Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.3 → 3.4

Time: 1.9min

Precision: binary64

Cost: 768

Math

\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

↓

\[\frac{\frac{-1}{z \cdot \left(y \cdot z\right) + y}}{-x} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

↓

(FPCore (x y z) :precision binary64 (/ (/ -1.0 (+ (* z (* y z)) y)) (- x)))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

double code(double x, double y, double z) {
	return (-1.0 / ((z * (y * z)) + y)) / -x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((-1.0d0) / ((z * (y * z)) + y)) / -x
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

public static double code(double x, double y, double z) {
	return (-1.0 / ((z * (y * z)) + y)) / -x;
}

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

↓

def code(x, y, z):
	return (-1.0 / ((z * (y * z)) + y)) / -x

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

↓

function code(x, y, z)
	return Float64(Float64(-1.0 / Float64(Float64(z * Float64(y * z)) + y)) / Float64(-x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

↓

function tmp = code(x, y, z)
	tmp = (-1.0 / ((z * (y * z)) + y)) / -x;
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(-1.0 / N[(N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]

Target

Original	6.3
Target	5.7
Herbie	3.4

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

1. Initial program 6.3

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

2. Applied egg-rr 3.4

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{z \cdot \left(y \cdot z\right) + y}} \]

3. Applied egg-rr 3.4

   \[\leadsto \color{blue}{\frac{\frac{-1}{\mathsf{fma}\left(z, y \cdot z, y\right)}}{-x}} \]

4. Applied egg-rr 3.4

   \[\leadsto \frac{\frac{-1}{\color{blue}{z \cdot \left(y \cdot z\right) + y}}}{-x} \]

Alternatives

Alternative 1
Error	20.0
Cost	904

\[\begin{array}{l} t_0 := \frac{-1}{y \cdot x}\\ \mathbf{if}\;z \leq -5500:\\ \;\;\;\;9 - \left(9 + t_0\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{-1 + z \cdot z}{y}}{-x}\\ \mathbf{else}:\\ \;\;\;\;1.5 - \left(1.5 + t_0\right)\\ \end{array} \]

Alternative 2
Error	20.3
Cost	840

\[\begin{array}{l} t_0 := 1.5 - \left(1.5 + \frac{-1}{y \cdot x}\right)\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.2
Cost	840

\[\begin{array}{l} t_0 := \frac{-1}{y \cdot x}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+26}:\\ \;\;\;\;9 - \left(9 + t_0\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;1.5 - \left(1.5 + t_0\right)\\ \end{array} \]

Alternative 4
Error	6.3
Cost	704

\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]

Alternative 5
Error	3.4
Cost	704

\[\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right) + y} \]

Alternative 6
Error	28.5
Cost	320

\[\frac{1}{y \cdot x} \]

Alternative 7
Error	28.5
Cost	320

\[\frac{\frac{1}{x}}{y} \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))