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

Average Accuracy: 90.3% → 97.4%

Time: 13.4s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -80000000:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(z \cdot y\right) \cdot \left(-z \cdot x\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -80000000.0)
   (/ (/ 1.0 z) (* x (* z y)))
   (if (<= z 1.3e+106)
     (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
     (/ -1.0 (* (* z y) (- (* z 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) {
	double tmp;
	if (z <= -80000000.0) {
		tmp = (1.0 / z) / (x * (z * y));
	} else if (z <= 1.3e+106) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = -1.0 / ((z * y) * -(z * x));
	}
	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 (z <= (-80000000.0d0)) then
        tmp = (1.0d0 / z) / (x * (z * y))
    else if (z <= 1.3d+106) then
        tmp = ((1.0d0 / x) / y) / (1.0d0 + (z * z))
    else
        tmp = (-1.0d0) / ((z * y) * -(z * x))
    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 (z <= -80000000.0) {
		tmp = (1.0 / z) / (x * (z * y));
	} else if (z <= 1.3e+106) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = -1.0 / ((z * y) * -(z * x));
	}
	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 z <= -80000000.0:
		tmp = (1.0 / z) / (x * (z * y))
	elif z <= 1.3e+106:
		tmp = ((1.0 / x) / y) / (1.0 + (z * z))
	else:
		tmp = -1.0 / ((z * y) * -(z * x))
	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 (z <= -80000000.0)
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(z * y)));
	elseif (z <= 1.3e+106)
		tmp = Float64(Float64(Float64(1.0 / x) / y) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(-1.0 / Float64(Float64(z * y) * Float64(-Float64(z * x))));
	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 (z <= -80000000.0)
		tmp = (1.0 / z) / (x * (z * y));
	elseif (z <= 1.3e+106)
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	else
		tmp = -1.0 / ((z * y) * -(z * x));
	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[z, -80000000.0], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e+106], N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(z * y), $MachinePrecision] * (-N[(z * x), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.3%
Target	92.3%
Herbie	97.4%

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

Derivation

1. Split input into 3 regimes

2. if z < -8e7

   1. Initial program 81.4%

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

   2. Taylor expanded in z around inf 81.4%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot {z}^{2}}} \]

   3. Simplified 89.8%

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

      [Start] 81.4	\[ \frac{\frac{1}{x}}{y \cdot {z}^{2}} \]
      *-commutative [=>] 81.4	\[ \frac{\frac{1}{x}}{\color{blue}{{z}^{2} \cdot y}} \]
      unpow2 [=>] 81.4	\[ \frac{\frac{1}{x}}{\color{blue}{\left(z \cdot z\right)} \cdot y} \]
      associate-*r* [<=] 89.8	\[ \frac{\frac{1}{x}}{\color{blue}{z \cdot \left(z \cdot y\right)}} \]

   4. Taylor expanded in x around 0 80.3%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left({z}^{2} \cdot x\right)}} \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{y}}{z}}{z}}{x}} \]
      Proof

      [Start] 80.3	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 80.5	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      unpow2 [=>] 80.5	\[ \frac{\frac{1}{y}}{\color{blue}{\left(z \cdot z\right)} \cdot x} \]
      associate-/r* [=>] 81.9	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{z \cdot z}}{x}} \]
      associate-/r* [=>] 90.7	\[ \frac{\color{blue}{\frac{\frac{\frac{1}{y}}{z}}{z}}}{x} \]

   6. Taylor expanded in y around 0 80.3%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left({z}^{2} \cdot x\right)}} \]

   7. Simplified 95.6%

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

      [Start] 80.3	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      unpow2 [=>] 80.3	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)} \]
      associate-*l* [=>] 88.3	\[ \frac{1}{y \cdot \color{blue}{\left(z \cdot \left(z \cdot x\right)\right)}} \]
      associate-*r* [=>] 95.0	\[ \frac{1}{\color{blue}{\left(y \cdot z\right) \cdot \left(z \cdot x\right)}} \]
      associate-/l/ [<=] 95.3	\[ \color{blue}{\frac{\frac{1}{z \cdot x}}{y \cdot z}} \]
      associate-/r* [=>] 95.9	\[ \frac{\color{blue}{\frac{\frac{1}{z}}{x}}}{y \cdot z} \]
      associate-/r* [<=] 95.6	\[ \color{blue}{\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}} \]

   if -8e7 < z < 1.3000000000000001e106

   1. Initial program 98.6%

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

   2. Simplified 98.5%

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

      [Start] 98.6	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 98.5	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   if 1.3000000000000001e106 < z

   1. Initial program 76.6%

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

   2. Taylor expanded in z around inf 76.6%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y \cdot {z}^{2}}} \]

   3. Simplified 89.2%

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

      [Start] 76.6	\[ \frac{\frac{1}{x}}{y \cdot {z}^{2}} \]
      *-commutative [=>] 76.6	\[ \frac{\frac{1}{x}}{\color{blue}{{z}^{2} \cdot y}} \]
      unpow2 [=>] 76.6	\[ \frac{\frac{1}{x}}{\color{blue}{\left(z \cdot z\right)} \cdot y} \]
      associate-*r* [<=] 89.2	\[ \frac{\frac{1}{x}}{\color{blue}{z \cdot \left(z \cdot y\right)}} \]

   4. Applied egg-rr 76.6%

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

      [Start] 89.2	\[ \frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)} \]
      associate-/l/ [=>] 88.7	\[ \color{blue}{\frac{1}{\left(z \cdot \left(z \cdot y\right)\right) \cdot x}} \]
      associate-/r* [=>] 89.2	\[ \color{blue}{\frac{\frac{1}{z \cdot \left(z \cdot y\right)}}{x}} \]
      /-rgt-identity [<=] 89.2	\[ \frac{\frac{1}{z \cdot \left(z \cdot y\right)}}{\color{blue}{\frac{x}{1}}} \]
      frac-2neg [=>] 89.2	\[ \frac{\frac{1}{z \cdot \left(z \cdot y\right)}}{\color{blue}{\frac{-x}{-1}}} \]
      associate-/r/ [=>] 89.2	\[ \color{blue}{\frac{\frac{1}{z \cdot \left(z \cdot y\right)}}{-x} \cdot \left(-1\right)} \]
      associate-*r* [=>] 76.6	\[ \frac{\frac{1}{\color{blue}{\left(z \cdot z\right) \cdot y}}}{-x} \cdot \left(-1\right) \]
      *-commutative [=>] 76.6	\[ \frac{\frac{1}{\color{blue}{y \cdot \left(z \cdot z\right)}}}{-x} \cdot \left(-1\right) \]
      metadata-eval [=>] 76.6	\[ \frac{\frac{1}{y \cdot \left(z \cdot z\right)}}{-x} \cdot \color{blue}{-1} \]

   5. Simplified 96.3%

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

      [Start] 76.6	\[ \frac{\frac{1}{y \cdot \left(z \cdot z\right)}}{-x} \cdot -1 \]
      associate-/l/ [=>] 76.4	\[ \color{blue}{\frac{1}{\left(-x\right) \cdot \left(y \cdot \left(z \cdot z\right)\right)}} \cdot -1 \]
      associate-*l/ [=>] 76.4	\[ \color{blue}{\frac{1 \cdot -1}{\left(-x\right) \cdot \left(y \cdot \left(z \cdot z\right)\right)}} \]
      metadata-eval [=>] 76.4	\[ \frac{\color{blue}{-1}}{\left(-x\right) \cdot \left(y \cdot \left(z \cdot z\right)\right)} \]
      distribute-lft-neg-out [=>] 76.4	\[ \frac{-1}{\color{blue}{-x \cdot \left(y \cdot \left(z \cdot z\right)\right)}} \]
      *-commutative [=>] 76.4	\[ \frac{-1}{-\color{blue}{\left(y \cdot \left(z \cdot z\right)\right) \cdot x}} \]
      associate-*r* [=>] 88.7	\[ \frac{-1}{-\color{blue}{\left(\left(y \cdot z\right) \cdot z\right)} \cdot x} \]
      *-commutative [<=] 88.7	\[ \frac{-1}{-\left(\color{blue}{\left(z \cdot y\right)} \cdot z\right) \cdot x} \]
      associate-*l* [=>] 96.3	\[ \frac{-1}{-\color{blue}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}} \]
      *-commutative [<=] 96.3	\[ \frac{-1}{-\left(z \cdot y\right) \cdot \color{blue}{\left(x \cdot z\right)}} \]
      distribute-rgt-neg-in [=>] 96.3	\[ \frac{-1}{\color{blue}{\left(z \cdot y\right) \cdot \left(-x \cdot z\right)}} \]
      *-commutative [=>] 96.3	\[ \frac{-1}{\color{blue}{\left(y \cdot z\right)} \cdot \left(-x \cdot z\right)} \]
      *-commutative [=>] 96.3	\[ \frac{-1}{\left(y \cdot z\right) \cdot \left(-\color{blue}{z \cdot x}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -80000000:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(z \cdot y\right) \cdot \left(-z \cdot x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.1%
Cost	13504

\[\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \left(\left(\mathsf{hypot}\left(1, z\right) \cdot y\right) \cdot x\right)} \]

Alternative 2
Accuracy	98.9%
Cost	1736

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;t_0 \leq 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + z \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	97.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+49} \lor \neg \left(z \leq 1.22 \cdot 10^{+64}\right):\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	97.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -520000000:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+105}:\\ \;\;\;\;\frac{1}{\left(1 + z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(z \cdot y\right) \cdot \left(-z \cdot x\right)}\\ \end{array} \]

Alternative 6
Accuracy	96.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \end{array} \]

Alternative 7
Accuracy	96.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	54.9%
Cost	320

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

Alternative 9
Accuracy	54.9%
Cost	320

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

Reproduce

herbie shell --seed 2023138 
(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)))))