num-traits/src/complex.rs

// Copyright 2013 The Rust Project Developers. See the COPYRIGHT
// file at the top-level directory of this distribution and at
// http://rust-lang.org/COPYRIGHT.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.


//! Complex numbers.

use std::fmt;
use std::ops::{Add, Div, Mul, Neg, Sub};

use {Zero, One, Num, Float};

// FIXME #1284: handle complex NaN & infinity etc. This
// probably doesn't map to C's _Complex correctly.

/// A complex number in Cartesian form.
#[derive(PartialEq, Copy, Clone, Hash, Debug)]
#[cfg_attr(feature = "rustc-serialize", derive(RustcEncodable, RustcDecodable))]
pub struct Complex<T> {
    /// Real portion of the complex number
    pub re: T,
    /// Imaginary portion of the complex number
    pub im: T
}

pub type Complex32 = Complex<f32>;
pub type Complex64 = Complex<f64>;

impl<T: Clone + Num> Complex<T> {
    /// Create a new Complex
    #[inline]
    pub fn new(re: T, im: T) -> Complex<T> {
        Complex { re: re, im: im }
    }

    /// Returns imaginary unit
    #[inline]
    pub fn i() -> Complex<T> {
        Self::new(T::zero(), T::one())
    }

    /// Returns the square of the norm (since `T` doesn't necessarily
    /// have a sqrt function), i.e. `re^2 + im^2`.
    #[inline]
    pub fn norm_sqr(&self) -> T {
        self.re.clone() * self.re.clone() + self.im.clone() * self.im.clone()
    }

    /// Multiplies `self` by the scalar `t`.
    #[inline]
    pub fn scale(&self, t: T) -> Complex<T> {
        Complex::new(self.re.clone() * t.clone(), self.im.clone() * t)
    }

    /// Divides `self` by the scalar `t`.
    #[inline]
    pub fn unscale(&self, t: T) -> Complex<T> {
        Complex::new(self.re.clone() / t.clone(), self.im.clone() / t)
    }
}

impl<T: Clone + Num + Neg<Output = T>> Complex<T> {
    /// Returns the complex conjugate. i.e. `re - i im`
    #[inline]
    pub fn conj(&self) -> Complex<T> {
        Complex::new(self.re.clone(), -self.im.clone())
    }

    /// Returns `1/self`
    #[inline]
    pub fn inv(&self) -> Complex<T> {
        let norm_sqr = self.norm_sqr();
        Complex::new(self.re.clone() / norm_sqr.clone(),
                     -self.im.clone() / norm_sqr)
    }
}

impl<T: Clone + Float> Complex<T> {
    /// Calculate |self|
    #[inline]
    pub fn norm(&self) -> T {
        self.re.hypot(self.im)
    }
    /// Calculate the principal Arg of self.
    #[inline]
    pub fn arg(&self) -> T {
        self.im.atan2(self.re)
    }
    /// Convert to polar form (r, theta), such that `self = r * exp(i
    /// * theta)`
    #[inline]
    pub fn to_polar(&self) -> (T, T) {
        (self.norm(), self.arg())
    }
    /// Convert a polar representation into a complex number.
    #[inline]
    pub fn from_polar(r: &T, theta: &T) -> Complex<T> {
        Complex::new(*r * theta.cos(), *r * theta.sin())
    }

    /// Computes `e^(self)`, where `e` is the base of the natural logarithm.
    #[inline]
    pub fn exp(&self) -> Complex<T> {
        // formula: e^(a + bi) = e^a (cos(b) + i*sin(b))
        Complex::new(self.im.cos(), self.im.sin()).scale(self.re.exp())
    }

    /// Computes the principal value of natural logarithm of `self`.
    ///
    /// This function has one branch cut:
    ///
    /// * `(-∞, 0]`, continuous from above.
    ///
    /// The branch satisfies `-π ≤ arg(ln(z)) ≤ π`.
    #[inline]
    pub fn ln(&self) -> Complex<T> {
        // formula: ln(z) = ln|z| + i*arg(z)
        Complex::new(self.norm().ln(), self.arg())
    }

    /// Computes the principal value of the square root of `self`.
    ///
    /// This function has one branch cut:
    ///
    /// * `(-∞, 0)`, continuous from above.
    ///
    /// The branch satisfies `-π/2 ≤ arg(sqrt(z)) ≤ π/2`.
    #[inline]
    pub fn sqrt(&self) -> Complex<T> {
        // formula: sqrt(r e^(it)) = sqrt(r) e^(it/2)
        let two = T::one() + T::one();
        let (r, theta) = self.to_polar();
        Complex::from_polar(&(r.sqrt()), &(theta/two))
    }

    /// Computes the sine of `self`.
    #[inline]
    pub fn sin(&self) -> Complex<T> {
        // formula: sin(a + bi) = sin(a)cosh(b) + i*cos(a)sinh(b)
        Complex::new(self.re.sin() * self.im.cosh(), self.re.cos() * self.im.sinh())
    }

    /// Computes the cosine of `self`.
    #[inline]
    pub fn cos(&self) -> Complex<T> {
        // formula: cos(a + bi) = cos(a)cosh(b) - i*sin(a)sinh(b)
        Complex::new(self.re.cos() * self.im.cosh(), -self.re.sin() * self.im.sinh())
    }

    /// Computes the tangent of `self`.
    #[inline]
    pub fn tan(&self) -> Complex<T> {
        // formula: tan(a + bi) = (sin(2a) + i*sinh(2b))/(cos(2a) + cosh(2b))
        let (two_re, two_im) = (self.re + self.re, self.im + self.im);
        Complex::new(two_re.sin(), two_im.sinh()).unscale(two_re.cos() + two_im.cosh())
    }

    /// Computes the principal value of the inverse sine of `self`.
    ///
    /// This function has two branch cuts:
    ///
    /// * `(-∞, -1)`, continuous from above.
    /// * `(1, ∞)`, continuous from below.
    ///
    /// The branch satisfies `-π/2 ≤ Re(asin(z)) ≤ π/2`.
    #[inline]
    pub fn asin(&self) -> Complex<T> {
        // formula: arcsin(z) = -i ln(sqrt(1-z^2) + iz)
        let i = Complex::i();
        -i*((Complex::one() - self*self).sqrt() + i*self).ln()
    }

    /// Computes the principal value of the inverse cosine of `self`.
    ///
    /// This function has two branch cuts:
    ///
    /// * `(-∞, -1)`, continuous from above.
    /// * `(1, ∞)`, continuous from below.
    ///
    /// The branch satisfies `0 ≤ Re(acos(z)) ≤ π`.
    #[inline]
    pub fn acos(&self) -> Complex<T> {
        // formula: arccos(z) = -i ln(i sqrt(1-z^2) + z)
        let i = Complex::i();
        -i*(i*(Complex::one() - self*self).sqrt() + self).ln()
    }

    /// Computes the principal value of the inverse tangent of `self`.
    ///
    /// This function has two branch cuts:
    ///
    /// * `(-∞i, -i]`, continuous from the left.
    /// * `[i, ∞i)`, continuous from the right.
    ///
    /// The branch satisfies `-π/2 ≤ Re(atan(z)) ≤ π/2`.
    #[inline]
    pub fn atan(&self) -> Complex<T> {
        // formula: arctan(z) = (ln(1+iz) - ln(1-iz))/(2i)
        let i = Complex::i();
        let one = Complex::one();
        let two = one + one;
        if *self == i {
            return Complex::new(T::zero(), T::infinity());
        }
        else if *self == -i {
            return Complex::new(T::zero(), -T::infinity());
        }
        ((one + i * self).ln() - (one - i * self).ln()) / (two * i)
    }

    /// Computes the hyperbolic sine of `self`.
    #[inline]
    pub fn sinh(&self) -> Complex<T> {
        // formula: sinh(a + bi) = sinh(a)cos(b) + i*cosh(a)sin(b)
        Complex::new(self.re.sinh() * self.im.cos(), self.re.cosh() * self.im.sin())
    }

    /// Computes the hyperbolic cosine of `self`.
    #[inline]
    pub fn cosh(&self) -> Complex<T> {
        // formula: cosh(a + bi) = cosh(a)cos(b) + i*sinh(a)sin(b)
        Complex::new(self.re.cosh() * self.im.cos(), self.re.sinh() * self.im.sin())
    }

    /// Computes the hyperbolic tangent of `self`.
    #[inline]
    pub fn tanh(&self) -> Complex<T> {
        // formula: tanh(a + bi) = (sinh(2a) + i*sin(2b))/(cosh(2a) + cos(2b))
        let (two_re, two_im) = (self.re + self.re, self.im + self.im);
        Complex::new(two_re.sinh(), two_im.sin()).unscale(two_re.cosh() + two_im.cos())
    }

    /// Computes the principal value of inverse hyperbolic sine of `self`.
    ///
    /// This function has two branch cuts:
    ///
    /// * `(-∞i, -i)`, continuous from the left.
    /// * `(i, ∞i)`, continuous from the right.
    ///
    /// The branch satisfies `-π/2 ≤ Im(asinh(z)) ≤ π/2`.
    #[inline]
    pub fn asinh(&self) -> Complex<T> {
        // formula: arcsinh(z) = ln(z + sqrt(1+z^2))
        let one = Complex::one();
        (self + (one + self * self).sqrt()).ln()
    }

    /// Computes the principal value of inverse hyperbolic cosine of `self`.
    ///
    /// This function has one branch cut:
    ///
    /// * `(-∞, 1)`, continuous from above.
    ///
    /// The branch satisfies `-π ≤ Im(acosh(z)) ≤ π` and `0 ≤ Re(acosh(z)) < ∞`.
    #[inline]
    pub fn acosh(&self) -> Complex<T> {
        // formula: arccosh(z) = 2 ln(sqrt((z+1)/2) + sqrt((z-1)/2))
        let one = Complex::one();
        let two = one + one;
        two * (((self + one)/two).sqrt() + ((self - one)/two).sqrt()).ln()
    }

    /// Computes the principal value of inverse hyperbolic tangent of `self`.
    ///
    /// This function has two branch cuts:
    ///
    /// * `(-∞, -1]`, continuous from above.
    /// * `[1, ∞)`, continuous from below.
    ///
    /// The branch satisfies `-π/2 ≤ Im(atanh(z)) ≤ π/2`.
    #[inline]
    pub fn atanh(&self) -> Complex<T> {
        // formula: arctanh(z) = (ln(1+z) - ln(1-z))/2
        let one = Complex::one();
        let two = one + one;
        if *self == one {
            return Complex::new(T::infinity(), T::zero());
        }
        else if *self == -one {
            return Complex::new(-T::infinity(), T::zero());
        }
        ((one + self).ln() - (one - self).ln()) / two
    }

    /// Checks if the given complex number is NaN
    #[inline]
    pub fn is_nan(self) -> bool {
        self.re.is_nan() || self.im.is_nan()
    }

    /// Checks if the given complex number is infinite
    #[inline]
    pub fn is_infinite(self) -> bool {
        !self.is_nan() && (self.re.is_infinite() || self.im.is_infinite())
    }

    /// Checks if the given complex number is finite
    #[inline]
    pub fn is_finite(self) -> bool {
        self.re.is_finite() && self.im.is_finite()
    }

    /// Checks if the given complex number is normal
    #[inline]
   pub fn is_normal(self) -> bool {
        self.re.is_normal() && self.im.is_normal()
    }
}

impl<T: Clone + Num> From<T> for Complex<T> {
    #[inline]
    fn from(re: T) -> Complex<T> {
        Complex { re: re, im: T::zero() }
    }
}

impl<'a, T: Clone + Num> From<&'a T> for Complex<T> {
    #[inline]
    fn from(re: &T) -> Complex<T> {
        From::from(re.clone())
    }
}

macro_rules! forward_ref_ref_binop {
    (impl $imp:ident, $method:ident) => {
        impl<'a, 'b, T: Clone + Num> $imp<&'b Complex<T>> for &'a Complex<T> {
            type Output = Complex<T>;

            #[inline]
            fn $method(self, other: &Complex<T>) -> Complex<T> {
                self.clone().$method(other.clone())
            }
        }
    }
}

macro_rules! forward_ref_val_binop {
    (impl $imp:ident, $method:ident) => {
        impl<'a, T: Clone + Num> $imp<Complex<T>> for &'a Complex<T> {
            type Output = Complex<T>;

            #[inline]
            fn $method(self, other: Complex<T>) -> Complex<T> {
                self.clone().$method(other)
            }
        }
    }
}

macro_rules! forward_val_ref_binop {
    (impl $imp:ident, $method:ident) => {
        impl<'a, T: Clone + Num> $imp<&'a Complex<T>> for Complex<T> {
            type Output = Complex<T>;

            #[inline]
            fn $method(self, other: &Complex<T>) -> Complex<T> {
                self.$method(other.clone())
            }
        }
    }
}

macro_rules! forward_all_binop {
    (impl $imp:ident, $method:ident) => {
        forward_ref_ref_binop!(impl $imp, $method);
        forward_ref_val_binop!(impl $imp, $method);
        forward_val_ref_binop!(impl $imp, $method);
    };
}

/* arithmetic */
forward_all_binop!(impl Add, add);

// (a + i b) + (c + i d) == (a + c) + i (b + d)
impl<T: Clone + Num> Add<Complex<T>> for Complex<T> {
    type Output = Complex<T>;

    #[inline]
    fn add(self, other: Complex<T>) -> Complex<T> {
        Complex::new(self.re + other.re, self.im + other.im)
    }
}

forward_all_binop!(impl Sub, sub);

// (a + i b) - (c + i d) == (a - c) + i (b - d)
impl<T: Clone + Num> Sub<Complex<T>> for Complex<T> {
    type Output = Complex<T>;

    #[inline]
    fn sub(self, other: Complex<T>) -> Complex<T> {
        Complex::new(self.re - other.re, self.im - other.im)
    }
}

forward_all_binop!(impl Mul, mul);

// (a + i b) * (c + i d) == (a*c - b*d) + i (a*d + b*c)
impl<T: Clone + Num> Mul<Complex<T>> for Complex<T> {
    type Output = Complex<T>;

    #[inline]
    fn mul(self, other: Complex<T>) -> Complex<T> {
        let re = self.re.clone() * other.re.clone() - self.im.clone() * other.im.clone();
        let im = self.re * other.im + self.im * other.re;
        Complex::new(re, im)
    }
}

forward_all_binop!(impl Div, div);

// (a + i b) / (c + i d) == [(a + i b) * (c - i d)] / (c*c + d*d)
//   == [(a*c + b*d) / (c*c + d*d)] + i [(b*c - a*d) / (c*c + d*d)]
impl<T: Clone + Num> Div<Complex<T>> for Complex<T> {
    type Output = Complex<T>;

    #[inline]
    fn div(self, other: Complex<T>) -> Complex<T> {
        let norm_sqr = other.norm_sqr();
        let re = self.re.clone() * other.re.clone() + self.im.clone() * other.im.clone();
        let im = self.im * other.re - self.re * other.im;
        Complex::new(re / norm_sqr.clone(), im / norm_sqr)
    }
}

impl<T: Clone + Num + Neg<Output = T>> Neg for Complex<T> {
    type Output = Complex<T>;

    #[inline]
    fn neg(self) -> Complex<T> {
        Complex::new(-self.re, -self.im)
    }
}

impl<'a, T: Clone + Num + Neg<Output = T>> Neg for &'a Complex<T> {
    type Output = Complex<T>;

    #[inline]
    fn neg(self) -> Complex<T> {
        -self.clone()
    }
}

macro_rules! real_arithmetic {
    (@forward $imp:ident::$method:ident for $($real:ident),*) => (
        impl<'a, T: Clone + Num> $imp<&'a T> for Complex<T> {
            type Output = Complex<T>;

            #[inline]
            fn $method(self, other: &T) -> Complex<T> {
                self.$method(other.clone())
            }
        }
        impl<'a, T: Clone + Num> $imp<T> for &'a Complex<T> {
            type Output = Complex<T>;

            #[inline]
            fn $method(self, other: T) -> Complex<T> {
                self.clone().$method(other)
            }
        }
        impl<'a, 'b, T: Clone + Num> $imp<&'a T> for &'b Complex<T> {
            type Output = Complex<T>;

            #[inline]
            fn $method(self, other: &T) -> Complex<T> {
                self.clone().$method(other.clone())
            }
        }
        $(
            impl<'a> $imp<&'a Complex<$real>> for $real {
                type Output = Complex<$real>;

                #[inline]
                fn $method(self, other: &Complex<$real>) -> Complex<$real> {
                    self.$method(other.clone())
                }
            }
            impl<'a> $imp<Complex<$real>> for &'a $real {
                type Output = Complex<$real>;

                #[inline]
                fn $method(self, other: Complex<$real>) -> Complex<$real> {
                    self.clone().$method(other)
                }
            }
            impl<'a, 'b> $imp<&'a Complex<$real>> for &'b $real {
                type Output = Complex<$real>;

                #[inline]
                fn $method(self, other: &Complex<$real>) -> Complex<$real> {
                    self.clone().$method(other.clone())
                }
            }
        )*
    );
    (@implement $imp:ident::$method:ident for $($real:ident),*) => (
        impl<T: Clone + Num> $imp<T> for Complex<T> {
            type Output = Complex<T>;

            #[inline]
            fn $method(self, other: T) -> Complex<T> {
                self.$method(Complex::from(other))
            }
        }
        $(
            impl $imp<Complex<$real>> for $real {
                type Output = Complex<$real>;

                #[inline]
                fn $method(self, other: Complex<$real>) -> Complex<$real> {
                    Complex::from(self).$method(other)
                }
            }
        )*
    );
    ($($real:ident),*) => (
        real_arithmetic!(@forward Add::add for $($real),*);
        real_arithmetic!(@forward Sub::sub for $($real),*);
        real_arithmetic!(@forward Mul::mul for $($real),*);
        real_arithmetic!(@forward Div::div for $($real),*);
        real_arithmetic!(@implement Add::add for $($real),*);
        real_arithmetic!(@implement Sub::sub for $($real),*);
        real_arithmetic!(@implement Mul::mul for $($real),*);
        real_arithmetic!(@implement Div::div for $($real),*);
    );
}

real_arithmetic!(usize, u8, u16, u32, u64, isize, i8, i16, i32, i64, f32, f64);

/* constants */
impl<T: Clone + Num> Zero for Complex<T> {
    #[inline]
    fn zero() -> Complex<T> {
        Complex::new(Zero::zero(), Zero::zero())
    }

    #[inline]
    fn is_zero(&self) -> bool {
        self.re.is_zero() && self.im.is_zero()
    }
}

impl<T: Clone + Num> One for Complex<T> {
    #[inline]
    fn one() -> Complex<T> {
        Complex::new(One::one(), Zero::zero())
    }
}

/* string conversions */
impl<T> fmt::Display for Complex<T> where
    T: fmt::Display + Num + PartialOrd + Clone
{
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        if self.im < Zero::zero() {
            write!(f, "{}-{}i", self.re, T::zero() - self.im.clone())
        } else {
            write!(f, "{}+{}i", self.re, self.im)
        }
    }
}

#[cfg(test)]
mod test {
    #![allow(non_upper_case_globals)]

    use super::{Complex64, Complex};
    use std::f64;

    use {Zero, One, Float};

    pub const _0_0i : Complex64 = Complex { re: 0.0, im: 0.0 };
    pub const _1_0i : Complex64 = Complex { re: 1.0, im: 0.0 };
    pub const _1_1i : Complex64 = Complex { re: 1.0, im: 1.0 };
    pub const _0_1i : Complex64 = Complex { re: 0.0, im: 1.0 };
    pub const _neg1_1i : Complex64 = Complex { re: -1.0, im: 1.0 };
    pub const _05_05i : Complex64 = Complex { re: 0.5, im: 0.5 };
    pub const all_consts : [Complex64; 5] = [_0_0i, _1_0i, _1_1i, _neg1_1i, _05_05i];

    #[test]
    fn test_consts() {
        // check our constants are what Complex::new creates
        fn test(c : Complex64, r : f64, i: f64) {
            assert_eq!(c, Complex::new(r,i));
        }
        test(_0_0i, 0.0, 0.0);
        test(_1_0i, 1.0, 0.0);
        test(_1_1i, 1.0, 1.0);
        test(_neg1_1i, -1.0, 1.0);
        test(_05_05i, 0.5, 0.5);

        assert_eq!(_0_0i, Zero::zero());
        assert_eq!(_1_0i, One::one());
    }

    #[test]
    #[cfg_attr(target_arch = "x86", ignore)]
    // FIXME #7158: (maybe?) currently failing on x86.
    fn test_norm() {
        fn test(c: Complex64, ns: f64) {
            assert_eq!(c.norm_sqr(), ns);
            assert_eq!(c.norm(), ns.sqrt())
        }
        test(_0_0i, 0.0);
        test(_1_0i, 1.0);
        test(_1_1i, 2.0);
        test(_neg1_1i, 2.0);
        test(_05_05i, 0.5);
    }

    #[test]
    fn test_scale_unscale() {
        assert_eq!(_05_05i.scale(2.0), _1_1i);
        assert_eq!(_1_1i.unscale(2.0), _05_05i);
        for &c in all_consts.iter() {
            assert_eq!(c.scale(2.0).unscale(2.0), c);
        }
    }

    #[test]
    fn test_conj() {
        for &c in all_consts.iter() {
            assert_eq!(c.conj(), Complex::new(c.re, -c.im));
            assert_eq!(c.conj().conj(), c);
        }
    }

    #[test]
    fn test_inv() {
        assert_eq!(_1_1i.inv(), _05_05i.conj());
        assert_eq!(_1_0i.inv(), _1_0i.inv());
    }

    #[test]
    #[should_panic]
    fn test_divide_by_zero_natural() {
        let n = Complex::new(2, 3);
        let d = Complex::new(0, 0);
        let _x = n / d;
    }

    #[test]
    fn test_inv_zero() {
        // FIXME #20: should this really fail, or just NaN?
        assert!(_0_0i.inv().is_nan());
    }

    #[test]
    fn test_arg() {
        fn test(c: Complex64, arg: f64) {
            assert!((c.arg() - arg).abs() < 1.0e-6)
        }
        test(_1_0i, 0.0);
        test(_1_1i, 0.25 * f64::consts::PI);
        test(_neg1_1i, 0.75 * f64::consts::PI);
        test(_05_05i, 0.25 * f64::consts::PI);
    }

    #[test]
    fn test_polar_conv() {
        fn test(c: Complex64) {
            let (r, theta) = c.to_polar();
            assert!((c - Complex::from_polar(&r, &theta)).norm() < 1e-6);
        }
        for &c in all_consts.iter() { test(c); }
    }

    fn close(a: Complex64, b: Complex64) -> bool {
        // returns true if a and b are reasonably close
        (a == b) || (a-b).norm() < 1e-10
    }

    #[test]
    fn test_exp() {
        assert!(close(_1_0i.exp(), _1_0i.scale(f64::consts::E)));
        assert!(close(_0_0i.exp(), _1_0i));
        assert!(close(_0_1i.exp(), Complex::new(1.0.cos(), 1.0.sin())));
        assert!(close(_05_05i.exp()*_05_05i.exp(), _1_1i.exp()));
        assert!(close(_0_1i.scale(-f64::consts::PI).exp(), _1_0i.scale(-1.0)));
        for &c in all_consts.iter() {
            // e^conj(z) = conj(e^z)
            assert!(close(c.conj().exp(), c.exp().conj()));
            // e^(z + 2 pi i) = e^z
            assert!(close(c.exp(), (c + _0_1i.scale(f64::consts::PI*2.0)).exp()));
        }
    }

    #[test]
    fn test_ln() {
        assert!(close(_1_0i.ln(), _0_0i));
        assert!(close(_0_1i.ln(), _0_1i.scale(f64::consts::PI/2.0)));
        assert!(close(_0_0i.ln(), Complex::new(f64::neg_infinity(), 0.0)));
        assert!(close((_neg1_1i * _05_05i).ln(), _neg1_1i.ln() + _05_05i.ln()));
        for &c in all_consts.iter() {
            // ln(conj(z() = conj(ln(z))
            assert!(close(c.conj().ln(), c.ln().conj()));
            // for this branch, -pi <= arg(ln(z)) <= pi
            assert!(-f64::consts::PI <= c.ln().arg() && c.ln().arg() <= f64::consts::PI);
        }
    }

    #[test]
    fn test_sqrt() {
        assert!(close(_0_0i.sqrt(), _0_0i));
        assert!(close(_1_0i.sqrt(), _1_0i));
        assert!(close(Complex::new(-1.0, 0.0).sqrt(), _0_1i));
        assert!(close(Complex::new(-1.0, -0.0).sqrt(), _0_1i.scale(-1.0)));
        assert!(close(_0_1i.sqrt(), _05_05i.scale(2.0.sqrt())));
        for &c in all_consts.iter() {
            // sqrt(conj(z() = conj(sqrt(z))
            assert!(close(c.conj().sqrt(), c.sqrt().conj()));
            // for this branch, -pi/2 <= arg(sqrt(z)) <= pi/2
            assert!(-f64::consts::PI/2.0 <= c.sqrt().arg() && c.sqrt().arg() <= f64::consts::PI/2.0);
            // sqrt(z) * sqrt(z) = z
            assert!(close(c.sqrt()*c.sqrt(), c));
        }
    }

    #[test]
    fn test_sin() {
        assert!(close(_0_0i.sin(), _0_0i));
        assert!(close(_1_0i.scale(f64::consts::PI*2.0).sin(), _0_0i));
        assert!(close(_0_1i.sin(), _0_1i.scale(1.0.sinh())));
        for &c in all_consts.iter() {
            // sin(conj(z)) = conj(sin(z))
            assert!(close(c.conj().sin(), c.sin().conj()));
            // sin(-z) = -sin(z)
            assert!(close(c.scale(-1.0).sin(), c.sin().scale(-1.0)));
        }
    }

    #[test]
    fn test_cos() {
        assert!(close(_0_0i.cos(), _1_0i));
        assert!(close(_1_0i.scale(f64::consts::PI*2.0).cos(), _1_0i));
        assert!(close(_0_1i.cos(), _1_0i.scale(1.0.cosh())));
        for &c in all_consts.iter() {
            // cos(conj(z)) = conj(cos(z))
            assert!(close(c.conj().cos(), c.cos().conj()));
            // cos(-z) = cos(z)
            assert!(close(c.scale(-1.0).cos(), c.cos()));
        }
    }

    #[test]
    fn test_tan() {
        assert!(close(_0_0i.tan(), _0_0i));
        assert!(close(_1_0i.scale(f64::consts::PI/4.0).tan(), _1_0i));
        assert!(close(_1_0i.scale(f64::consts::PI).tan(), _0_0i));
        for &c in all_consts.iter() {
            // tan(conj(z)) = conj(tan(z))
            assert!(close(c.conj().tan(), c.tan().conj()));
            // tan(-z) = -tan(z)
            assert!(close(c.scale(-1.0).tan(), c.tan().scale(-1.0)));
        }
    }

    #[test]
    fn test_asin() {
        assert!(close(_0_0i.asin(), _0_0i));
        assert!(close(_1_0i.asin(), _1_0i.scale(f64::consts::PI/2.0)));
        assert!(close(_1_0i.scale(-1.0).asin(), _1_0i.scale(-f64::consts::PI/2.0)));
        assert!(close(_0_1i.asin(), _0_1i.scale((1.0 + 2.0.sqrt()).ln())));
        for &c in all_consts.iter() {
            // asin(conj(z)) = conj(asin(z))
            assert!(close(c.conj().asin(), c.asin().conj()));
            // asin(-z) = -asin(z)
            assert!(close(c.scale(-1.0).asin(), c.asin().scale(-1.0)));
            // for this branch, -pi/2 <= asin(z).re <= pi/2
            assert!(-f64::consts::PI/2.0 <= c.asin().re && c.asin().re <= f64::consts::PI/2.0);
        }
    }

    #[test]
    fn test_acos() {
        assert!(close(_0_0i.acos(), _1_0i.scale(f64::consts::PI/2.0)));
        assert!(close(_1_0i.acos(), _0_0i));
        assert!(close(_1_0i.scale(-1.0).acos(), _1_0i.scale(f64::consts::PI)));
        assert!(close(_0_1i.acos(), Complex::new(f64::consts::PI/2.0, (2.0.sqrt() - 1.0).ln())));
        for &c in all_consts.iter() {
            // acos(conj(z)) = conj(acos(z))
            assert!(close(c.conj().acos(), c.acos().conj()));
            // for this branch, 0 <= acos(z).re <= pi
            assert!(0.0 <= c.acos().re && c.acos().re <= f64::consts::PI);
        }
    }

    #[test]
    fn test_atan() {
        assert!(close(_0_0i.atan(), _0_0i));
        assert!(close(_1_0i.atan(), _1_0i.scale(f64::consts::PI/4.0)));
        assert!(close(_1_0i.scale(-1.0).atan(), _1_0i.scale(-f64::consts::PI/4.0)));
        assert!(close(_0_1i.atan(), Complex::new(0.0, f64::infinity())));
        for &c in all_consts.iter() {
            // atan(conj(z)) = conj(atan(z))
            assert!(close(c.conj().atan(), c.atan().conj()));
            // atan(-z) = -atan(z)
            assert!(close(c.scale(-1.0).atan(), c.atan().scale(-1.0)));
            // for this branch, -pi/2 <= atan(z).re <= pi/2
            assert!(-f64::consts::PI/2.0 <= c.atan().re && c.atan().re <= f64::consts::PI/2.0);
        }
    }

    #[test]
    fn test_sinh() {
        assert!(close(_0_0i.sinh(), _0_0i));
        assert!(close(_1_0i.sinh(), _1_0i.scale((f64::consts::E - 1.0/f64::consts::E)/2.0)));
        assert!(close(_0_1i.sinh(), _0_1i.scale(1.0.sin())));
        for &c in all_consts.iter() {
            // sinh(conj(z)) = conj(sinh(z))
            assert!(close(c.conj().sinh(), c.sinh().conj()));
            // sinh(-z) = -sinh(z)
            assert!(close(c.scale(-1.0).sinh(), c.sinh().scale(-1.0)));
        }
    }

    #[test]
    fn test_cosh() {
        assert!(close(_0_0i.cosh(), _1_0i));
        assert!(close(_1_0i.cosh(), _1_0i.scale((f64::consts::E + 1.0/f64::consts::E)/2.0)));
        assert!(close(_0_1i.cosh(), _1_0i.scale(1.0.cos())));
        for &c in all_consts.iter() {
            // cosh(conj(z)) = conj(cosh(z))
            assert!(close(c.conj().cosh(), c.cosh().conj()));
            // cosh(-z) = cosh(z)
            assert!(close(c.scale(-1.0).cosh(), c.cosh()));
        }
    }

    #[test]
    fn test_tanh() {
        assert!(close(_0_0i.tanh(), _0_0i));
        assert!(close(_1_0i.tanh(), _1_0i.scale((f64::consts::E.powi(2) - 1.0)/(f64::consts::E.powi(2) + 1.0))));
        assert!(close(_0_1i.tanh(), _0_1i.scale(1.0.tan())));
        for &c in all_consts.iter() {
            // tanh(conj(z)) = conj(tanh(z))
            assert!(close(c.conj().tanh(), c.conj().tanh()));
            // tanh(-z) = -tanh(z)
            assert!(close(c.scale(-1.0).tanh(), c.tanh().scale(-1.0)));
        }
    }

    #[test]
    fn test_asinh() {
        assert!(close(_0_0i.asinh(), _0_0i));
        assert!(close(_1_0i.asinh(), _1_0i.scale(1.0 + 2.0.sqrt()).ln()));
        assert!(close(_0_1i.asinh(), _0_1i.scale(f64::consts::PI/2.0)));
        assert!(close(_0_1i.asinh().scale(-1.0), _0_1i.scale(-f64::consts::PI/2.0)));
        for &c in all_consts.iter() {
            // asinh(conj(z)) = conj(asinh(z))
            assert!(close(c.conj().asinh(), c.conj().asinh()));
            // asinh(-z) = -asinh(z)
            assert!(close(c.scale(-1.0).asinh(), c.asinh().scale(-1.0)));
            // for this branch, -pi/2 <= asinh(z).im <= pi/2
            assert!(-f64::consts::PI/2.0 <= c.asinh().im && c.asinh().im <= f64::consts::PI/2.0);
        }
    }

    #[test]
    fn test_acosh() {
        assert!(close(_0_0i.acosh(), _0_1i.scale(f64::consts::PI/2.0)));
        assert!(close(_1_0i.acosh(), _0_0i));
        assert!(close(_1_0i.scale(-1.0).acosh(), _0_1i.scale(f64::consts::PI)));
        for &c in all_consts.iter() {
            // acosh(conj(z)) = conj(acosh(z))
            assert!(close(c.conj().acosh(), c.conj().acosh()));
            // for this branch, -pi <= acosh(z).im <= pi and 0 <= acosh(z).re
            assert!(-f64::consts::PI <= c.acosh().im && c.acosh().im <= f64::consts::PI && 0.0 <= c.cosh().re);
        }
    }

    #[test]
    fn test_atanh() {
        assert!(close(_0_0i.atanh(), _0_0i));
        assert!(close(_0_1i.atanh(), _0_1i.scale(f64::consts::PI/4.0)));
        assert!(close(_1_0i.atanh(), Complex::new(f64::infinity(), 0.0)));
        for &c in all_consts.iter() {
            // atanh(conj(z)) = conj(atanh(z))
            assert!(close(c.conj().atanh(), c.conj().atanh()));
            // atanh(-z) = -atanh(z)
            assert!(close(c.scale(-1.0).atanh(), c.atanh().scale(-1.0)));
            // for this branch, -pi/2 <= atanh(z).im <= pi/2
            assert!(-f64::consts::PI/2.0 <= c.atanh().im && c.atanh().im <= f64::consts::PI/2.0);
        }
    }

    #[test]
    fn test_exp_ln() {
        for &c in all_consts.iter() {
            // e^ln(z) = z
            assert!(close(c.ln().exp(), c));
        }
    }

    #[test]
    fn test_trig_to_hyperbolic() {
        for &c in all_consts.iter() {
            // sin(iz) = i sinh(z)
            assert!(close((_0_1i * c).sin(), _0_1i * c.sinh()));
            // cos(iz) = cosh(z)
            assert!(close((_0_1i * c).cos(), c.cosh()));
            // tan(iz) = i tanh(z)
            assert!(close((_0_1i * c).tan(), _0_1i * c.tanh()));
        }
    }

    #[test]
    fn test_trig_identities() {
        for &c in all_consts.iter() {
            // tan(z) = sin(z)/cos(z)
            assert!(close(c.tan(), c.sin()/c.cos()));
            // sin(z)^2 + cos(z)^2 = 1
            assert!(close(c.sin()*c.sin() + c.cos()*c.cos(), _1_0i));

            // sin(asin(z)) = z
            assert!(close(c.asin().sin(), c));
            // cos(acos(z)) = z
            assert!(close(c.acos().cos(), c));
            // tan(atan(z)) = z
            // i and -i are branch points
            if c != _0_1i && c != _0_1i.scale(-1.0) {
                assert!(close(c.atan().tan(), c));
            }

            // sin(z) = (e^(iz) - e^(-iz))/(2i)
            assert!(close(((_0_1i*c).exp() - (_0_1i*c).exp().inv())/_0_1i.scale(2.0), c.sin()));
            // cos(z) = (e^(iz) + e^(-iz))/2
            assert!(close(((_0_1i*c).exp() + (_0_1i*c).exp().inv()).unscale(2.0), c.cos()));
            // tan(z) = i (1 - e^(2iz))/(1 + e^(2iz))
            assert!(close(_0_1i * (_1_0i - (_0_1i*c).scale(2.0).exp())/(_1_0i + (_0_1i*c).scale(2.0).exp()), c.tan()));
        }
    }

    #[test]
    fn test_hyperbolic_identites() {
        for &c in all_consts.iter() {
            // tanh(z) = sinh(z)/cosh(z)
            assert!(close(c.tanh(), c.sinh()/c.cosh()));
            // cosh(z)^2 - sinh(z)^2 = 1
            assert!(close(c.cosh()*c.cosh() - c.sinh()*c.sinh(), _1_0i));

            // sinh(asinh(z)) = z
            assert!(close(c.asinh().sinh(), c));
            // cosh(acosh(z)) = z
            assert!(close(c.acosh().cosh(), c));
            // tanh(atanh(z)) = z
            // 1 and -1 are branch points
            if c != _1_0i && c != _1_0i.scale(-1.0) {
                assert!(close(c.atanh().tanh(), c));
            }

            // sinh(z) = (e^z - e^(-z))/2
            assert!(close((c.exp() - c.exp().inv()).unscale(2.0), c.sinh()));
            // cosh(z) = (e^z + e^(-z))/2
            assert!(close((c.exp() + c.exp().inv()).unscale(2.0), c.cosh()));
            // tanh(z) = ( e^(2z) - 1)/(e^(2z) + 1)
            assert!(close((c.scale(2.0).exp() - _1_0i)/(c.scale(2.0).exp() + _1_0i), c.tanh()));
        }
    }

    mod complex_arithmetic {
        use super::{_0_0i, _1_0i, _1_1i, _0_1i, _neg1_1i, _05_05i, all_consts};
        use Zero;

        #[test]
        fn test_add() {
            assert_eq!(_05_05i + _05_05i, _1_1i);
            assert_eq!(_0_1i + _1_0i, _1_1i);
            assert_eq!(_1_0i + _neg1_1i, _0_1i);

            for &c in all_consts.iter() {
                assert_eq!(_0_0i + c, c);
                assert_eq!(c + _0_0i, c);
            }
        }

        #[test]
        fn test_sub() {
            assert_eq!(_05_05i - _05_05i, _0_0i);
            assert_eq!(_0_1i - _1_0i, _neg1_1i);
            assert_eq!(_0_1i - _neg1_1i, _1_0i);

            for &c in all_consts.iter() {
                assert_eq!(c - _0_0i, c);
                assert_eq!(c - c, _0_0i);
            }
        }

        #[test]
        fn test_mul() {
            assert_eq!(_05_05i * _05_05i, _0_1i.unscale(2.0));
            assert_eq!(_1_1i * _0_1i, _neg1_1i);

            // i^2 & i^4
            assert_eq!(_0_1i * _0_1i, -_1_0i);
            assert_eq!(_0_1i * _0_1i * _0_1i * _0_1i, _1_0i);

            for &c in all_consts.iter() {
                assert_eq!(c * _1_0i, c);
                assert_eq!(_1_0i * c, c);
            }
        }

        #[test]
        fn test_div() {
            assert_eq!(_neg1_1i / _0_1i, _1_1i);
            for &c in all_consts.iter() {
                if c != Zero::zero() {
                    assert_eq!(c / c, _1_0i);
                }
            }
        }

        #[test]
        fn test_neg() {
            assert_eq!(-_1_0i + _0_1i, _neg1_1i);
            assert_eq!((-_0_1i) * _0_1i, _1_0i);
            for &c in all_consts.iter() {
                assert_eq!(-(-c), c);
            }
        }
    }

    mod real_arithmetic {
        use super::super::Complex;

        #[test]
        fn test_add() {
            assert_eq!(Complex::new(4.0, 2.0) + 0.5, Complex::new(4.5, 2.0));
            assert_eq!(0.5 + Complex::new(4.0, 2.0), Complex::new(4.5, 2.0));
        }

        #[test]
        fn test_sub() {
            assert_eq!(Complex::new(4.0, 2.0) - 0.5, Complex::new(3.5, 2.0));
            assert_eq!(0.5 - Complex::new(4.0, 2.0), Complex::new(-3.5, -2.0));
        }

        #[test]
        fn test_mul() {
            assert_eq!(Complex::new(4.0, 2.0) * 0.5, Complex::new(2.0, 1.0));
            assert_eq!(0.5 * Complex::new(4.0, 2.0), Complex::new(2.0, 1.0));
        }

        #[test]
        fn test_div() {
            assert_eq!(Complex::new(4.0, 2.0) / 0.5, Complex::new(8.0, 4.0));
            assert_eq!(0.5 / Complex::new(4.0, 2.0), Complex::new(0.1, -0.05));
        }
    }

    #[test]
    fn test_to_string() {
        fn test(c : Complex64, s: String) {
            assert_eq!(c.to_string(), s);
        }
        test(_0_0i, "0+0i".to_string());
        test(_1_0i, "1+0i".to_string());
        test(_0_1i, "0+1i".to_string());
        test(_1_1i, "1+1i".to_string());
        test(_neg1_1i, "-1+1i".to_string());
        test(-_neg1_1i, "1-1i".to_string());
        test(_05_05i, "0.5+0.5i".to_string());
    }

    #[test]
    fn test_hash() {
        let a = Complex::new(0i32, 0i32);
        let b = Complex::new(1i32, 0i32);
        let c = Complex::new(0i32, 1i32);
        assert!(::hash(&a) != ::hash(&b));
        assert!(::hash(&b) != ::hash(&c));
        assert!(::hash(&c) != ::hash(&a));
    }

    #[test]
    fn test_is_nan() {
        assert!(!_1_1i.is_nan());
        let a = Complex::new(f64::NAN, f64::NAN);
        assert!(a.is_nan());
    }

    #[test]
    fn test_is_nan_special_cases() {
        let a = Complex::new(0f64, f64::NAN);
        let b = Complex::new(f64::NAN, 0f64);
        assert!(a.is_nan());
        assert!(b.is_nan());
    }

    #[test]
    fn test_is_infinite() {
        let a = Complex::new(2f64, f64::INFINITY);
        assert!(a.is_infinite());
    }

    #[test]
    fn test_is_finite() {
        assert!(_1_1i.is_finite())
    }

    #[test]
    fn test_is_normal() {
        let a = Complex::new(0f64, f64::NAN);
        let b = Complex::new(2f64, f64::INFINITY);
        assert!(!a.is_normal());
        assert!(!b.is_normal());
        assert!(_1_1i.is_normal());
    }
}