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// 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.
//!
//! ## Compatibility
//!
//! The `num-complex` crate is tested for rustc 1.60 and greater.
#![doc(html_root_url = "https://docs.rs/num-complex/0.4")]
#![no_std]
#[cfg(any(test, feature = "std"))]
#[cfg_attr(test, macro_use)]
extern crate std;
use core::fmt;
#[cfg(test)]
use core::hash;
use core::iter::{Product, Sum};
use core::ops::{Add, Div, Mul, Neg, Rem, Sub};
use core::str::FromStr;
#[cfg(feature = "std")]
use std::error::Error;
use num_traits::{ConstOne, ConstZero, Inv, MulAdd, Num, One, Pow, Signed, Zero};
use num_traits::float::FloatCore;
#[cfg(any(feature = "std", feature = "libm"))]
use num_traits::float::{Float, FloatConst};
mod cast;
mod pow;
#[cfg(any(feature = "std", feature = "libm"))]
mod complex_float;
#[cfg(any(feature = "std", feature = "libm"))]
pub use crate::complex_float::ComplexFloat;
#[cfg(feature = "rand")]
mod crand;
#[cfg(feature = "rand")]
pub use crate::crand::ComplexDistribution;
// FIXME #1284: handle complex NaN & infinity etc. This
// probably doesn't map to C's _Complex correctly.
/// A complex number in Cartesian form.
///
/// ## Representation and Foreign Function Interface Compatibility
///
/// `Complex<T>` is memory layout compatible with an array `[T; 2]`.
///
/// Note that `Complex<F>` where F is a floating point type is **only** memory
/// layout compatible with C's complex types, **not** necessarily calling
/// convention compatible. This means that for FFI you can only pass
/// `Complex<F>` behind a pointer, not as a value.
///
/// ## Examples
///
/// Example of extern function declaration.
///
/// ```
/// use num_complex::Complex;
/// use std::os::raw::c_int;
///
/// extern "C" {
/// fn zaxpy_(n: *const c_int, alpha: *const Complex<f64>,
/// x: *const Complex<f64>, incx: *const c_int,
/// y: *mut Complex<f64>, incy: *const c_int);
/// }
/// ```
#[derive(PartialEq, Eq, Copy, Clone, Hash, Debug, Default)]
#[repr(C)]
#[cfg_attr(
feature = "rkyv",
derive(rkyv::Archive, rkyv::Serialize, rkyv::Deserialize)
)]
#[cfg_attr(feature = "rkyv", archive(as = "Complex<T::Archived>"))]
#[cfg_attr(feature = "bytecheck", derive(bytecheck::CheckBytes))]
pub struct Complex<T> {
/// Real portion of the complex number
pub re: T,
/// Imaginary portion of the complex number
pub im: T,
}
/// Alias for a [`Complex<f32>`]
pub type Complex32 = Complex<f32>;
/// Create a new [`Complex<f32>`] with arguments that can convert [`Into<f32>`].
///
/// ```
/// use num_complex::{c32, Complex32};
/// assert_eq!(c32(1u8, 2), Complex32::new(1.0, 2.0));
/// ```
///
/// Note: ambiguous integer literals in Rust will [default] to `i32`, which does **not** implement
/// `Into<f32>`, so a call like `c32(1, 2)` will result in a type error. The example above uses a
/// suffixed `1u8` to set its type, and then the `2` can be inferred as the same type.
///
/// [default]: https://doc.rust-lang.org/reference/expressions/literal-expr.html#integer-literal-expressions
#[inline]
pub fn c32<T: Into<f32>>(re: T, im: T) -> Complex32 {
Complex::new(re.into(), im.into())
}
/// Alias for a [`Complex<f64>`]
pub type Complex64 = Complex<f64>;
/// Create a new [`Complex<f64>`] with arguments that can convert [`Into<f64>`].
///
/// ```
/// use num_complex::{c64, Complex64};
/// assert_eq!(c64(1, 2), Complex64::new(1.0, 2.0));
/// ```
#[inline]
pub fn c64<T: Into<f64>>(re: T, im: T) -> Complex64 {
Complex::new(re.into(), im.into())
}
impl<T> Complex<T> {
/// Create a new `Complex`
#[inline]
pub const fn new(re: T, im: T) -> Self {
Complex { re, im }
}
}
impl<T: Clone + Num> Complex<T> {
/// Returns the imaginary unit.
///
/// See also [`Complex::I`].
#[inline]
pub fn i() -> Self {
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) -> Self {
Self::new(self.re.clone() * t.clone(), self.im.clone() * t)
}
/// Divides `self` by the scalar `t`.
#[inline]
pub fn unscale(&self, t: T) -> Self {
Self::new(self.re.clone() / t.clone(), self.im.clone() / t)
}
/// Raises `self` to an unsigned integer power.
#[inline]
pub fn powu(&self, exp: u32) -> Self {
Pow::pow(self, exp)
}
}
impl<T: Clone + Num + Neg<Output = T>> Complex<T> {
/// Returns the complex conjugate. i.e. `re - i im`
#[inline]
pub fn conj(&self) -> Self {
Self::new(self.re.clone(), -self.im.clone())
}
/// Returns `1/self`
#[inline]
pub fn inv(&self) -> Self {
let norm_sqr = self.norm_sqr();
Self::new(
self.re.clone() / norm_sqr.clone(),
-self.im.clone() / norm_sqr,
)
}
/// Raises `self` to a signed integer power.
#[inline]
pub fn powi(&self, exp: i32) -> Self {
Pow::pow(self, exp)
}
}
impl<T: Clone + Signed> Complex<T> {
/// Returns the L1 norm `|re| + |im|` -- the [Manhattan distance] from the origin.
///
/// [Manhattan distance]: https://en.wikipedia.org/wiki/Taxicab_geometry
#[inline]
pub fn l1_norm(&self) -> T {
self.re.abs() + self.im.abs()
}
}
#[cfg(any(feature = "std", feature = "libm"))]
impl<T: Float> Complex<T> {
/// Create a new Complex with a given phase: `exp(i * phase)`.
/// See [cis (mathematics)](https://en.wikipedia.org/wiki/Cis_(mathematics)).
#[inline]
pub fn cis(phase: T) -> Self {
Self::new(phase.cos(), phase.sin())
}
/// 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) -> Self {
Self::new(r * theta.cos(), r * theta.sin())
}
/// Computes `e^(self)`, where `e` is the base of the natural logarithm.
#[inline]
pub fn exp(self) -> Self {
// formula: e^(a + bi) = e^a (cos(b) + i*sin(b)) = from_polar(e^a, b)
let Complex { re, mut im } = self;
// Treat the corner cases +∞, -∞, and NaN
if re.is_infinite() {
if re < T::zero() {
if !im.is_finite() {
return Self::new(T::zero(), T::zero());
}
} else if im == T::zero() || !im.is_finite() {
if im.is_infinite() {
im = T::nan();
}
return Self::new(re, im);
}
} else if re.is_nan() && im == T::zero() {
return self;
}
Self::from_polar(re.exp(), im)
}
/// 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) -> Self {
// formula: ln(z) = ln|z| + i*arg(z)
let (r, theta) = self.to_polar();
Self::new(r.ln(), theta)
}
/// 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) -> Self {
if self.im.is_zero() {
if self.re.is_sign_positive() {
// simple positive real √r, and copy `im` for its sign
Self::new(self.re.sqrt(), self.im)
} else {
// √(r e^(iπ)) = √r e^(iπ/2) = i√r
// √(r e^(-iπ)) = √r e^(-iπ/2) = -i√r
let re = T::zero();
let im = (-self.re).sqrt();
if self.im.is_sign_positive() {
Self::new(re, im)
} else {
Self::new(re, -im)
}
}
} else if self.re.is_zero() {
// √(r e^(iπ/2)) = √r e^(iπ/4) = √(r/2) + i√(r/2)
// √(r e^(-iπ/2)) = √r e^(-iπ/4) = √(r/2) - i√(r/2)
let one = T::one();
let two = one + one;
let x = (self.im.abs() / two).sqrt();
if self.im.is_sign_positive() {
Self::new(x, x)
} else {
Self::new(x, -x)
}
} else {
// formula: sqrt(r e^(it)) = sqrt(r) e^(it/2)
let one = T::one();
let two = one + one;
let (r, theta) = self.to_polar();
Self::from_polar(r.sqrt(), theta / two)
}
}
/// Computes the principal value of the cube root of `self`.
///
/// This function has one branch cut:
///
/// * `(-∞, 0)`, continuous from above.
///
/// The branch satisfies `-π/3 ≤ arg(cbrt(z)) ≤ π/3`.
///
/// Note that this does not match the usual result for the cube root of
/// negative real numbers. For example, the real cube root of `-8` is `-2`,
/// but the principal complex cube root of `-8` is `1 + i√3`.
#[inline]
pub fn cbrt(self) -> Self {
if self.im.is_zero() {
if self.re.is_sign_positive() {
// simple positive real ∛r, and copy `im` for its sign
Self::new(self.re.cbrt(), self.im)
} else {
// ∛(r e^(iπ)) = ∛r e^(iπ/3) = ∛r/2 + i∛r√3/2
// ∛(r e^(-iπ)) = ∛r e^(-iπ/3) = ∛r/2 - i∛r√3/2
let one = T::one();
let two = one + one;
let three = two + one;
let re = (-self.re).cbrt() / two;
let im = three.sqrt() * re;
if self.im.is_sign_positive() {
Self::new(re, im)
} else {
Self::new(re, -im)
}
}
} else if self.re.is_zero() {
// ∛(r e^(iπ/2)) = ∛r e^(iπ/6) = ∛r√3/2 + i∛r/2
// ∛(r e^(-iπ/2)) = ∛r e^(-iπ/6) = ∛r√3/2 - i∛r/2
let one = T::one();
let two = one + one;
let three = two + one;
let im = self.im.abs().cbrt() / two;
let re = three.sqrt() * im;
if self.im.is_sign_positive() {
Self::new(re, im)
} else {
Self::new(re, -im)
}
} else {
// formula: cbrt(r e^(it)) = cbrt(r) e^(it/3)
let one = T::one();
let three = one + one + one;
let (r, theta) = self.to_polar();
Self::from_polar(r.cbrt(), theta / three)
}
}
/// Raises `self` to a floating point power.
#[inline]
pub fn powf(self, exp: T) -> Self {
if exp.is_zero() {
return Self::one();
}
// formula: x^y = (ρ e^(i θ))^y = ρ^y e^(i θ y)
// = from_polar(ρ^y, θ y)
let (r, theta) = self.to_polar();
Self::from_polar(r.powf(exp), theta * exp)
}
/// Returns the logarithm of `self` with respect to an arbitrary base.
#[inline]
pub fn log(self, base: T) -> Self {
// formula: log_y(x) = log_y(ρ e^(i θ))
// = log_y(ρ) + log_y(e^(i θ)) = log_y(ρ) + ln(e^(i θ)) / ln(y)
// = log_y(ρ) + i θ / ln(y)
let (r, theta) = self.to_polar();
Self::new(r.log(base), theta / base.ln())
}
/// Raises `self` to a complex power.
#[inline]
pub fn powc(self, exp: Self) -> Self {
if exp.is_zero() {
return Self::one();
}
// formula: x^y = exp(y * ln(x))
(exp * self.ln()).exp()
}
/// Raises a floating point number to the complex power `self`.
#[inline]
pub fn expf(self, base: T) -> Self {
// formula: x^(a+bi) = x^a x^bi = x^a e^(b ln(x) i)
// = from_polar(x^a, b ln(x))
Self::from_polar(base.powf(self.re), self.im * base.ln())
}
/// Computes the sine of `self`.
#[inline]
pub fn sin(self) -> Self {
// formula: sin(a + bi) = sin(a)cosh(b) + i*cos(a)sinh(b)
Self::new(
self.re.sin() * self.im.cosh(),
self.re.cos() * self.im.sinh(),
)
}
/// Computes the cosine of `self`.
#[inline]
pub fn cos(self) -> Self {
// formula: cos(a + bi) = cos(a)cosh(b) - i*sin(a)sinh(b)
Self::new(
self.re.cos() * self.im.cosh(),
-self.re.sin() * self.im.sinh(),
)
}
/// Computes the tangent of `self`.
#[inline]
pub fn tan(self) -> Self {
// 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);
Self::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) -> Self {
// formula: arcsin(z) = -i ln(sqrt(1-z^2) + iz)
let i = Self::i();
-i * ((Self::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) -> Self {
// formula: arccos(z) = -i ln(i sqrt(1-z^2) + z)
let i = Self::i();
-i * (i * (Self::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) -> Self {
// formula: arctan(z) = (ln(1+iz) - ln(1-iz))/(2i)
let i = Self::i();
let one = Self::one();
let two = one + one;
if self == i {
return Self::new(T::zero(), T::infinity());
} else if self == -i {
return Self::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) -> Self {
// formula: sinh(a + bi) = sinh(a)cos(b) + i*cosh(a)sin(b)
Self::new(
self.re.sinh() * self.im.cos(),
self.re.cosh() * self.im.sin(),
)
}
/// Computes the hyperbolic cosine of `self`.
#[inline]
pub fn cosh(self) -> Self {
// formula: cosh(a + bi) = cosh(a)cos(b) + i*sinh(a)sin(b)
Self::new(
self.re.cosh() * self.im.cos(),
self.re.sinh() * self.im.sin(),
)
}
/// Computes the hyperbolic tangent of `self`.
#[inline]
pub fn tanh(self) -> Self {
// 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);
Self::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) -> Self {
// formula: arcsinh(z) = ln(z + sqrt(1+z^2))
let one = Self::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) -> Self {
// formula: arccosh(z) = 2 ln(sqrt((z+1)/2) + sqrt((z-1)/2))
let one = Self::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) -> Self {
// formula: arctanh(z) = (ln(1+z) - ln(1-z))/2
let one = Self::one();
let two = one + one;
if self == one {
return Self::new(T::infinity(), T::zero());
} else if self == -one {
return Self::new(-T::infinity(), T::zero());
}
((one + self).ln() - (one - self).ln()) / two
}
/// Returns `1/self` using floating-point operations.
///
/// This may be more accurate than the generic `self.inv()` in cases
/// where `self.norm_sqr()` would overflow to ∞ or underflow to 0.
///
/// # Examples
///
/// ```
/// use num_complex::Complex64;
/// let c = Complex64::new(1e300, 1e300);
///
/// // The generic `inv()` will overflow.
/// assert!(!c.inv().is_normal());
///
/// // But we can do better for `Float` types.
/// let inv = c.finv();
/// assert!(inv.is_normal());
/// println!("{:e}", inv);
///
/// let expected = Complex64::new(5e-301, -5e-301);
/// assert!((inv - expected).norm() < 1e-315);
/// ```
#[inline]
pub fn finv(self) -> Complex<T> {
let norm = self.norm();
self.conj() / norm / norm
}
/// Returns `self/other` using floating-point operations.
///
/// This may be more accurate than the generic `Div` implementation in cases
/// where `other.norm_sqr()` would overflow to ∞ or underflow to 0.
///
/// # Examples
///
/// ```
/// use num_complex::Complex64;
/// let a = Complex64::new(2.0, 3.0);
/// let b = Complex64::new(1e300, 1e300);
///
/// // Generic division will overflow.
/// assert!(!(a / b).is_normal());
///
/// // But we can do better for `Float` types.
/// let quotient = a.fdiv(b);
/// assert!(quotient.is_normal());
/// println!("{:e}", quotient);
///
/// let expected = Complex64::new(2.5e-300, 5e-301);
/// assert!((quotient - expected).norm() < 1e-315);
/// ```
#[inline]
pub fn fdiv(self, other: Complex<T>) -> Complex<T> {
self * other.finv()
}
}
#[cfg(any(feature = "std", feature = "libm"))]
impl<T: Float + FloatConst> Complex<T> {
/// Computes `2^(self)`.
#[inline]
pub fn exp2(self) -> Self {
// formula: 2^(a + bi) = 2^a (cos(b*log2) + i*sin(b*log2))
// = from_polar(2^a, b*log2)
Self::from_polar(self.re.exp2(), self.im * T::LN_2())
}
/// Computes the principal value of log base 2 of `self`.
#[inline]
pub fn log2(self) -> Self {
Self::ln(self) / T::LN_2()
}
/// Computes the principal value of log base 10 of `self`.
#[inline]
pub fn log10(self) -> Self {
Self::ln(self) / T::LN_10()
}
}
impl<T: FloatCore> Complex<T> {
/// 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()
}
}
// Safety: `Complex<T>` is `repr(C)` and contains only instances of `T`, so we
// can guarantee it contains no *added* padding. Thus, if `T: Zeroable`,
// `Complex<T>` is also `Zeroable`
#[cfg(feature = "bytemuck")]
unsafe impl<T: bytemuck::Zeroable> bytemuck::Zeroable for Complex<T> {}
// Safety: `Complex<T>` is `repr(C)` and contains only instances of `T`, so we
// can guarantee it contains no *added* padding. Thus, if `T: Pod`,
// `Complex<T>` is also `Pod`
#[cfg(feature = "bytemuck")]
unsafe impl<T: bytemuck::Pod> bytemuck::Pod for Complex<T> {}
impl<T: Clone + Num> From<T> for Complex<T> {
#[inline]
fn from(re: T) -> Self {
Self::new(re, T::zero())
}
}
impl<'a, T: Clone + Num> From<&'a T> for Complex<T> {
#[inline]
fn from(re: &T) -> Self {
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 $method(self, other: &Complex<T>) -> Self::Output {
self.clone().$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 $method(self, other: Complex<T>) -> Self::Output {
self.clone().$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 $method(self, other: &Complex<T>) -> Self::Output {
self.$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 = Self;
#[inline]
fn add(self, other: Self) -> Self::Output {
Self::Output::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 = Self;
#[inline]
fn sub(self, other: Self) -> Self::Output {
Self::Output::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 = Self;
#[inline]
fn mul(self, other: Self) -> Self::Output {
let re = self.re.clone() * other.re.clone() - self.im.clone() * other.im.clone();
let im = self.re * other.im + self.im * other.re;
Self::Output::new(re, im)
}
}
// (a + i b) * (c + i d) + (e + i f) == ((a*c + e) - b*d) + i (a*d + (b*c + f))
impl<T: Clone + Num + MulAdd<Output = T>> MulAdd<Complex<T>> for Complex<T> {
type Output = Complex<T>;
#[inline]
fn mul_add(self, other: Complex<T>, add: Complex<T>) -> Complex<T> {
let re = self.re.clone().mul_add(other.re.clone(), add.re)
- (self.im.clone() * other.im.clone()); // FIXME: use mulsub when available in rust
let im = self.re.mul_add(other.im, self.im.mul_add(other.re, add.im));
Complex::new(re, im)
}
}
impl<'a, 'b, T: Clone + Num + MulAdd<Output = T>> MulAdd<&'b Complex<T>> for &'a Complex<T> {
type Output = Complex<T>;
#[inline]
fn mul_add(self, other: &Complex<T>, add: &Complex<T>) -> Complex<T> {
self.clone().mul_add(other.clone(), add.clone())
}
}
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 = Self;
#[inline]
fn div(self, other: Self) -> Self::Output {
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;
Self::Output::new(re / norm_sqr.clone(), im / norm_sqr)
}
}
forward_all_binop!(impl Rem, rem);
impl<T: Clone + Num> Complex<T> {
/// Find the gaussian integer corresponding to the true ratio rounded towards zero.
fn div_trunc(&self, divisor: &Self) -> Self {
let Complex { re, im } = self / divisor;
Complex::new(re.clone() - re % T::one(), im.clone() - im % T::one())
}
}
impl<T: Clone + Num> Rem<Complex<T>> for Complex<T> {
type Output = Self;
#[inline]
fn rem(self, modulus: Self) -> Self::Output {
let gaussian = self.div_trunc(&modulus);
self - modulus * gaussian
}
}
// Op Assign
mod opassign {
use core::ops::{AddAssign, DivAssign, MulAssign, RemAssign, SubAssign};
use num_traits::{MulAddAssign, NumAssign};
use crate::Complex;
impl<T: Clone + NumAssign> AddAssign for Complex<T> {
fn add_assign(&mut self, other: Self) {
self.re += other.re;
self.im += other.im;
}
}
impl<T: Clone + NumAssign> SubAssign for Complex<T> {
fn sub_assign(&mut self, other: Self) {
self.re -= other.re;
self.im -= other.im;
}
}
// (a + i b) * (c + i d) == (a*c - b*d) + i (a*d + b*c)
impl<T: Clone + NumAssign> MulAssign for Complex<T> {
fn mul_assign(&mut self, other: Self) {
let a = self.re.clone();
self.re *= other.re.clone();
self.re -= self.im.clone() * other.im.clone();
self.im *= other.re;
self.im += a * other.im;
}
}
// (a + i b) * (c + i d) + (e + i f) == ((a*c + e) - b*d) + i (b*c + (a*d + f))
impl<T: Clone + NumAssign + MulAddAssign> MulAddAssign for Complex<T> {
fn mul_add_assign(&mut self, other: Complex<T>, add: Complex<T>) {
let a = self.re.clone();
self.re.mul_add_assign(other.re.clone(), add.re); // (a*c + e)
self.re -= self.im.clone() * other.im.clone(); // ((a*c + e) - b*d)
let mut adf = a;
adf.mul_add_assign(other.im, add.im); // (a*d + f)
self.im.mul_add_assign(other.re, adf); // (b*c + (a*d + f))
}
}
impl<'a, 'b, T: Clone + NumAssign + MulAddAssign> MulAddAssign<&'a Complex<T>, &'b Complex<T>>
for Complex<T>
{
fn mul_add_assign(&mut self, other: &Complex<T>, add: &Complex<T>) {
self.mul_add_assign(other.clone(), add.clone());
}
}
// (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 + NumAssign> DivAssign for Complex<T> {
fn div_assign(&mut self, other: Self) {
let a = self.re.clone();
let norm_sqr = other.norm_sqr();
self.re *= other.re.clone();
self.re += self.im.clone() * other.im.clone();
self.re /= norm_sqr.clone();
self.im *= other.re;
self.im -= a * other.im;
self.im /= norm_sqr;
}
}
impl<T: Clone + NumAssign> RemAssign for Complex<T> {
fn rem_assign(&mut self, modulus: Self) {
let gaussian = self.div_trunc(&modulus);
*self -= modulus * gaussian;
}
}
impl<T: Clone + NumAssign> AddAssign<T> for Complex<T> {
fn add_assign(&mut self, other: T) {
self.re += other;
}
}
impl<T: Clone + NumAssign> SubAssign<T> for Complex<T> {
fn sub_assign(&mut self, other: T) {
self.re -= other;
}
}
impl<T: Clone + NumAssign> MulAssign<T> for Complex<T> {
fn mul_assign(&mut self, other: T) {
self.re *= other.clone();
self.im *= other;
}
}
impl<T: Clone + NumAssign> DivAssign<T> for Complex<T> {
fn div_assign(&mut self, other: T) {
self.re /= other.clone();
self.im /= other;
}
}
impl<T: Clone + NumAssign> RemAssign<T> for Complex<T> {
fn rem_assign(&mut self, other: T) {
self.re %= other.clone();
self.im %= other;
}
}
macro_rules! forward_op_assign {
(impl $imp:ident, $method:ident) => {
impl<'a, T: Clone + NumAssign> $imp<&'a Complex<T>> for Complex<T> {
#[inline]
fn $method(&mut self, other: &Self) {
self.$method(other.clone())
}
}
impl<'a, T: Clone + NumAssign> $imp<&'a T> for Complex<T> {
#[inline]
fn $method(&mut self, other: &T) {
self.$method(other.clone())
}
}
};
}
forward_op_assign!(impl AddAssign, add_assign);
forward_op_assign!(impl SubAssign, sub_assign);
forward_op_assign!(impl MulAssign, mul_assign);
forward_op_assign!(impl DivAssign, div_assign);
forward_op_assign!(impl RemAssign, rem_assign);
}
impl<T: Clone + Num + Neg<Output = T>> Neg for Complex<T> {
type Output = Self;
#[inline]
fn neg(self) -> Self::Output {
Self::Output::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) -> Self::Output {
-self.clone()
}
}
impl<T: Clone + Num + Neg<Output = T>> Inv for Complex<T> {
type Output = Self;
#[inline]
fn inv(self) -> Self::Output {
Complex::inv(&self)
}
}
impl<'a, T: Clone + Num + Neg<Output = T>> Inv for &'a Complex<T> {
type Output = Complex<T>;
#[inline]
fn inv(self) -> Self::Output {
Complex::inv(self)
}
}
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 $method(self, other: &T) -> Self::Output {
self.$method(other.clone())
}
}
impl<'a, T: Clone + Num> $imp<T> for &'a Complex<T> {
type Output = Complex<T>;
#[inline]
fn $method(self, other: T) -> Self::Output {
self.clone().$method(other)
}
}
impl<'a, 'b, T: Clone + Num> $imp<&'a T> for &'b Complex<T> {
type Output = Complex<T>;
#[inline]
fn $method(self, other: &T) -> Self::Output {
self.clone().$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())
}
}
)*
);
($($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!(@forward Rem::rem for $($real),*);
$(
impl Add<Complex<$real>> for $real {
type Output = Complex<$real>;
#[inline]
fn add(self, other: Complex<$real>) -> Self::Output {
Self::Output::new(self + other.re, other.im)
}
}
impl Sub<Complex<$real>> for $real {
type Output = Complex<$real>;
#[inline]
fn sub(self, other: Complex<$real>) -> Self::Output {
Self::Output::new(self - other.re, $real::zero() - other.im)
}
}
impl Mul<Complex<$real>> for $real {
type Output = Complex<$real>;
#[inline]
fn mul(self, other: Complex<$real>) -> Self::Output {
Self::Output::new(self * other.re, self * other.im)
}
}
impl Div<Complex<$real>> for $real {
type Output = Complex<$real>;
#[inline]
fn div(self, other: Complex<$real>) -> Self::Output {
// a / (c + i d) == [a * (c - i d)] / (c*c + d*d)
let norm_sqr = other.norm_sqr();
Self::Output::new(self * other.re / norm_sqr.clone(),
$real::zero() - self * other.im / norm_sqr)
}
}
impl Rem<Complex<$real>> for $real {
type Output = Complex<$real>;
#[inline]
fn rem(self, other: Complex<$real>) -> Self::Output {
Self::Output::new(self, Self::zero()) % other
}
}
)*
);
}
impl<T: Clone + Num> Add<T> for Complex<T> {
type Output = Complex<T>;
#[inline]
fn add(self, other: T) -> Self::Output {
Self::Output::new(self.re + other, self.im)
}
}
impl<T: Clone + Num> Sub<T> for Complex<T> {
type Output = Complex<T>;
#[inline]
fn sub(self, other: T) -> Self::Output {
Self::Output::new(self.re - other, self.im)
}
}
impl<T: Clone + Num> Mul<T> for Complex<T> {
type Output = Complex<T>;
#[inline]
fn mul(self, other: T) -> Self::Output {
Self::Output::new(self.re * other.clone(), self.im * other)
}
}
impl<T: Clone + Num> Div<T> for Complex<T> {
type Output = Self;
#[inline]
fn div(self, other: T) -> Self::Output {
Self::Output::new(self.re / other.clone(), self.im / other)
}
}
impl<T: Clone + Num> Rem<T> for Complex<T> {
type Output = Complex<T>;
#[inline]
fn rem(self, other: T) -> Self::Output {
Self::Output::new(self.re % other.clone(), self.im % other)
}
}
real_arithmetic!(usize, u8, u16, u32, u64, u128, isize, i8, i16, i32, i64, i128, f32, f64);
// constants
impl<T: ConstZero> Complex<T> {
/// A constant `Complex` 0.
pub const ZERO: Self = Self::new(T::ZERO, T::ZERO);
}
impl<T: Clone + Num + ConstZero> ConstZero for Complex<T> {
const ZERO: Self = Self::ZERO;
}
impl<T: Clone + Num> Zero for Complex<T> {
#[inline]
fn zero() -> Self {
Self::new(Zero::zero(), Zero::zero())
}
#[inline]
fn is_zero(&self) -> bool {
self.re.is_zero() && self.im.is_zero()
}
#[inline]
fn set_zero(&mut self) {
self.re.set_zero();
self.im.set_zero();
}
}
impl<T: ConstOne + ConstZero> Complex<T> {
/// A constant `Complex` 1.
pub const ONE: Self = Self::new(T::ONE, T::ZERO);
/// A constant `Complex` _i_, the imaginary unit.
pub const I: Self = Self::new(T::ZERO, T::ONE);
}
impl<T: Clone + Num + ConstOne + ConstZero> ConstOne for Complex<T> {
const ONE: Self = Self::ONE;
}
impl<T: Clone + Num> One for Complex<T> {
#[inline]
fn one() -> Self {
Self::new(One::one(), Zero::zero())
}
#[inline]
fn is_one(&self) -> bool {
self.re.is_one() && self.im.is_zero()
}
#[inline]
fn set_one(&mut self) {
self.re.set_one();
self.im.set_zero();
}
}
macro_rules! write_complex {
($f:ident, $t:expr, $prefix:expr, $re:expr, $im:expr, $T:ident) => {{
let abs_re = if $re < Zero::zero() {
$T::zero() - $re.clone()
} else {
$re.clone()
};
let abs_im = if $im < Zero::zero() {
$T::zero() - $im.clone()
} else {
$im.clone()
};
return if let Some(prec) = $f.precision() {
fmt_re_im(
$f,
$re < $T::zero(),
$im < $T::zero(),
format_args!(concat!("{:.1$", $t, "}"), abs_re, prec),
format_args!(concat!("{:.1$", $t, "}"), abs_im, prec),
)
} else {
fmt_re_im(
$f,
$re < $T::zero(),
$im < $T::zero(),
format_args!(concat!("{:", $t, "}"), abs_re),
format_args!(concat!("{:", $t, "}"), abs_im),
)
};
fn fmt_re_im(
f: &mut fmt::Formatter<'_>,
re_neg: bool,
im_neg: bool,
real: fmt::Arguments<'_>,
imag: fmt::Arguments<'_>,
) -> fmt::Result {
let prefix = if f.alternate() { $prefix } else { "" };
let sign = if re_neg {
"-"
} else if f.sign_plus() {
"+"
} else {
""
};
if im_neg {
fmt_complex(
f,
format_args!(
"{}{pre}{re}-{pre}{im}i",
sign,
re = real,
im = imag,
pre = prefix
),
)
} else {
fmt_complex(
f,
format_args!(
"{}{pre}{re}+{pre}{im}i",
sign,
re = real,
im = imag,
pre = prefix
),
)
}
}
#[cfg(feature = "std")]
// Currently, we can only apply width using an intermediate `String` (and thus `std`)
fn fmt_complex(f: &mut fmt::Formatter<'_>, complex: fmt::Arguments<'_>) -> fmt::Result {
use std::string::ToString;
if let Some(width) = f.width() {
write!(f, "{0: >1$}", complex.to_string(), width)
} else {
write!(f, "{}", complex)
}
}
#[cfg(not(feature = "std"))]
fn fmt_complex(f: &mut fmt::Formatter<'_>, complex: fmt::Arguments<'_>) -> fmt::Result {
write!(f, "{}", complex)
}
}};
}
// 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 {
write_complex!(f, "", "", self.re, self.im, T)
}
}
impl<T> fmt::LowerExp for Complex<T>
where
T: fmt::LowerExp + Num + PartialOrd + Clone,
{
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write_complex!(f, "e", "", self.re, self.im, T)
}
}
impl<T> fmt::UpperExp for Complex<T>
where
T: fmt::UpperExp + Num + PartialOrd + Clone,
{
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write_complex!(f, "E", "", self.re, self.im, T)
}
}
impl<T> fmt::LowerHex for Complex<T>
where
T: fmt::LowerHex + Num + PartialOrd + Clone,
{
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write_complex!(f, "x", "0x", self.re, self.im, T)
}
}
impl<T> fmt::UpperHex for Complex<T>
where
T: fmt::UpperHex + Num + PartialOrd + Clone,
{
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write_complex!(f, "X", "0x", self.re, self.im, T)
}
}
impl<T> fmt::Octal for Complex<T>
where
T: fmt::Octal + Num + PartialOrd + Clone,
{
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write_complex!(f, "o", "0o", self.re, self.im, T)
}
}
impl<T> fmt::Binary for Complex<T>
where
T: fmt::Binary + Num + PartialOrd + Clone,
{
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
write_complex!(f, "b", "0b", self.re, self.im, T)
}
}
fn from_str_generic<T, E, F>(s: &str, from: F) -> Result<Complex<T>, ParseComplexError<E>>
where
F: Fn(&str) -> Result<T, E>,
T: Clone + Num,
{
let imag = match s.rfind('j') {
None => 'i',
_ => 'j',
};
let mut neg_b = false;
let mut a = s;
let mut b = "";
for (i, w) in s.as_bytes().windows(2).enumerate() {
let p = w[0];
let c = w[1];
// ignore '+'/'-' if part of an exponent
if (c == b'+' || c == b'-') && !(p == b'e' || p == b'E') {
// trim whitespace around the separator
a = s[..=i].trim_end_matches(char::is_whitespace);
b = s[i + 2..].trim_start_matches(char::is_whitespace);
neg_b = c == b'-';
if b.is_empty() || (neg_b && b.starts_with('-')) {
return Err(ParseComplexError::expr_error());
}
break;
}
}
// split off real and imaginary parts
if b.is_empty() {
// input was either pure real or pure imaginary
b = if a.ends_with(imag) { "0" } else { "0i" };
}
let re;
let neg_re;
let im;
let neg_im;
if a.ends_with(imag) {
im = a;
neg_im = false;
re = b;
neg_re = neg_b;
} else if b.ends_with(imag) {
re = a;
neg_re = false;
im = b;
neg_im = neg_b;
} else {
return Err(ParseComplexError::expr_error());
}
// parse re
let re = from(re).map_err(ParseComplexError::from_error)?;
let re = if neg_re { T::zero() - re } else { re };
// pop imaginary unit off
let mut im = &im[..im.len() - 1];
// handle im == "i" or im == "-i"
if im.is_empty() || im == "+" {
im = "1";
} else if im == "-" {
im = "-1";
}
// parse im
let im = from(im).map_err(ParseComplexError::from_error)?;
let im = if neg_im { T::zero() - im } else { im };
Ok(Complex::new(re, im))
}
impl<T> FromStr for Complex<T>
where
T: FromStr + Num + Clone,
{
type Err = ParseComplexError<T::Err>;
/// Parses `a +/- bi`; `ai +/- b`; `a`; or `bi` where `a` and `b` are of type `T`
fn from_str(s: &str) -> Result<Self, Self::Err> {
from_str_generic(s, T::from_str)
}
}
impl<T: Num + Clone> Num for Complex<T> {
type FromStrRadixErr = ParseComplexError<T::FromStrRadixErr>;
/// Parses `a +/- bi`; `ai +/- b`; `a`; or `bi` where `a` and `b` are of type `T`
///
/// `radix` must be <= 18; larger radix would include *i* and *j* as digits,
/// which cannot be supported.
///
/// The conversion returns an error if 18 <= radix <= 36; it panics if radix > 36.
///
/// The elements of `T` are parsed using `Num::from_str_radix` too, and errors
/// (or panics) from that are reflected here as well.
fn from_str_radix(s: &str, radix: u32) -> Result<Self, Self::FromStrRadixErr> {
assert!(
radix <= 36,
"from_str_radix: radix is too high (maximum 36)"
);
// larger radix would include 'i' and 'j' as digits, which cannot be supported
if radix > 18 {
return Err(ParseComplexError::unsupported_radix());
}
from_str_generic(s, |x| -> Result<T, T::FromStrRadixErr> {
T::from_str_radix(x, radix)
})
}
}
impl<T: Num + Clone> Sum for Complex<T> {
fn sum<I>(iter: I) -> Self
where
I: Iterator<Item = Self>,
{
iter.fold(Self::zero(), |acc, c| acc + c)
}
}
impl<'a, T: 'a + Num + Clone> Sum<&'a Complex<T>> for Complex<T> {
fn sum<I>(iter: I) -> Self
where
I: Iterator<Item = &'a Complex<T>>,
{
iter.fold(Self::zero(), |acc, c| acc + c)
}
}
impl<T: Num + Clone> Product for Complex<T> {
fn product<I>(iter: I) -> Self
where
I: Iterator<Item = Self>,
{
iter.fold(Self::one(), |acc, c| acc * c)
}
}
impl<'a, T: 'a + Num + Clone> Product<&'a Complex<T>> for Complex<T> {
fn product<I>(iter: I) -> Self
where
I: Iterator<Item = &'a Complex<T>>,
{
iter.fold(Self::one(), |acc, c| acc * c)
}
}
#[cfg(feature = "serde")]
impl<T> serde::Serialize for Complex<T>
where
T: serde::Serialize,
{
fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
where
S: serde::Serializer,
{
(&self.re, &self.im).serialize(serializer)
}
}
#[cfg(feature = "serde")]
impl<'de, T> serde::Deserialize<'de> for Complex<T>
where
T: serde::Deserialize<'de>,
{
fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>
where
D: serde::Deserializer<'de>,
{
let (re, im) = serde::Deserialize::deserialize(deserializer)?;
Ok(Self::new(re, im))
}
}
#[derive(Debug, PartialEq)]
pub struct ParseComplexError<E> {
kind: ComplexErrorKind<E>,
}
#[derive(Debug, PartialEq)]
enum ComplexErrorKind<E> {
ParseError(E),
ExprError,
UnsupportedRadix,
}
impl<E> ParseComplexError<E> {
fn expr_error() -> Self {
ParseComplexError {
kind: ComplexErrorKind::ExprError,
}
}
fn unsupported_radix() -> Self {
ParseComplexError {
kind: ComplexErrorKind::UnsupportedRadix,
}
}
fn from_error(error: E) -> Self {
ParseComplexError {
kind: ComplexErrorKind::ParseError(error),
}
}
}
#[cfg(feature = "std")]
impl<E: Error> Error for ParseComplexError<E> {
#[allow(deprecated)]
fn description(&self) -> &str {
match self.kind {
ComplexErrorKind::ParseError(ref e) => e.description(),
ComplexErrorKind::ExprError => "invalid or unsupported complex expression",
ComplexErrorKind::UnsupportedRadix => "unsupported radix for conversion",
}
}
}
impl<E: fmt::Display> fmt::Display for ParseComplexError<E> {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
match self.kind {
ComplexErrorKind::ParseError(ref e) => e.fmt(f),
ComplexErrorKind::ExprError => "invalid or unsupported complex expression".fmt(f),
ComplexErrorKind::UnsupportedRadix => "unsupported radix for conversion".fmt(f),
}
}
}
#[cfg(test)]
fn hash<T: hash::Hash>(x: &T) -> u64 {
use std::collections::hash_map::RandomState;
use std::hash::{BuildHasher, Hasher};
let mut hasher = <RandomState as BuildHasher>::Hasher::new();
x.hash(&mut hasher);
hasher.finish()
}
#[cfg(test)]
pub(crate) mod test {
#![allow(non_upper_case_globals)]
use super::{Complex, Complex64};
use super::{ComplexErrorKind, ParseComplexError};
use core::f64;
use core::str::FromStr;
use std::string::{String, ToString};
use num_traits::{Num, One, Zero};
pub const _0_0i: Complex64 = Complex::new(0.0, 0.0);
pub const _1_0i: Complex64 = Complex::new(1.0, 0.0);
pub const _1_1i: Complex64 = Complex::new(1.0, 1.0);
pub const _0_1i: Complex64 = Complex::new(0.0, 1.0);
pub const _neg1_1i: Complex64 = Complex::new(-1.0, 1.0);
pub const _05_05i: Complex64 = Complex::new(0.5, 0.5);
pub const all_consts: [Complex64; 5] = [_0_0i, _1_0i, _1_1i, _neg1_1i, _05_05i];
pub const _4_2i: Complex64 = Complex::new(4.0, 2.0);
pub const _1_infi: Complex64 = Complex::new(1.0, f64::INFINITY);
pub const _neg1_infi: Complex64 = Complex::new(-1.0, f64::INFINITY);
pub const _1_nani: Complex64 = Complex::new(1.0, f64::NAN);
pub const _neg1_nani: Complex64 = Complex::new(-1.0, f64::NAN);
pub const _inf_0i: Complex64 = Complex::new(f64::INFINITY, 0.0);
pub const _neginf_1i: Complex64 = Complex::new(f64::NEG_INFINITY, 1.0);
pub const _neginf_neg1i: Complex64 = Complex::new(f64::NEG_INFINITY, -1.0);
pub const _inf_1i: Complex64 = Complex::new(f64::INFINITY, 1.0);
pub const _inf_neg1i: Complex64 = Complex::new(f64::INFINITY, -1.0);
pub const _neginf_infi: Complex64 = Complex::new(f64::NEG_INFINITY, f64::INFINITY);
pub const _inf_infi: Complex64 = Complex::new(f64::INFINITY, f64::INFINITY);
pub const _neginf_nani: Complex64 = Complex::new(f64::NEG_INFINITY, f64::NAN);
pub const _inf_nani: Complex64 = Complex::new(f64::INFINITY, f64::NAN);
pub const _nan_0i: Complex64 = Complex::new(f64::NAN, 0.0);
pub const _nan_1i: Complex64 = Complex::new(f64::NAN, 1.0);
pub const _nan_neg1i: Complex64 = Complex::new(f64::NAN, -1.0);
pub const _nan_nani: Complex64 = Complex::new(f64::NAN, f64::NAN);
#[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]
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]
#[allow(clippy::float_cmp)]
fn test_l1_norm() {
assert_eq!(_0_0i.l1_norm(), 0.0);
assert_eq!(_1_0i.l1_norm(), 1.0);
assert_eq!(_1_1i.l1_norm(), 2.0);
assert_eq!(_0_1i.l1_norm(), 1.0);
assert_eq!(_neg1_1i.l1_norm(), 2.0);
assert_eq!(_05_05i.l1_norm(), 1.0);
assert_eq!(_4_2i.l1_norm(), 6.0);
}
#[test]
fn test_pow() {
for c in all_consts.iter() {
assert_eq!(c.powi(0), _1_0i);
let mut pos = _1_0i;
let mut neg = _1_0i;
for i in 1i32..20 {
pos *= c;
assert_eq!(pos, c.powi(i));
if c.is_zero() {
assert!(c.powi(-i).is_nan());
} else {
neg /= c;
assert_eq!(neg, c.powi(-i));
}
}
}
}
#[cfg(any(feature = "std", feature = "libm"))]
pub(crate) mod float {
use core::f64::INFINITY;
use super::*;
use num_traits::{Float, Pow};
#[test]
fn test_cis() {
assert!(close(Complex::cis(0.0 * f64::consts::PI), _1_0i));
assert!(close(Complex::cis(0.5 * f64::consts::PI), _0_1i));
assert!(close(Complex::cis(1.0 * f64::consts::PI), -_1_0i));
assert!(close(Complex::cis(1.5 * f64::consts::PI), -_0_1i));
assert!(close(Complex::cis(2.0 * f64::consts::PI), _1_0i));
}
#[test]
#[cfg_attr(target_arch = "x86", ignore)]
// FIXME #7158: (maybe?) currently failing on x86.
#[allow(clippy::float_cmp)]
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_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);
}
}
pub(crate) fn close(a: Complex64, b: Complex64) -> bool {
close_to_tol(a, b, 1e-10)
}
fn close_to_tol(a: Complex64, b: Complex64, tol: f64) -> bool {
// returns true if a and b are reasonably close
let close = (a == b) || (a - b).norm() < tol;
if !close {
println!("{:?} != {:?}", a, b);
}
close
}
// Version that also works if re or im are +inf, -inf, or nan
fn close_naninf(a: Complex64, b: Complex64) -> bool {
close_naninf_to_tol(a, b, 1.0e-10)
}
fn close_naninf_to_tol(a: Complex64, b: Complex64, tol: f64) -> bool {
let mut close = true;
// Compare the real parts
if a.re.is_finite() {
if b.re.is_finite() {
close = (a.re == b.re) || (a.re - b.re).abs() < tol;
} else {
close = false;
}
} else if (a.re.is_nan() && !b.re.is_nan())
|| (a.re.is_infinite()
&& a.re.is_sign_positive()
&& !(b.re.is_infinite() && b.re.is_sign_positive()))
|| (a.re.is_infinite()
&& a.re.is_sign_negative()
&& !(b.re.is_infinite() && b.re.is_sign_negative()))
{
close = false;
}
// Compare the imaginary parts
if a.im.is_finite() {
if b.im.is_finite() {
close &= (a.im == b.im) || (a.im - b.im).abs() < tol;
} else {
close = false;
}
} else if (a.im.is_nan() && !b.im.is_nan())
|| (a.im.is_infinite()
&& a.im.is_sign_positive()
&& !(b.im.is_infinite() && b.im.is_sign_positive()))
|| (a.im.is_infinite()
&& a.im.is_sign_negative()
&& !(b.im.is_infinite() && b.im.is_sign_negative()))
{
close = false;
}
if close == false {
println!("{:?} != {:?}", a, b);
}
close
}
#[test]
fn test_exp2() {
assert!(close(_0_0i.exp2(), _1_0i));
}
#[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()
));
}
// The test values below were taken from https://en.cppreference.com/w/cpp/numeric/complex/exp
assert!(close_naninf(_1_infi.exp(), _nan_nani));
assert!(close_naninf(_neg1_infi.exp(), _nan_nani));
assert!(close_naninf(_1_nani.exp(), _nan_nani));
assert!(close_naninf(_neg1_nani.exp(), _nan_nani));
assert!(close_naninf(_inf_0i.exp(), _inf_0i));
assert!(close_naninf(_neginf_1i.exp(), 0.0 * Complex::cis(1.0)));
assert!(close_naninf(_neginf_neg1i.exp(), 0.0 * Complex::cis(-1.0)));
assert!(close_naninf(
_inf_1i.exp(),
f64::INFINITY * Complex::cis(1.0)
));
assert!(close_naninf(
_inf_neg1i.exp(),
f64::INFINITY * Complex::cis(-1.0)
));
assert!(close_naninf(_neginf_infi.exp(), _0_0i)); // Note: ±0±0i: signs of zeros are unspecified
assert!(close_naninf(_inf_infi.exp(), _inf_nani)); // Note: ±∞+NaN*i: sign of the real part is unspecified
assert!(close_naninf(_neginf_nani.exp(), _0_0i)); // Note: ±0±0i: signs of zeros are unspecified
assert!(close_naninf(_inf_nani.exp(), _inf_nani)); // Note: ±∞+NaN*i: sign of the real part is unspecified
assert!(close_naninf(_nan_0i.exp(), _nan_0i));
assert!(close_naninf(_nan_1i.exp(), _nan_nani));
assert!(close_naninf(_nan_neg1i.exp(), _nan_nani));
assert!(close_naninf(_nan_nani.exp(), _nan_nani));
}
#[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_powc() {
let a = Complex::new(2.0, -3.0);
let b = Complex::new(3.0, 0.0);
assert!(close(a.powc(b), a.powf(b.re)));
assert!(close(b.powc(a), a.expf(b.re)));
let c = Complex::new(1.0 / 3.0, 0.1);
assert!(close_to_tol(
a.powc(c),
Complex::new(1.65826, -0.33502),
1e-5
));
let z = Complex::new(0.0, 0.0);
assert!(close(z.powc(b), z));
assert!(z.powc(Complex64::new(0., INFINITY)).is_nan());
assert!(z.powc(Complex64::new(10., INFINITY)).is_nan());
assert!(z.powc(Complex64::new(INFINITY, INFINITY)).is_nan());
assert!(close(z.powc(Complex64::new(INFINITY, 0.)), z));
assert!(z.powc(Complex64::new(-1., 0.)).re.is_infinite());
assert!(z.powc(Complex64::new(-1., 0.)).im.is_nan());
for c in all_consts.iter() {
assert_eq!(c.powc(_0_0i), _1_0i);
}
assert_eq!(_nan_nani.powc(_0_0i), _1_0i);
}
#[test]
fn test_powf() {
let c = Complex64::new(2.0, -1.0);
let expected = Complex64::new(-0.8684746, -16.695934);
assert!(close_to_tol(c.powf(3.5), expected, 1e-5));
assert!(close_to_tol(Pow::pow(c, 3.5_f64), expected, 1e-5));
assert!(close_to_tol(Pow::pow(c, 3.5_f32), expected, 1e-5));
for c in all_consts.iter() {
assert_eq!(c.powf(0.0), _1_0i);
}
assert_eq!(_nan_nani.powf(0.0), _1_0i);
}
#[test]
fn test_log() {
let c = Complex::new(2.0, -1.0);
let r = c.log(10.0);
assert!(close_to_tol(r, Complex::new(0.349485, -0.20135958), 1e-5));
}
#[test]
fn test_log2() {
assert!(close(_1_0i.log2(), _0_0i));
}
#[test]
fn test_log10() {
assert!(close(_1_0i.log10(), _0_0i));
}
#[test]
fn test_some_expf_cases() {
let c = Complex::new(2.0, -1.0);
let r = c.expf(10.0);
assert!(close_to_tol(r, Complex::new(-66.82015, -74.39803), 1e-5));
let c = Complex::new(5.0, -2.0);
let r = c.expf(3.4);
assert!(close_to_tol(r, Complex::new(-349.25, -290.63), 1e-2));
let c = Complex::new(-1.5, 2.0 / 3.0);
let r = c.expf(1.0 / 3.0);
assert!(close_to_tol(r, Complex::new(3.8637, -3.4745), 1e-2));
}
#[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::FRAC_PI_2 <= c.sqrt().arg()
&& c.sqrt().arg() <= f64::consts::FRAC_PI_2
);
// sqrt(z) * sqrt(z) = z
assert!(close(c.sqrt() * c.sqrt(), c));
}
}
#[test]
fn test_sqrt_real() {
for n in (0..100).map(f64::from) {
// √(n² + 0i) = n + 0i
let n2 = n * n;
assert_eq!(Complex64::new(n2, 0.0).sqrt(), Complex64::new(n, 0.0));
// √(-n² + 0i) = 0 + ni
assert_eq!(Complex64::new(-n2, 0.0).sqrt(), Complex64::new(0.0, n));
// √(-n² - 0i) = 0 - ni
assert_eq!(Complex64::new(-n2, -0.0).sqrt(), Complex64::new(0.0, -n));
}
}
#[test]
fn test_sqrt_imag() {
for n in (0..100).map(f64::from) {
// √(0 + n²i) = n e^(iπ/4)
let n2 = n * n;
assert!(close(
Complex64::new(0.0, n2).sqrt(),
Complex64::from_polar(n, f64::consts::FRAC_PI_4)
));
// √(0 - n²i) = n e^(-iπ/4)
assert!(close(
Complex64::new(0.0, -n2).sqrt(),
Complex64::from_polar(n, -f64::consts::FRAC_PI_4)
));
}
}
#[test]
fn test_cbrt() {
assert!(close(_0_0i.cbrt(), _0_0i));
assert!(close(_1_0i.cbrt(), _1_0i));
assert!(close(
Complex::new(-1.0, 0.0).cbrt(),
Complex::new(0.5, 0.75.sqrt())
));
assert!(close(
Complex::new(-1.0, -0.0).cbrt(),
Complex::new(0.5, -(0.75.sqrt()))
));
assert!(close(_0_1i.cbrt(), Complex::new(0.75.sqrt(), 0.5)));
assert!(close(_0_1i.conj().cbrt(), Complex::new(0.75.sqrt(), -0.5)));
for &c in all_consts.iter() {
// cbrt(conj(z() = conj(cbrt(z))
assert!(close(c.conj().cbrt(), c.cbrt().conj()));
// for this branch, -pi/3 <= arg(cbrt(z)) <= pi/3
assert!(
-f64::consts::FRAC_PI_3 <= c.cbrt().arg()
&& c.cbrt().arg() <= f64::consts::FRAC_PI_3
);
// cbrt(z) * cbrt(z) cbrt(z) = z
assert!(close(c.cbrt() * c.cbrt() * c.cbrt(), c));
}
}
#[test]
fn test_cbrt_real() {
for n in (0..100).map(f64::from) {
// ∛(n³ + 0i) = n + 0i
let n3 = n * n * n;
assert!(close(
Complex64::new(n3, 0.0).cbrt(),
Complex64::new(n, 0.0)
));
// ∛(-n³ + 0i) = n e^(iπ/3)
assert!(close(
Complex64::new(-n3, 0.0).cbrt(),
Complex64::from_polar(n, f64::consts::FRAC_PI_3)
));
// ∛(-n³ - 0i) = n e^(-iπ/3)
assert!(close(
Complex64::new(-n3, -0.0).cbrt(),
Complex64::from_polar(n, -f64::consts::FRAC_PI_3)
));
}
}
#[test]
fn test_cbrt_imag() {
for n in (0..100).map(f64::from) {
// ∛(0 + n³i) = n e^(iπ/6)
let n3 = n * n * n;
assert!(close(
Complex64::new(0.0, n3).cbrt(),
Complex64::from_polar(n, f64::consts::FRAC_PI_6)
));
// ∛(0 - n³i) = n e^(-iπ/6)
assert!(close(
Complex64::new(0.0, -n3).cbrt(),
Complex64::from_polar(n, -f64::consts::FRAC_PI_6)
));
}
}
#[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_exp2_log() {
for &c in all_consts.iter() {
// 2^log2(z) = z
assert!(close(c.log2().exp2(), 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()
));
}
}
}
// Test both a + b and a += b
macro_rules! test_a_op_b {
($a:ident + $b:expr, $answer:expr) => {
assert_eq!($a + $b, $answer);
assert_eq!(
{
let mut x = $a;
x += $b;
x
},
$answer
);
};
($a:ident - $b:expr, $answer:expr) => {
assert_eq!($a - $b, $answer);
assert_eq!(
{
let mut x = $a;
x -= $b;
x
},
$answer
);
};
($a:ident * $b:expr, $answer:expr) => {
assert_eq!($a * $b, $answer);
assert_eq!(
{
let mut x = $a;
x *= $b;
x
},
$answer
);
};
($a:ident / $b:expr, $answer:expr) => {
assert_eq!($a / $b, $answer);
assert_eq!(
{
let mut x = $a;
x /= $b;
x
},
$answer
);
};
($a:ident % $b:expr, $answer:expr) => {
assert_eq!($a % $b, $answer);
assert_eq!(
{
let mut x = $a;
x %= $b;
x
},
$answer
);
};
}
// Test both a + b and a + &b
macro_rules! test_op {
($a:ident $op:tt $b:expr, $answer:expr) => {
test_a_op_b!($a $op $b, $answer);
test_a_op_b!($a $op &$b, $answer);
};
}
mod complex_arithmetic {
use super::{_05_05i, _0_0i, _0_1i, _1_0i, _1_1i, _4_2i, _neg1_1i, all_consts};
use num_traits::{MulAdd, MulAddAssign, Zero};
#[test]
fn test_add() {
test_op!(_05_05i + _05_05i, _1_1i);
test_op!(_0_1i + _1_0i, _1_1i);
test_op!(_1_0i + _neg1_1i, _0_1i);
for &c in all_consts.iter() {
test_op!(_0_0i + c, c);
test_op!(c + _0_0i, c);
}
}
#[test]
fn test_sub() {
test_op!(_05_05i - _05_05i, _0_0i);
test_op!(_0_1i - _1_0i, _neg1_1i);
test_op!(_0_1i - _neg1_1i, _1_0i);
for &c in all_consts.iter() {
test_op!(c - _0_0i, c);
test_op!(c - c, _0_0i);
}
}
#[test]
fn test_mul() {
test_op!(_05_05i * _05_05i, _0_1i.unscale(2.0));
test_op!(_1_1i * _0_1i, _neg1_1i);
// i^2 & i^4
test_op!(_0_1i * _0_1i, -_1_0i);
assert_eq!(_0_1i * _0_1i * _0_1i * _0_1i, _1_0i);
for &c in all_consts.iter() {
test_op!(c * _1_0i, c);
test_op!(_1_0i * c, c);
}
}
#[test]
#[cfg(any(feature = "std", feature = "libm"))]
fn test_mul_add_float() {
assert_eq!(_05_05i.mul_add(_05_05i, _0_0i), _05_05i * _05_05i + _0_0i);
assert_eq!(_05_05i * _05_05i + _0_0i, _05_05i.mul_add(_05_05i, _0_0i));
assert_eq!(_0_1i.mul_add(_0_1i, _0_1i), _neg1_1i);
assert_eq!(_1_0i.mul_add(_1_0i, _1_0i), _1_0i * _1_0i + _1_0i);
assert_eq!(_1_0i * _1_0i + _1_0i, _1_0i.mul_add(_1_0i, _1_0i));
let mut x = _1_0i;
x.mul_add_assign(_1_0i, _1_0i);
assert_eq!(x, _1_0i * _1_0i + _1_0i);
for &a in &all_consts {
for &b in &all_consts {
for &c in &all_consts {
let abc = a * b + c;
assert_eq!(a.mul_add(b, c), abc);
let mut x = a;
x.mul_add_assign(b, c);
assert_eq!(x, abc);
}
}
}
}
#[test]
fn test_mul_add() {
use super::Complex;
const _0_0i: Complex<i32> = Complex { re: 0, im: 0 };
const _1_0i: Complex<i32> = Complex { re: 1, im: 0 };
const _1_1i: Complex<i32> = Complex { re: 1, im: 1 };
const _0_1i: Complex<i32> = Complex { re: 0, im: 1 };
const _neg1_1i: Complex<i32> = Complex { re: -1, im: 1 };
const all_consts: [Complex<i32>; 5] = [_0_0i, _1_0i, _1_1i, _0_1i, _neg1_1i];
assert_eq!(_1_0i.mul_add(_1_0i, _0_0i), _1_0i * _1_0i + _0_0i);
assert_eq!(_1_0i * _1_0i + _0_0i, _1_0i.mul_add(_1_0i, _0_0i));
assert_eq!(_0_1i.mul_add(_0_1i, _0_1i), _neg1_1i);
assert_eq!(_1_0i.mul_add(_1_0i, _1_0i), _1_0i * _1_0i + _1_0i);
assert_eq!(_1_0i * _1_0i + _1_0i, _1_0i.mul_add(_1_0i, _1_0i));
let mut x = _1_0i;
x.mul_add_assign(_1_0i, _1_0i);
assert_eq!(x, _1_0i * _1_0i + _1_0i);
for &a in &all_consts {
for &b in &all_consts {
for &c in &all_consts {
let abc = a * b + c;
assert_eq!(a.mul_add(b, c), abc);
let mut x = a;
x.mul_add_assign(b, c);
assert_eq!(x, abc);
}
}
}
}
#[test]
fn test_div() {
test_op!(_neg1_1i / _0_1i, _1_1i);
for &c in all_consts.iter() {
if c != Zero::zero() {
test_op!(c / c, _1_0i);
}
}
}
#[test]
fn test_rem() {
test_op!(_neg1_1i % _0_1i, _0_0i);
test_op!(_4_2i % _0_1i, _0_0i);
test_op!(_05_05i % _0_1i, _05_05i);
test_op!(_05_05i % _1_1i, _05_05i);
assert_eq!((_4_2i + _05_05i) % _0_1i, _05_05i);
assert_eq!((_4_2i + _05_05i) % _1_1i, _05_05i);
}
#[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;
use super::{_4_2i, _neg1_1i};
#[test]
fn test_add() {
test_op!(_4_2i + 0.5, Complex::new(4.5, 2.0));
assert_eq!(0.5 + _4_2i, Complex::new(4.5, 2.0));
}
#[test]
fn test_sub() {
test_op!(_4_2i - 0.5, Complex::new(3.5, 2.0));
assert_eq!(0.5 - _4_2i, Complex::new(-3.5, -2.0));
}
#[test]
fn test_mul() {
assert_eq!(_4_2i * 0.5, Complex::new(2.0, 1.0));
assert_eq!(0.5 * _4_2i, Complex::new(2.0, 1.0));
}
#[test]
fn test_div() {
assert_eq!(_4_2i / 0.5, Complex::new(8.0, 4.0));
assert_eq!(0.5 / _4_2i, Complex::new(0.1, -0.05));
}
#[test]
fn test_rem() {
assert_eq!(_4_2i % 2.0, Complex::new(0.0, 0.0));
assert_eq!(_4_2i % 3.0, Complex::new(1.0, 2.0));
assert_eq!(3.0 % _4_2i, Complex::new(3.0, 0.0));
assert_eq!(_neg1_1i % 2.0, _neg1_1i);
assert_eq!(-_4_2i % 3.0, Complex::new(-1.0, -2.0));
}
#[test]
fn test_div_rem_gaussian() {
// These would overflow with `norm_sqr` division.
let max = Complex::new(255u8, 255u8);
assert_eq!(max / 200, Complex::new(1, 1));
assert_eq!(max % 200, Complex::new(55, 55));
}
}
#[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_string_formatting() {
let a = Complex::new(1.23456, 123.456);
assert_eq!(format!("{}", a), "1.23456+123.456i");
assert_eq!(format!("{:.2}", a), "1.23+123.46i");
assert_eq!(format!("{:.2e}", a), "1.23e0+1.23e2i");
assert_eq!(format!("{:+.2E}", a), "+1.23E0+1.23E2i");
#[cfg(feature = "std")]
assert_eq!(format!("{:+20.2E}", a), " +1.23E0+1.23E2i");
let b = Complex::new(0x80, 0xff);
assert_eq!(format!("{:X}", b), "80+FFi");
assert_eq!(format!("{:#x}", b), "0x80+0xffi");
assert_eq!(format!("{:+#b}", b), "+0b10000000+0b11111111i");
assert_eq!(format!("{:+#o}", b), "+0o200+0o377i");
#[cfg(feature = "std")]
assert_eq!(format!("{:+#16o}", b), " +0o200+0o377i");
let c = Complex::new(-10, -10000);
assert_eq!(format!("{}", c), "-10-10000i");
#[cfg(feature = "std")]
assert_eq!(format!("{:16}", c), " -10-10000i");
}
#[test]
fn test_hash() {
let a = Complex::new(0i32, 0i32);
let b = Complex::new(1i32, 0i32);
let c = Complex::new(0i32, 1i32);
assert!(crate::hash(&a) != crate::hash(&b));
assert!(crate::hash(&b) != crate::hash(&c));
assert!(crate::hash(&c) != crate::hash(&a));
}
#[test]
fn test_hashset() {
use std::collections::HashSet;
let a = Complex::new(0i32, 0i32);
let b = Complex::new(1i32, 0i32);
let c = Complex::new(0i32, 1i32);
let set: HashSet<_> = [a, b, c].iter().cloned().collect();
assert!(set.contains(&a));
assert!(set.contains(&b));
assert!(set.contains(&c));
assert!(!set.contains(&(a + b + c)));
}
#[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());
}
#[test]
fn test_from_str() {
fn test(z: Complex64, s: &str) {
assert_eq!(FromStr::from_str(s), Ok(z));
}
test(_0_0i, "0 + 0i");
test(_0_0i, "0+0j");
test(_0_0i, "0 - 0j");
test(_0_0i, "0-0i");
test(_0_0i, "0i + 0");
test(_0_0i, "0");
test(_0_0i, "-0");
test(_0_0i, "0i");
test(_0_0i, "0j");
test(_0_0i, "+0j");
test(_0_0i, "-0i");
test(_1_0i, "1 + 0i");
test(_1_0i, "1+0j");
test(_1_0i, "1 - 0j");
test(_1_0i, "+1-0i");
test(_1_0i, "-0j+1");
test(_1_0i, "1");
test(_1_1i, "1 + i");
test(_1_1i, "1+j");
test(_1_1i, "1 + 1j");
test(_1_1i, "1+1i");
test(_1_1i, "i + 1");
test(_1_1i, "1i+1");
test(_1_1i, "+j+1");
test(_0_1i, "0 + i");
test(_0_1i, "0+j");
test(_0_1i, "-0 + j");
test(_0_1i, "-0+i");
test(_0_1i, "0 + 1i");
test(_0_1i, "0+1j");
test(_0_1i, "-0 + 1j");
test(_0_1i, "-0+1i");
test(_0_1i, "j + 0");
test(_0_1i, "i");
test(_0_1i, "j");
test(_0_1i, "1j");
test(_neg1_1i, "-1 + i");
test(_neg1_1i, "-1+j");
test(_neg1_1i, "-1 + 1j");
test(_neg1_1i, "-1+1i");
test(_neg1_1i, "1i-1");
test(_neg1_1i, "j + -1");
test(_05_05i, "0.5 + 0.5i");
test(_05_05i, "0.5+0.5j");
test(_05_05i, "5e-1+0.5j");
test(_05_05i, "5E-1 + 0.5j");
test(_05_05i, "5E-1i + 0.5");
test(_05_05i, "0.05e+1j + 50E-2");
}
#[test]
fn test_from_str_radix() {
fn test(z: Complex64, s: &str, radix: u32) {
let res: Result<Complex64, <Complex64 as Num>::FromStrRadixErr> =
Num::from_str_radix(s, radix);
assert_eq!(res.unwrap(), z)
}
test(_4_2i, "4+2i", 10);
test(Complex::new(15.0, 32.0), "F+20i", 16);
test(Complex::new(15.0, 32.0), "1111+100000i", 2);
test(Complex::new(-15.0, -32.0), "-F-20i", 16);
test(Complex::new(-15.0, -32.0), "-1111-100000i", 2);
fn test_error(s: &str, radix: u32) -> ParseComplexError<<f64 as Num>::FromStrRadixErr> {
let res = Complex64::from_str_radix(s, radix);
res.expect_err(&format!("Expected failure on input {:?}", s))
}
let err = test_error("1ii", 19);
if let ComplexErrorKind::UnsupportedRadix = err.kind {
/* pass */
} else {
panic!("Expected failure on invalid radix, got {:?}", err);
}
let err = test_error("1 + 0", 16);
if let ComplexErrorKind::ExprError = err.kind {
/* pass */
} else {
panic!("Expected failure on expr error, got {:?}", err);
}
}
#[test]
#[should_panic(expected = "radix is too high")]
fn test_from_str_radix_fail() {
// ensure we preserve the underlying panic on radix > 36
let _complex = Complex64::from_str_radix("1", 37);
}
#[test]
fn test_from_str_fail() {
fn test(s: &str) {
let complex: Result<Complex64, _> = FromStr::from_str(s);
assert!(
complex.is_err(),
"complex {:?} -> {:?} should be an error",
s,
complex
);
}
test("foo");
test("6E");
test("0 + 2.718");
test("1 - -2i");
test("314e-2ij");
test("4.3j - i");
test("1i - 2i");
test("+ 1 - 3.0i");
}
#[test]
fn test_sum() {
let v = vec![_0_1i, _1_0i];
assert_eq!(v.iter().sum::<Complex64>(), _1_1i);
assert_eq!(v.into_iter().sum::<Complex64>(), _1_1i);
}
#[test]
fn test_prod() {
let v = vec![_0_1i, _1_0i];
assert_eq!(v.iter().product::<Complex64>(), _0_1i);
assert_eq!(v.into_iter().product::<Complex64>(), _0_1i);
}
#[test]
fn test_zero() {
let zero = Complex64::zero();
assert!(zero.is_zero());
let mut c = Complex::new(1.23, 4.56);
assert!(!c.is_zero());
assert_eq!(c + zero, c);
c.set_zero();
assert!(c.is_zero());
}
#[test]
fn test_one() {
let one = Complex64::one();
assert!(one.is_one());
let mut c = Complex::new(1.23, 4.56);
assert!(!c.is_one());
assert_eq!(c * one, c);
c.set_one();
assert!(c.is_one());
}
#[test]
#[allow(clippy::float_cmp)]
fn test_const() {
const R: f64 = 12.3;
const I: f64 = -4.5;
const C: Complex64 = Complex::new(R, I);
assert_eq!(C.re, 12.3);
assert_eq!(C.im, -4.5);
}
}