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

Average Error: 6.5 → 5.1

Time: 24.5s

Precision: binary64

Cost: 836

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y}}{x \cdot \left(-1 - z \cdot z\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-33)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ -1.0 y) (* x (- -1.0 (* z 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) {
	double tmp;
	if (y <= 5e-33) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (-1.0 / y) / (x * (-1.0 - (z * z)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= 5d-33) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = ((-1.0d0) / y) / (x * ((-1.0d0) - (z * z)))
    end if
    code = tmp
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) {
	double tmp;
	if (y <= 5e-33) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (-1.0 / y) / (x * (-1.0 - (z * z)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= 5e-33:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (-1.0 / y) / (x * (-1.0 - (z * z)))
	return tmp

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)
	tmp = 0.0
	if (y <= 5e-33)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(-1.0 / y) / Float64(x * Float64(-1.0 - Float64(z * z))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-33)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (-1.0 / y) / (x * (-1.0 - (z * z)));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[y, 5e-33], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / y), $MachinePrecision] / N[(x * N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.5
Target	5.1
Herbie	5.1

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

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000028e-33

   1. Initial program 8.4

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

   if 5.00000000000000028e-33 < y

   1. Initial program 4.6

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

   2. Simplified 4.8

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

      [Start] 4.6	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      rational.json-simplify-47 [=>] 4.8	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]

   3. Applied egg-rr 2.2

      \[\leadsto \color{blue}{\frac{\frac{-1}{-1 - z \cdot z}}{y \cdot x} + 0} \]

   4. Simplified 2.0

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

      [Start] 2.2	\[ \frac{\frac{-1}{-1 - z \cdot z}}{y \cdot x} + 0 \]
      rational.json-simplify-4 [=>] 2.2	\[ \color{blue}{\frac{\frac{-1}{-1 - z \cdot z}}{y \cdot x}} \]
      rational.json-simplify-46 [=>] 4.4	\[ \color{blue}{\frac{\frac{\frac{-1}{-1 - z \cdot z}}{y}}{x}} \]
      rational.json-simplify-44 [=>] 4.4	\[ \frac{\color{blue}{\frac{\frac{-1}{y}}{-1 - z \cdot z}}}{x} \]
      rational.json-simplify-44 [<=] 2.0	\[ \color{blue}{\frac{\frac{\frac{-1}{y}}{x}}{-1 - z \cdot z}} \]
      rational.json-simplify-47 [=>] 2.0	\[ \color{blue}{\frac{\frac{-1}{y}}{x \cdot \left(-1 - z \cdot z\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y}}{x \cdot \left(-1 - z \cdot z\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.4
Cost	904

\[\begin{array}{l} t_0 := -1 + \left(1 - \frac{\frac{1}{x}}{-y}\right)\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	5.4
Cost	836

\[\begin{array}{l} t_0 := 1 + z \cdot z\\ \mathbf{if}\;y \leq 10^{+15}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot t_0\right)}\\ \end{array} \]

Alternative 3
Error	5.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq 0.5:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{y}}{x \cdot \left(-1 - z \cdot z\right)}\\ \end{array} \]

Alternative 4
Error	26.4
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{x \cdot \left(x \cdot y\right)}\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	25.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot \left(x \cdot y\right)}\\ \end{array} \]

Alternative 6
Error	25.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{y}{x \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot \left(x \cdot y\right)}\\ \end{array} \]

Alternative 7
Error	6.6
Cost	704

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

Alternative 8
Error	29.2
Cost	320

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

Alternative 9
Error	29.3
Cost	320

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

Alternative 10
Error	29.3
Cost	320

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

Reproduce

herbie shell --seed 2023064 
(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)))))