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

Average Error: 6.6 → 3.6

Time: 4.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

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 / x) / (y + (z * (y * z)));
}

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 / x) / (y + (z * (y * z)))
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 / x) / (y + (z * (y * z)));
}

Python

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

↓

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

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 / x) / Float64(y + Float64(z * Float64(y * z))))
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 / x) / (y + (z * (y * z)));
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 / x), $MachinePrecision] / N[(y + N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	6.6
Target	6.0
Herbie	3.6

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

Derivation

1. Initial program 6.6

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

2. Simplified 3.6

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

   [Start] 6.6	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-37 [=>] 6.6	\[ \frac{\frac{1}{x}}{\color{blue}{1 \cdot y + y \cdot \left(z \cdot z\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 6.6	\[ \frac{\frac{1}{x}}{\color{blue}{y \cdot 1} + y \cdot \left(z \cdot z\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 6.6	\[ \frac{\frac{1}{x}}{\color{blue}{y} + y \cdot \left(z \cdot z\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 3.6	\[ \frac{\frac{1}{x}}{y + \color{blue}{z \cdot \left(y \cdot z\right)}} \]

3. Final simplification 3.6

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

Alternatives

Alternative 1
Error	6.6
Cost	704

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

Alternative 2
Error	28.8
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(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)))))