), 0.0f);
+        assert_fuzzy_eq!((-0.0f).round(), -0.0f);
+        assert_fuzzy_eq!((-1.0f).round(), -1.0f);
+        assert_fuzzy_eq!((-1.3f).round(), -1.0f);
+        assert_fuzzy_eq!((-1.5f).round(), -2.0f);
+        assert_fuzzy_eq!((-1.7f).round(), -2.0f);
+    }
+
+    #[test]
+    fn test_trunc() {
+        assert_fuzzy_eq!(1.0f.trunc(), 1.0f);
+        assert_fuzzy_eq!(1.3f.trunc(), 1.0f);
+        assert_fuzzy_eq!(1.5f.trunc(), 1.0f);
+        assert_fuzzy_eq!(1.7f.trunc(), 1.0f);
+        assert_fuzzy_eq!(0.0f.trunc(), 0.0f);
+        assert_fuzzy_eq!((-0.0f).trunc(), -0.0f);
+        assert_fuzzy_eq!((-1.0f).trunc(), -1.0f);
+        assert_fuzzy_eq!((-1.3f).trunc(), -1.0f);
+        assert_fuzzy_eq!((-1.5f).trunc(), -1.0f);
+        assert_fuzzy_eq!((-1.7f).trunc(), -1.0f);
+    }
+
+    #[test]
+    fn test_fract() {
+        assert_fuzzy_eq!(1.0f.fract(), 0.0f);
+        assert_fuzzy_eq!(1.3f.fract(), 0.3f);
+        assert_fuzzy_eq!(1.5f.fract(), 0.5f);
+        assert_fuzzy_eq!(1.7f.fract(), 0.7f);
+        assert_fuzzy_eq!(0.0f.fract(), 0.0f);
+        assert_fuzzy_eq!((-0.0f).fract(), -0.0f);
+        assert_fuzzy_eq!((-1.0f).fract(), -0.0f);
+        assert_fuzzy_eq!((-1.3f).fract(), -0.3f);
+        assert_fuzzy_eq!((-1.5f).fract(), -0.5f);
+        assert_fuzzy_eq!((-1.7f).fract(), -0.7f);
+    }
+
+    #[test]
+    fn test_real_consts() {
+        assert_fuzzy_eq!(Real::two_pi::<float>(), 2f * Real::pi::<float>());
+        assert_fuzzy_eq!(Real::frac_pi_2::<float>(), Real::pi::<float>() / 2f);
+        assert_fuzzy_eq!(Real::frac_pi_3::<float>(), Real::pi::<float>() / 3f);
+        assert_fuzzy_eq!(Real::frac_pi_4::<float>(), Real::pi::<float>() / 4f);
+        assert_fuzzy_eq!(Real::frac_pi_6::<float>(), Real::pi::<float>() / 6f);
+        assert_fuzzy_eq!(Real::frac_pi_8::<float>(), Real::pi::<float>() / 8f);
+        assert_fuzzy_eq!(Real::frac_1_pi::<float>(), 1f / Real::pi::<float>());
+        assert_fuzzy_eq!(Real::frac_2_pi::<float>(), 2f / Real::pi::<float>());
+        assert_fuzzy_eq!(Real::frac_2_sqrtpi::<float>(), 2f / Real::pi::<float>().sqrt());
+        assert_fuzzy_eq!(Real::sqrt2::<float>(), 2f.sqrt());
+        assert_fuzzy_eq!(Real::frac_1_sqrt2::<float>(), 1f / 2f.sqrt());
+        assert_fuzzy_eq!(Real::log2_e::<float>(), Real::e::<float>().log2());
+        assert_fuzzy_eq!(Real::log10_e::<float>(), Real::e::<float>().log10());
+        assert_fuzzy_eq!(Real::log_2::<float>(), 2f.log());
+        assert_fuzzy_eq!(Real::log_10::<float>(), 10f.log());
+    }
+
     #[test]
     pub fn test_signed() {
         assert_eq!(infinity.abs(), infinity);
@@ -704,15 +975,6 @@ mod tests {
         assert_eq!(to_str_digits(infinity, 10u), ~"inf");
         assert_eq!(to_str_digits(-infinity, 10u), ~"-inf");
     }
-
-    #[test]
-    pub fn test_round() {
-        assert_eq!(round(5.8), 6.0);
-        assert_eq!(round(5.2), 5.0);
-        assert_eq!(round(3.0), 3.0);
-        assert_eq!(round(2.5), 3.0);
-        assert_eq!(round(-3.5), -4.0);
-    }
 }
 
 //
diff --git a/src/libcore/num/int-template.rs b/src/libcore/num/int-template.rs
index 426ed8a8b0f..f9edf1cefc8 100644
--- a/src/libcore/num/int-template.rs
+++ b/src/libcore/num/int-template.rs
@@ -10,12 +10,9 @@
 
 use T = self::inst::T;
 
-use to_str::ToStr;
 use from_str::FromStr;
 use num::{ToStrRadix, FromStrRadix};
 use num::strconv;
-use num::Signed;
-use num;
 use prelude::*;
 
 pub use cmp::{min, max};
@@ -133,6 +130,8 @@ pub fn compl(i: T) -> T {
 #[inline(always)]
 pub fn abs(i: T) -> T { i.abs() }
 
+impl Num for T {}
+
 #[cfg(notest)]
 impl Ord for T {
     #[inline(always)]
@@ -186,10 +185,7 @@ impl Div<T,T> for T {
     #[inline(always)]
     fn div(&self, other: &T) -> T { *self / *other }
 }
-
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Quot<T,T> for T {
     /**
      * Returns the integer quotient, truncated towards 0. As this behaviour reflects
@@ -218,10 +214,7 @@ impl Modulo<T,T> for T {
     #[inline(always)]
     fn modulo(&self, other: &T) -> T { *self % *other }
 }
-
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Rem<T,T> for T {
     /**
      * Returns the integer remainder after division, satisfying:
@@ -286,7 +279,7 @@ impl Signed for T {
     fn is_negative(&self) -> bool { *self < 0 }
 }
 
-impl Natural for T {
+impl Integer for T {
     /**
      * Floored integer division
      *
@@ -305,13 +298,13 @@ impl Natural for T {
      * ~~~
      */
     #[inline(always)]
-    fn div(&self, other: T) -> T {
+    fn div(&self, other: &T) -> T {
         // Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
         // December 2001](http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
         match self.quot_rem(other) {
-            (q, r) if (r > 0 && other < 0)
-                   || (r < 0 && other > 0) => q - 1,
-            (q, _)                         => q,
+            (q, r) if (r > 0 && *other < 0)
+                   || (r < 0 && *other > 0) => q - 1,
+            (q, _)                          => q,
         }
     }
 
@@ -337,32 +330,32 @@ impl Natural for T {
      * ~~~
      */
     #[inline(always)]
-    fn modulo(&self, other: T) -> T {
+    fn modulo(&self, other: &T) -> T {
         // Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
         // December 2001](http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
-        match *self % other {
-            r if (r > 0 && other < 0)
-              || (r < 0 && other > 0) => r + other,
-            r                         => r,
+        match *self % *other {
+            r if (r > 0 && *other < 0)
+              || (r < 0 && *other > 0) => r + *other,
+            r                          => r,
         }
     }
 
     /// Calculates `div` and `modulo` simultaneously
     #[inline(always)]
-    fn div_mod(&self, other: T) -> (T,T) {
+    fn div_mod(&self, other: &T) -> (T,T) {
         // Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
         // December 2001](http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
         match self.quot_rem(other) {
-            (q, r) if (r > 0 && other < 0)
-                   || (r < 0 && other > 0) => (q - 1, r + other),
-            (q, r)                         => (q, r),
+            (q, r) if (r > 0 && *other < 0)
+                   || (r < 0 && *other > 0) => (q - 1, r + *other),
+            (q, r)                          => (q, r),
         }
     }
 
     /// Calculates `quot` (`\`) and `rem` (`%`) simultaneously
     #[inline(always)]
-    fn quot_rem(&self, other: T) -> (T,T) {
-        (*self / other, *self % other)
+    fn quot_rem(&self, other: &T) -> (T,T) {
+        (*self / *other, *self % *other)
     }
 
     /**
@@ -371,9 +364,9 @@ impl Natural for T {
      * The result is always positive
      */
     #[inline(always)]
-    fn gcd(&self, other: T) -> T {
+    fn gcd(&self, other: &T) -> T {
         // Use Euclid's algorithm
-        let mut m = *self, n = other;
+        let mut m = *self, n = *other;
         while m != 0 {
             let temp = m;
             m = n % temp;
@@ -386,17 +379,17 @@ impl Natural for T {
      * Calculates the Lowest Common Multiple (LCM) of the number and `other`
      */
     #[inline(always)]
-    fn lcm(&self, other: T) -> T {
-        ((*self * other) / self.gcd(other)).abs() // should not have to recaluculate abs
+    fn lcm(&self, other: &T) -> T {
+        ((*self * *other) / self.gcd(other)).abs() // should not have to recaluculate abs
     }
 
     /// Returns `true` if the number can be divided by `other` without leaving a remainder
     #[inline(always)]
-    fn divisible_by(&self, other: T) -> bool { *self % other == 0 }
+    fn divisible_by(&self, other: &T) -> bool { *self % *other == 0 }
 
     /// Returns `true` if the number is divisible by `2`
     #[inline(always)]
-    fn is_even(&self) -> bool { self.divisible_by(2) }
+    fn is_even(&self) -> bool { self.divisible_by(&2) }
 
     /// Returns `true` if the number is not divisible by `2`
     #[inline(always)]
@@ -523,6 +516,11 @@ mod tests {
     use prelude::*;
 
     #[test]
+    fn test_num() {
+        num::test_num(10 as T, 2 as T);
+    }
+
+    #[test]
     pub fn test_signed() {
         assert_eq!((1 as T).abs(), 1 as T);
         assert_eq!((0 as T).abs(), 0 as T);
@@ -564,7 +562,7 @@ mod tests {
         fn test_nd_qr(nd: (T,T), qr: (T,T)) {
             let (n,d) = nd;
             let separate_quot_rem = (n / d, n % d);
-            let combined_quot_rem = n.quot_rem(d);
+            let combined_quot_rem = n.quot_rem(&d);
 
             assert_eq!(separate_quot_rem, qr);
             assert_eq!(combined_quot_rem, qr);
@@ -588,8 +586,8 @@ mod tests {
     fn test_div_mod() {
         fn test_nd_dm(nd: (T,T), dm: (T,T)) {
             let (n,d) = nd;
-            let separate_div_mod = (n.div(d), n.modulo(d));
-            let combined_div_mod = n.div_mod(d);
+            let separate_div_mod = (n.div(&d), n.modulo(&d));
+            let combined_div_mod = n.div_mod(&d);
 
             assert_eq!(separate_div_mod, dm);
             assert_eq!(combined_div_mod, dm);
@@ -611,26 +609,26 @@ mod tests {
 
     #[test]
     fn test_gcd() {
-        assert_eq!((10 as T).gcd(2), 2 as T);
-        assert_eq!((10 as T).gcd(3), 1 as T);
-        assert_eq!((0 as T).gcd(3), 3 as T);
-        assert_eq!((3 as T).gcd(3), 3 as T);
-        assert_eq!((56 as T).gcd(42), 14 as T);
-        assert_eq!((3 as T).gcd(-3), 3 as T);
-        assert_eq!((-6 as T).gcd(3), 3 as T);
-        assert_eq!((-4 as T).gcd(-2), 2 as T);
+        assert_eq!((10 as T).gcd(&2), 2 as T);
+        assert_eq!((10 as T).gcd(&3), 1 as T);
+        assert_eq!((0 as T).gcd(&3), 3 as T);
+        assert_eq!((3 as T).gcd(&3), 3 as T);
+        assert_eq!((56 as T).gcd(&42), 14 as T);
+        assert_eq!((3 as T).gcd(&-3), 3 as T);
+        assert_eq!((-6 as T).gcd(&3), 3 as T);
+        assert_eq!((-4 as T).gcd(&-2), 2 as T);
     }
 
     #[test]
     fn test_lcm() {
-        assert_eq!((1 as T).lcm(0), 0 as T);
-        assert_eq!((0 as T).lcm(1), 0 as T);
-        assert_eq!((1 as T).lcm(1), 1 as T);
-        assert_eq!((-1 as T).lcm(1), 1 as T);
-        assert_eq!((1 as T).lcm(-1), 1 as T);
-        assert_eq!((-1 as T).lcm(-1), 1 as T);
-        assert_eq!((8 as T).lcm(9), 72 as T);
-        assert_eq!((11 as T).lcm(5), 55 as T);
+        assert_eq!((1 as T).lcm(&0), 0 as T);
+        assert_eq!((0 as T).lcm(&1), 0 as T);
+        assert_eq!((1 as T).lcm(&1), 1 as T);
+        assert_eq!((-1 as T).lcm(&1), 1 as T);
+        assert_eq!((1 as T).lcm(&-1), 1 as T);
+        assert_eq!((-1 as T).lcm(&-1), 1 as T);
+        assert_eq!((8 as T).lcm(&9), 72 as T);
+        assert_eq!((11 as T).lcm(&5), 55 as T);
     }
 
     #[test]
diff --git a/src/libcore/num/num.rs b/src/libcore/num/num.rs
index 577bb3f0f15..e19afdc69c3 100644
--- a/src/libcore/num/num.rs
+++ b/src/libcore/num/num.rs
@@ -16,9 +16,7 @@ use ops::{Add, Sub, Mul, Neg};
 use Quot = ops::Div;
 #[cfg(stage0)]
 use Rem = ops::Modulo;
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 use ops::{Add, Sub, Mul, Quot, Rem, Neg};
 use option::Option;
 use kinds::Copy;
@@ -33,30 +31,18 @@ pub trait Num: Eq + Zero + One
              + Quot<Self,Self>
              + Rem<Self,Self> {}
 
-impl Num for u8 {}
-impl Num for u16 {}
-impl Num for u32 {}
-impl Num for u64 {}
-impl Num for uint {}
-impl Num for i8 {}
-impl Num for i16 {}
-impl Num for i32 {}
-impl Num for i64 {}
-impl Num for int {}
-impl Num for f32 {}
-impl Num for f64 {}
-impl Num for float {}
-
 pub trait IntConvertible {
     fn to_int(&self) -> int;
     fn from_int(n: int) -> Self;
 }
 
 pub trait Zero {
+    // FIXME (#5527): These should be associated constants
     fn zero() -> Self;
 }
 
 pub trait One {
+    // FIXME (#5527): These should be associated constants
     fn one() -> Self;
 }
 
@@ -75,35 +61,115 @@ pub fn abs<T:Ord + Zero + Neg<T>>(v: T) -> T {
     if v < Zero::zero() { v.neg() } else { v }
 }
 
-pub trait Natural: Num
+pub trait Integer: Num
                  + Ord
                  + Quot<Self,Self>
                  + Rem<Self,Self> {
-    fn div(&self, other: Self) -> Self;
-    fn modulo(&self, other: Self) -> Self;
-    fn div_mod(&self, other: Self) -> (Self,Self);
-    fn quot_rem(&self, other: Self) -> (Self,Self);
-
-    fn gcd(&self, other: Self) -> Self;
-    fn lcm(&self, other: Self) -> Self;
-    fn divisible_by(&self, other: Self) -> bool;
+    fn div(&self, other: &Self) -> Self;
+    fn modulo(&self, other: &Self) -> Self;
+    fn div_mod(&self, other: &Self) -> (Self,Self);
+    fn quot_rem(&self, other: &Self) -> (Self,Self);
+
+    fn gcd(&self, other: &Self) -> Self;
+    fn lcm(&self, other: &Self) -> Self;
+    fn divisible_by(&self, other: &Self) -> bool;
     fn is_even(&self) -> bool;
     fn is_odd(&self) -> bool;
 }
 
 pub trait Round {
-    fn round(&self, mode: RoundMode) -> Self;
-
     fn floor(&self) -> Self;
-    fn ceil(&self)  -> Self;
+    fn ceil(&self) -> Self;
+    fn round(&self) -> Self;
+    fn trunc(&self) -> Self;
     fn fract(&self) -> Self;
 }
 
-pub enum RoundMode {
-    RoundDown,
-    RoundUp,
-    RoundToZero,
-    RoundFromZero
+pub trait Fractional: Num
+                    + Ord
+                    + Round
+                    + Quot<Self,Self> {
+    fn recip(&self) -> Self;
+}
+
+pub trait Real: Signed
+              + Fractional {
+    // FIXME (#5527): usages of `int` should be replaced with an associated
+    // integer type once these are implemented
+
+    // Common Constants
+    // FIXME (#5527): These should be associated 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 {
+    // FIXME (#5527): usages of `int` should be replaced with an associated
+    // integer type once these are implemented
+
+    // 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;
 }
 
 /**
@@ -230,8 +296,9 @@ pub fn pow_with_uint<T:NumCast+One+Zero+Copy+Quot<T,T>+Mul<T,T>>(
     total
 }
 
+/// Helper function for testing numeric operations
 #[cfg(stage0,test)]
-fn test_num<T:Num + NumCast>(ten: T, two: T) {
+pub fn test_num<T:Num + NumCast>(ten: T, two: T) {
     assert_eq!(ten.add(&two),    cast(12));
     assert_eq!(ten.sub(&two),    cast(8));
     assert_eq!(ten.mul(&two),    cast(20));
@@ -247,7 +314,7 @@ fn test_num<T:Num + NumCast>(ten: T, two: T) {
 #[cfg(stage1,test)]
 #[cfg(stage2,test)]
 #[cfg(stage3,test)]
-fn test_num<T:Num + NumCast>(ten: T, two: T) {
+pub fn test_num<T:Num + NumCast>(ten: T, two: T) {
     assert_eq!(ten.add(&two),  cast(12));
     assert_eq!(ten.sub(&two),  cast(8));
     assert_eq!(ten.mul(&two),  cast(20));
@@ -261,20 +328,6 @@ fn test_num<T:Num + NumCast>(ten: T, two: T) {
     assert_eq!(ten.rem(&two),  ten % two);
 }
 
-#[test] fn test_u8_num()    { test_num(10u8,  2u8)  }
-#[test] fn test_u16_num()   { test_num(10u16, 2u16) }
-#[test] fn test_u32_num()   { test_num(10u32, 2u32) }
-#[test] fn test_u64_num()   { test_num(10u64, 2u64) }
-#[test] fn test_uint_num()  { test_num(10u,   2u)   }
-#[test] fn test_i8_num()    { test_num(10i8,  2i8)  }
-#[test] fn test_i16_num()   { test_num(10i16, 2i16) }
-#[test] fn test_i32_num()   { test_num(10i32, 2i32) }
-#[test] fn test_i64_num()   { test_num(10i64, 2i64) }
-#[test] fn test_int_num()   { test_num(10i,   2i)   }
-#[test] fn test_f32_num()   { test_num(10f32, 2f32) }
-#[test] fn test_f64_num()   { test_num(10f64, 2f64) }
-#[test] fn test_float_num() { test_num(10f,   2f)   }
-
 macro_rules! test_cast_20(
     ($_20:expr) => ({
         let _20 = $_20;
diff --git a/src/libcore/num/uint-template.rs b/src/libcore/num/uint-template.rs
index a0da84a8c53..96019ddd564 100644
--- a/src/libcore/num/uint-template.rs
+++ b/src/libcore/num/uint-template.rs
@@ -11,13 +11,9 @@
 use T = self::inst::T;
 use T_SIGNED = self::inst::T_SIGNED;
 
-use to_str::ToStr;
 use from_str::FromStr;
 use num::{ToStrRadix, FromStrRadix};
 use num::strconv;
-use num::Unsigned;
-use num;
-use option::Option;
 use prelude::*;
 
 pub use cmp::{min, max};
@@ -100,6 +96,8 @@ pub fn compl(i: T) -> T {
     max_value ^ i
 }
 
+impl Num for T {}
+
 #[cfg(notest)]
 impl Ord for T {
     #[inline(always)]
@@ -153,10 +151,7 @@ impl Div<T,T> for T {
     #[inline(always)]
     fn div(&self, other: &T) -> T { *self / *other }
 }
-
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Quot<T,T> for T {
     #[inline(always)]
     fn quot(&self, other: &T) -> T { *self / *other }
@@ -167,10 +162,7 @@ impl Modulo<T,T> for T {
     #[inline(always)]
     fn modulo(&self, other: &T) -> T { *self % *other }
 }
-
-#[cfg(stage1,notest)]
-#[cfg(stage2,notest)]
-#[cfg(stage3,notest)]
+#[cfg(not(stage0),notest)]
 impl Rem<T,T> for T {
     #[inline(always)]
     fn rem(&self, other: &T) -> T { *self % *other }
@@ -184,32 +176,32 @@ impl Neg<T> for T {
 
 impl Unsigned for T {}
 
-impl Natural for T {
+impl Integer for T {
     /// Unsigned integer division. Returns the same result as `quot` (`/`).
     #[inline(always)]
-    fn div(&self, other: T) -> T { *self / other }
+    fn div(&self, other: &T) -> T { *self / *other }
 
     /// Unsigned integer modulo operation. Returns the same result as `rem` (`%`).
     #[inline(always)]
-    fn modulo(&self, other: T) -> T { *self / other }
+    fn modulo(&self, other: &T) -> T { *self / *other }
 
     /// Calculates `div` and `modulo` simultaneously
     #[inline(always)]
-    fn div_mod(&self, other: T) -> (T,T) {
-        (*self / other, *self % other)
+    fn div_mod(&self, other: &T) -> (T,T) {
+        (*self / *other, *self % *other)
     }
 
     /// Calculates `quot` (`\`) and `rem` (`%`) simultaneously
     #[inline(always)]
-    fn quot_rem(&self, other: T) -> (T,T) {
-        (*self / other, *self % other)
+    fn quot_rem(&self, other: &T) -> (T,T) {
+        (*self / *other, *self % *other)
     }
 
     /// Calculates the Greatest Common Divisor (GCD) of the number and `other`
     #[inline(always)]
-    fn gcd(&self, other: T) -> T {
+    fn gcd(&self, other: &T) -> T {
         // Use Euclid's algorithm
-        let mut m = *self, n = other;
+        let mut m = *self, n = *other;
         while m != 0 {
             let temp = m;
             m = n % temp;
@@ -220,17 +212,17 @@ impl Natural for T {
 
     /// Calculates the Lowest Common Multiple (LCM) of the number and `other`
     #[inline(always)]
-    fn lcm(&self, other: T) -> T {
-        (*self * other) / self.gcd(other)
+    fn lcm(&self, other: &T) -> T {
+        (*self * *other) / self.gcd(other)
     }
 
     /// Returns `true` if the number can be divided by `other` without leaving a remainder
     #[inline(always)]
-    fn divisible_by(&self, other: T) -> bool { *self % other == 0 }
+    fn divisible_by(&self, other: &T) -> bool { *self % *other == 0 }
 
     /// Returns `true` if the number is divisible by `2`
     #[inline(always)]
-    fn is_even(&self) -> bool { self.divisible_by(2) }
+    fn is_even(&self) -> bool { self.divisible_by(&2) }
 
     /// Returns `true` if the number is not divisible by `2`
     #[inline(always)]
@@ -357,22 +349,27 @@ mod tests {
     use prelude::*;
 
     #[test]
+    fn test_num() {
+        num::test_num(10 as T, 2 as T);
+    }
+
+    #[test]
     fn test_gcd() {
-        assert_eq!((10 as T).gcd(2), 2 as T);
-        assert_eq!((10 as T).gcd(3), 1 as T);
-        assert_eq!((0 as T).gcd(3), 3 as T);
-        assert_eq!((3 as T).gcd(3), 3 as T);
-        assert_eq!((56 as T).gcd(42), 14 as T);
+        assert_eq!((10 as T).gcd(&2), 2 as T);
+        assert_eq!((10 as T).gcd(&3), 1 as T);
+        assert_eq!((0 as T).gcd(&3), 3 as T);
+        assert_eq!((3 as T).gcd(&3), 3 as T);
+        assert_eq!((56 as T).gcd(&42), 14 as T);
     }
 
     #[test]
     fn test_lcm() {
-        assert_eq!((1 as T).lcm(0), 0 as T);
-        assert_eq!((0 as T).lcm(1), 0 as T);
-        assert_eq!((1 as T).lcm(1), 1 as T);
-        assert_eq!((8 as T).lcm(9), 72 as T);
-        assert_eq!((11 as T).lcm(5), 55 as T);
-        assert_eq!((99 as T).lcm(17), 1683 as T);
+        assert_eq!((1 as T).lcm(&0), 0 as T);
+        assert_eq!((0 as T).lcm(&1), 0 as T);
+        assert_eq!((1 as T).lcm(&1), 1 as T);
+        assert_eq!((8 as T).lcm(&9), 72 as T);
+        assert_eq!((11 as T).lcm(&5), 55 as T);
+        assert_eq!((99 as T).lcm(&17), 1683 as T);
     }
 
     #[test]
diff --git a/src/libcore/ops.rs b/src/libcore/ops.rs
index 465a9330f74..1aa7aada05c 100644
--- a/src/libcore/ops.rs
+++ b/src/libcore/ops.rs
@@ -36,9 +36,7 @@ pub trait Div<RHS,Result> {
     fn div(&self, rhs: &RHS) -> Result;
 }
 #[lang="quot"]
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 pub trait Quot<RHS,Result> {
     fn quot(&self, rhs: &RHS) -> Result;
 }
@@ -49,9 +47,7 @@ pub trait Modulo<RHS,Result> {
     fn modulo(&self, rhs: &RHS) -> Result;
 }
 #[lang="rem"]
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 pub trait Rem<RHS,Result> {
     fn rem(&self, rhs: &RHS) -> Result;
 }
diff --git a/src/libcore/prelude.rs b/src/libcore/prelude.rs
index 03e6065a85c..553bb826810 100644
--- a/src/libcore/prelude.rs
+++ b/src/libcore/prelude.rs
@@ -16,9 +16,7 @@ pub use either::{Either, Left, Right};
 pub use kinds::{Const, Copy, Owned, Durable};
 #[cfg(stage0)]
 pub use ops::{Add, Sub, Mul, Div, Modulo, Neg, Not};
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 pub use ops::{Add, Sub, Mul, Quot, Rem, Neg, Not};
 pub use ops::{BitAnd, BitOr, BitXor};
 pub use ops::{Drop};
@@ -39,7 +37,9 @@ pub use hash::Hash;
 pub use iter::{BaseIter, ReverseIter, MutableIter, ExtendedIter, EqIter};
 pub use iter::{CopyableIter, CopyableOrderedIter, CopyableNonstrictIter};
 pub use iter::{Times, ExtendedMutableIter};
-pub use num::{Num, Signed, Unsigned, Natural, NumCast};
+pub use num::{Num, NumCast};
+pub use num::{Signed, Unsigned, Integer};
+pub use num::{Round, Fractional, Real, RealExt};
 pub use path::GenericPath;
 pub use path::Path;
 pub use path::PosixPath;
diff --git a/src/libstd/base64.rs b/src/libstd/base64.rs
index cbdd2b19d27..e90f0fb3c81 100644
--- a/src/libstd/base64.rs
+++ b/src/libstd/base64.rs
@@ -118,7 +118,7 @@ pub trait FromBase64 {
 impl FromBase64 for ~[u8] {
     /**
      * Convert base64 `u8` vector into u8 byte values.
-     * Every 4 encoded characters is converted into 3 octets, rem padding.
+     * Every 4 encoded characters is converted into 3 octets, modulo padding.
      *
      * *Example*:
      *
diff --git a/src/libstd/num/rational.rs b/src/libstd/num/rational.rs
index 36652380bff..8af1d99fa47 100644
--- a/src/libstd/num/rational.rs
+++ b/src/libstd/num/rational.rs
@@ -204,20 +204,6 @@ impl<T: Copy + Num + Ord>
 /* Utils */
 impl<T: Copy + Num + Ord>
     Round for Ratio<T> {
-    fn round(&self, mode: num::RoundMode) -> Ratio<T> {
-        match mode {
-            num::RoundUp => { self.ceil() }
-            num::RoundDown => { self.floor()}
-            num::RoundToZero => { Ratio::from_integer(self.numer / self.denom) }
-            num::RoundFromZero => {
-                if *self < Zero::zero() {
-                    Ratio::from_integer((self.numer - self.denom + One::one()) / self.denom)
-                } else {
-                    Ratio::from_integer((self.numer + self.denom - One::one()) / self.denom)
-                }
-            }
-        }
-    }
 
     fn floor(&self) -> Ratio<T> {
         if *self < Zero::zero() {
@@ -226,6 +212,7 @@ impl<T: Copy + Num + Ord>
             Ratio::from_integer(self.numer / self.denom)
         }
     }
+
     fn ceil(&self) -> Ratio<T> {
         if *self < Zero::zero() {
             Ratio::from_integer(self.numer / self.denom)
@@ -233,6 +220,21 @@ impl<T: Copy + Num + Ord>
             Ratio::from_integer((self.numer + self.denom - One::one()) / self.denom)
         }
     }
+
+    #[inline(always)]
+    fn round(&self) -> Ratio<T> {
+        if *self < Zero::zero() {
+            Ratio::from_integer((self.numer - self.denom + One::one()) / self.denom)
+        } else {
+            Ratio::from_integer((self.numer + self.denom - One::one()) / self.denom)
+        }
+    }
+
+    #[inline(always)]
+    fn trunc(&self) -> Ratio<T> {
+        Ratio::from_integer(self.numer / self.denom)
+    }
+
     fn fract(&self) -> Ratio<T> {
         Ratio::new_raw(self.numer % self.denom, self.denom)
     }
@@ -421,18 +423,18 @@ mod test {
     fn test_round() {
         assert_eq!(_1_2.ceil(), _1);
         assert_eq!(_1_2.floor(), _0);
-        assert_eq!(_1_2.round(num::RoundToZero), _0);
-        assert_eq!(_1_2.round(num::RoundFromZero), _1);
+        assert_eq!(_1_2.round(), _1);
+        assert_eq!(_1_2.trunc(), _0);
 
         assert_eq!(_neg1_2.ceil(), _0);
         assert_eq!(_neg1_2.floor(), -_1);
-        assert_eq!(_neg1_2.round(num::RoundToZero), _0);
-        assert_eq!(_neg1_2.round(num::RoundFromZero), -_1);
+        assert_eq!(_neg1_2.round(), -_1);
+        assert_eq!(_neg1_2.trunc(), _0);
 
         assert_eq!(_1.ceil(), _1);
         assert_eq!(_1.floor(), _1);
-        assert_eq!(_1.round(num::RoundToZero), _1);
-        assert_eq!(_1.round(num::RoundFromZero), _1);
+        assert_eq!(_1.round(), _1);
+        assert_eq!(_1.trunc(), _1);
     }
 
     #[test]
diff --git a/src/libstd/std.rc b/src/libstd/std.rc
index 7bedef0f841..07c679409cf 100644
--- a/src/libstd/std.rc
+++ b/src/libstd/std.rc
@@ -76,9 +76,7 @@ pub mod rope;
 pub mod smallintmap;
 pub mod sort;
 pub mod dlist;
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 pub mod treemap;
 
 // And ... other stuff
@@ -98,19 +96,13 @@ pub mod cmp;
 pub mod base64;
 pub mod rl;
 pub mod workcache;
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 #[path="num/bigint.rs"]
 pub mod bigint;
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 #[path="num/rational.rs"]
 pub mod rational;
-#[cfg(stage1)]
-#[cfg(stage2)]
-#[cfg(stage3)]
+#[cfg(not(stage0))]
 #[path="num/complex.rs"]
 pub mod complex;
 pub mod stats;