use std::ops::Neg; use {Num, NumCast, Float}; // NOTE: These doctests have the same issue as those in src/float.rs. // They're testing the inherent methods directly, and not those of `Real`. /// A trait for real number types that do not necessarily have /// floating-point-specific characteristics such as NaN and infinity. /// /// See [this Wikipedia article](https://en.wikipedia.org/wiki/Real_data_type) /// for a list of data types that could meaningfully implement this trait. /// /// This trait is only available with the `std` feature. pub trait Real : Num + Copy + NumCast + PartialOrd + Neg { /// Returns the smallest finite value that this type can represent. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x: f64 = Real::min_value(); /// /// assert_eq!(x, f64::MIN); /// ``` fn min_value() -> Self; /// Returns the smallest positive, normalized value that this type can represent. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x: f64 = Real::min_positive_value(); /// /// assert_eq!(x, f64::MIN_POSITIVE); /// ``` fn min_positive_value() -> Self; /// Returns epsilon, a small positive value. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x: f64 = Real::epsilon(); /// /// assert_eq!(x, f64::EPSILON); /// ``` /// /// # Panics /// /// The default implementation will panic if `f32::EPSILON` cannot /// be cast to `Self`. fn epsilon() -> Self; /// Returns the largest finite value that this type can represent. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x: f64 = Real::max_value(); /// assert_eq!(x, f64::MAX); /// ``` fn max_value() -> Self; /// Returns the largest integer less than or equal to a number. /// /// ``` /// use num_traits::real::Real; /// /// let f = 3.99; /// let g = 3.0; /// /// assert_eq!(f.floor(), 3.0); /// assert_eq!(g.floor(), 3.0); /// ``` fn floor(self) -> Self; /// Returns the smallest integer greater than or equal to a number. /// /// ``` /// use num_traits::real::Real; /// /// let f = 3.01; /// let g = 4.0; /// /// assert_eq!(f.ceil(), 4.0); /// assert_eq!(g.ceil(), 4.0); /// ``` fn ceil(self) -> Self; /// Returns the nearest integer to a number. Round half-way cases away from /// `0.0`. /// /// ``` /// use num_traits::real::Real; /// /// let f = 3.3; /// let g = -3.3; /// /// assert_eq!(f.round(), 3.0); /// assert_eq!(g.round(), -3.0); /// ``` fn round(self) -> Self; /// Return the integer part of a number. /// /// ``` /// use num_traits::real::Real; /// /// let f = 3.3; /// let g = -3.7; /// /// assert_eq!(f.trunc(), 3.0); /// assert_eq!(g.trunc(), -3.0); /// ``` fn trunc(self) -> Self; /// Returns the fractional part of a number. /// /// ``` /// use num_traits::real::Real; /// /// let x = 3.5; /// let y = -3.5; /// let abs_difference_x = (x.fract() - 0.5).abs(); /// let abs_difference_y = (y.fract() - (-0.5)).abs(); /// /// assert!(abs_difference_x < 1e-10); /// assert!(abs_difference_y < 1e-10); /// ``` fn fract(self) -> Self; /// Computes the absolute value of `self`. Returns `Float::nan()` if the /// number is `Float::nan()`. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x = 3.5; /// let y = -3.5; /// /// let abs_difference_x = (x.abs() - x).abs(); /// let abs_difference_y = (y.abs() - (-y)).abs(); /// /// assert!(abs_difference_x < 1e-10); /// assert!(abs_difference_y < 1e-10); /// /// assert!(::num_traits::Float::is_nan(f64::NAN.abs())); /// ``` fn abs(self) -> Self; /// Returns a number that represents the sign of `self`. /// /// - `1.0` if the number is positive, `+0.0` or `Float::infinity()` /// - `-1.0` if the number is negative, `-0.0` or `Float::neg_infinity()` /// - `Float::nan()` if the number is `Float::nan()` /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let f = 3.5; /// /// assert_eq!(f.signum(), 1.0); /// assert_eq!(f64::NEG_INFINITY.signum(), -1.0); /// /// assert!(f64::NAN.signum().is_nan()); /// ``` fn signum(self) -> Self; /// Returns `true` if `self` is positive, including `+0.0`, /// `Float::infinity()`, and with newer versions of Rust `f64::NAN`. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let neg_nan: f64 = -f64::NAN; /// /// let f = 7.0; /// let g = -7.0; /// /// assert!(f.is_sign_positive()); /// assert!(!g.is_sign_positive()); /// assert!(!neg_nan.is_sign_positive()); /// ``` fn is_sign_positive(self) -> bool; /// Returns `true` if `self` is negative, including `-0.0`, /// `Float::neg_infinity()`, and with newer versions of Rust `-f64::NAN`. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let nan: f64 = f64::NAN; /// /// let f = 7.0; /// let g = -7.0; /// /// assert!(!f.is_sign_negative()); /// assert!(g.is_sign_negative()); /// assert!(!nan.is_sign_negative()); /// ``` fn is_sign_negative(self) -> bool; /// Fused multiply-add. Computes `(self * a) + b` with only one rounding /// error, yielding a more accurate result than an unfused multiply-add. /// /// Using `mul_add` can be more performant than an unfused multiply-add if /// the target architecture has a dedicated `fma` CPU instruction. /// /// ``` /// use num_traits::real::Real; /// /// let m = 10.0; /// let x = 4.0; /// let b = 60.0; /// /// // 100.0 /// let abs_difference = (m.mul_add(x, b) - (m*x + b)).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn mul_add(self, a: Self, b: Self) -> Self; /// Take the reciprocal (inverse) of a number, `1/x`. /// /// ``` /// use num_traits::real::Real; /// /// let x = 2.0; /// let abs_difference = (x.recip() - (1.0/x)).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn recip(self) -> Self; /// Raise a number to an integer power. /// /// Using this function is generally faster than using `powf` /// /// ``` /// use num_traits::real::Real; /// /// let x = 2.0; /// let abs_difference = (x.powi(2) - x*x).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn powi(self, n: i32) -> Self; /// Raise a number to a real number power. /// /// ``` /// use num_traits::real::Real; /// /// let x = 2.0; /// let abs_difference = (x.powf(2.0) - x*x).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn powf(self, n: Self) -> Self; /// Take the square root of a number. /// /// Returns NaN if `self` is a negative floating-point number. /// /// # Panics /// /// If the implementing type doesn't support NaN, this method should panic if `self < 0`. /// /// ``` /// use num_traits::real::Real; /// /// let positive = 4.0; /// let negative = -4.0; /// /// let abs_difference = (positive.sqrt() - 2.0).abs(); /// /// assert!(abs_difference < 1e-10); /// assert!(::num_traits::Float::is_nan(negative.sqrt())); /// ``` fn sqrt(self) -> Self; /// Returns `e^(self)`, (the exponential function). /// /// ``` /// use num_traits::real::Real; /// /// let one = 1.0; /// // e^1 /// let e = one.exp(); /// /// // ln(e) - 1 == 0 /// let abs_difference = (e.ln() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn exp(self) -> Self; /// Returns `2^(self)`. /// /// ``` /// use num_traits::real::Real; /// /// let f = 2.0; /// /// // 2^2 - 4 == 0 /// let abs_difference = (f.exp2() - 4.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn exp2(self) -> Self; /// Returns the natural logarithm of the number. /// /// # Panics /// /// If `self <= 0` and this type does not support a NaN representation, this function should panic. /// /// ``` /// use num_traits::real::Real; /// /// let one = 1.0; /// // e^1 /// let e = one.exp(); /// /// // ln(e) - 1 == 0 /// let abs_difference = (e.ln() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn ln(self) -> Self; /// Returns the logarithm of the number with respect to an arbitrary base. /// /// # Panics /// /// If `self <= 0` and this type does not support a NaN representation, this function should panic. /// /// ``` /// use num_traits::real::Real; /// /// let ten = 10.0; /// let two = 2.0; /// /// // log10(10) - 1 == 0 /// let abs_difference_10 = (ten.log(10.0) - 1.0).abs(); /// /// // log2(2) - 1 == 0 /// let abs_difference_2 = (two.log(2.0) - 1.0).abs(); /// /// assert!(abs_difference_10 < 1e-10); /// assert!(abs_difference_2 < 1e-10); /// ``` fn log(self, base: Self) -> Self; /// Returns the base 2 logarithm of the number. /// /// # Panics /// /// If `self <= 0` and this type does not support a NaN representation, this function should panic. /// /// ``` /// use num_traits::real::Real; /// /// let two = 2.0; /// /// // log2(2) - 1 == 0 /// let abs_difference = (two.log2() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn log2(self) -> Self; /// Returns the base 10 logarithm of the number. /// /// # Panics /// /// If `self <= 0` and this type does not support a NaN representation, this function should panic. /// /// /// ``` /// use num_traits::real::Real; /// /// let ten = 10.0; /// /// // log10(10) - 1 == 0 /// let abs_difference = (ten.log10() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn log10(self) -> Self; /// Converts radians to degrees. /// /// ``` /// use std::f64::consts; /// /// let angle = consts::PI; /// /// let abs_difference = (angle.to_degrees() - 180.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn to_degrees(self) -> Self; /// Converts degrees to radians. /// /// ``` /// use std::f64::consts; /// /// let angle = 180.0_f64; /// /// let abs_difference = (angle.to_radians() - consts::PI).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn to_radians(self) -> Self; /// Returns the maximum of the two numbers. /// /// ``` /// use num_traits::real::Real; /// /// let x = 1.0; /// let y = 2.0; /// /// assert_eq!(x.max(y), y); /// ``` fn max(self, other: Self) -> Self; /// Returns the minimum of the two numbers. /// /// ``` /// use num_traits::real::Real; /// /// let x = 1.0; /// let y = 2.0; /// /// assert_eq!(x.min(y), x); /// ``` fn min(self, other: Self) -> Self; /// The positive difference of two numbers. /// /// * If `self <= other`: `0:0` /// * Else: `self - other` /// /// ``` /// use num_traits::real::Real; /// /// let x = 3.0; /// let y = -3.0; /// /// let abs_difference_x = (x.abs_sub(1.0) - 2.0).abs(); /// let abs_difference_y = (y.abs_sub(1.0) - 0.0).abs(); /// /// assert!(abs_difference_x < 1e-10); /// assert!(abs_difference_y < 1e-10); /// ``` fn abs_sub(self, other: Self) -> Self; /// Take the cubic root of a number. /// /// ``` /// use num_traits::real::Real; /// /// let x = 8.0; /// /// // x^(1/3) - 2 == 0 /// let abs_difference = (x.cbrt() - 2.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn cbrt(self) -> Self; /// Calculate the length of the hypotenuse of a right-angle triangle given /// legs of length `x` and `y`. /// /// ``` /// use num_traits::real::Real; /// /// let x = 2.0; /// let y = 3.0; /// /// // sqrt(x^2 + y^2) /// let abs_difference = (x.hypot(y) - (x.powi(2) + y.powi(2)).sqrt()).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn hypot(self, other: Self) -> Self; /// Computes the sine of a number (in radians). /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x = f64::consts::PI/2.0; /// /// let abs_difference = (x.sin() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn sin(self) -> Self; /// Computes the cosine of a number (in radians). /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x = 2.0*f64::consts::PI; /// /// let abs_difference = (x.cos() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn cos(self) -> Self; /// Computes the tangent of a number (in radians). /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x = f64::consts::PI/4.0; /// let abs_difference = (x.tan() - 1.0).abs(); /// /// assert!(abs_difference < 1e-14); /// ``` fn tan(self) -> Self; /// Computes the arcsine of a number. Return value is in radians in /// the range [-pi/2, pi/2] or NaN if the number is outside the range /// [-1, 1]. /// /// # Panics /// /// If this type does not support a NaN representation, this function should panic /// if the number is outside the range [-1, 1]. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let f = f64::consts::PI / 2.0; /// /// // asin(sin(pi/2)) /// let abs_difference = (f.sin().asin() - f64::consts::PI / 2.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn asin(self) -> Self; /// Computes the arccosine of a number. Return value is in radians in /// the range [0, pi] or NaN if the number is outside the range /// [-1, 1]. /// /// # Panics /// /// If this type does not support a NaN representation, this function should panic /// if the number is outside the range [-1, 1]. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let f = f64::consts::PI / 4.0; /// /// // acos(cos(pi/4)) /// let abs_difference = (f.cos().acos() - f64::consts::PI / 4.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn acos(self) -> Self; /// Computes the arctangent of a number. Return value is in radians in the /// range [-pi/2, pi/2]; /// /// ``` /// use num_traits::real::Real; /// /// let f = 1.0; /// /// // atan(tan(1)) /// let abs_difference = (f.tan().atan() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn atan(self) -> Self; /// Computes the four quadrant arctangent of `self` (`y`) and `other` (`x`). /// /// * `x = 0`, `y = 0`: `0` /// * `x >= 0`: `arctan(y/x)` -> `[-pi/2, pi/2]` /// * `y >= 0`: `arctan(y/x) + pi` -> `(pi/2, pi]` /// * `y < 0`: `arctan(y/x) - pi` -> `(-pi, -pi/2)` /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let pi = f64::consts::PI; /// // All angles from horizontal right (+x) /// // 45 deg counter-clockwise /// let x1 = 3.0; /// let y1 = -3.0; /// /// // 135 deg clockwise /// let x2 = -3.0; /// let y2 = 3.0; /// /// let abs_difference_1 = (y1.atan2(x1) - (-pi/4.0)).abs(); /// let abs_difference_2 = (y2.atan2(x2) - 3.0*pi/4.0).abs(); /// /// assert!(abs_difference_1 < 1e-10); /// assert!(abs_difference_2 < 1e-10); /// ``` fn atan2(self, other: Self) -> Self; /// Simultaneously computes the sine and cosine of the number, `x`. Returns /// `(sin(x), cos(x))`. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x = f64::consts::PI/4.0; /// let f = x.sin_cos(); /// /// let abs_difference_0 = (f.0 - x.sin()).abs(); /// let abs_difference_1 = (f.1 - x.cos()).abs(); /// /// assert!(abs_difference_0 < 1e-10); /// assert!(abs_difference_0 < 1e-10); /// ``` fn sin_cos(self) -> (Self, Self); /// Returns `e^(self) - 1` in a way that is accurate even if the /// number is close to zero. /// /// ``` /// use num_traits::real::Real; /// /// let x = 7.0; /// /// // e^(ln(7)) - 1 /// let abs_difference = (x.ln().exp_m1() - 6.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn exp_m1(self) -> Self; /// Returns `ln(1+n)` (natural logarithm) more accurately than if /// the operations were performed separately. /// /// # Panics /// /// If this type does not support a NaN representation, this function should panic /// if `self-1 <= 0`. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let x = f64::consts::E - 1.0; /// /// // ln(1 + (e - 1)) == ln(e) == 1 /// let abs_difference = (x.ln_1p() - 1.0).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn ln_1p(self) -> Self; /// Hyperbolic sine function. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let e = f64::consts::E; /// let x = 1.0; /// /// let f = x.sinh(); /// // Solving sinh() at 1 gives `(e^2-1)/(2e)` /// let g = (e*e - 1.0)/(2.0*e); /// let abs_difference = (f - g).abs(); /// /// assert!(abs_difference < 1e-10); /// ``` fn sinh(self) -> Self; /// Hyperbolic cosine function. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let e = f64::consts::E; /// let x = 1.0; /// let f = x.cosh(); /// // Solving cosh() at 1 gives this result /// let g = (e*e + 1.0)/(2.0*e); /// let abs_difference = (f - g).abs(); /// /// // Same result /// assert!(abs_difference < 1.0e-10); /// ``` fn cosh(self) -> Self; /// Hyperbolic tangent function. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let e = f64::consts::E; /// let x = 1.0; /// /// let f = x.tanh(); /// // Solving tanh() at 1 gives `(1 - e^(-2))/(1 + e^(-2))` /// let g = (1.0 - e.powi(-2))/(1.0 + e.powi(-2)); /// let abs_difference = (f - g).abs(); /// /// assert!(abs_difference < 1.0e-10); /// ``` fn tanh(self) -> Self; /// Inverse hyperbolic sine function. /// /// ``` /// use num_traits::real::Real; /// /// let x = 1.0; /// let f = x.sinh().asinh(); /// /// let abs_difference = (f - x).abs(); /// /// assert!(abs_difference < 1.0e-10); /// ``` fn asinh(self) -> Self; /// Inverse hyperbolic cosine function. /// /// ``` /// use num_traits::real::Real; /// /// let x = 1.0; /// let f = x.cosh().acosh(); /// /// let abs_difference = (f - x).abs(); /// /// assert!(abs_difference < 1.0e-10); /// ``` fn acosh(self) -> Self; /// Inverse hyperbolic tangent function. /// /// ``` /// use num_traits::real::Real; /// use std::f64; /// /// let e = f64::consts::E; /// let f = e.tanh().atanh(); /// /// let abs_difference = (f - e).abs(); /// /// assert!(abs_difference < 1.0e-10); /// ``` fn atanh(self) -> Self; } impl Real for T { forward! { Float::min_value() -> Self; Float::min_positive_value() -> Self; Float::epsilon() -> Self; Float::max_value() -> Self; } forward! { Float::floor(self) -> Self; Float::ceil(self) -> Self; Float::round(self) -> Self; Float::trunc(self) -> Self; Float::fract(self) -> Self; Float::abs(self) -> Self; Float::signum(self) -> Self; Float::is_sign_positive(self) -> bool; Float::is_sign_negative(self) -> bool; Float::mul_add(self, a: Self, b: Self) -> Self; Float::recip(self) -> Self; Float::powi(self, n: i32) -> Self; Float::powf(self, n: Self) -> Self; Float::sqrt(self) -> Self; Float::exp(self) -> Self; Float::exp2(self) -> Self; Float::ln(self) -> Self; Float::log(self, base: Self) -> Self; Float::log2(self) -> Self; Float::log10(self) -> Self; Float::to_degrees(self) -> Self; Float::to_radians(self) -> Self; Float::max(self, other: Self) -> Self; Float::min(self, other: Self) -> Self; Float::abs_sub(self, other: Self) -> Self; Float::cbrt(self) -> Self; Float::hypot(self, other: Self) -> Self; Float::sin(self) -> Self; Float::cos(self) -> Self; Float::tan(self) -> Self; Float::asin(self) -> Self; Float::acos(self) -> Self; Float::atan(self) -> Self; Float::atan2(self, other: Self) -> Self; Float::sin_cos(self) -> (Self, Self); Float::exp_m1(self) -> Self; Float::ln_1p(self) -> Self; Float::sinh(self) -> Self; Float::cosh(self) -> Self; Float::tanh(self) -> Self; Float::asinh(self) -> Self; Float::acosh(self) -> Self; Float::atanh(self) -> Self; } }