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

Average Error: 10.09% → 1.14%

Time: 12.2s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot z + 1\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;t_0 \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + \left(z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (* z z) 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ 1.0 (* (* z x) (* z y)))
     (if (<= t_0 1e+301)
       (/ (/ 1.0 x) (+ y (* (* z z) y)))
       (/ (/ 1.0 (* y (* z x))) 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 t_0 = y * ((z * z) + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 1.0 / ((z * x) * (z * y));
	} else if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / (y + ((z * z) * y));
	} else {
		tmp = (1.0 / (y * (z * x))) / z;
	}
	return tmp;
}

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 t_0 = y * ((z * z) + 1.0);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 1.0 / ((z * x) * (z * y));
	} else if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / (y + ((z * z) * y));
	} else {
		tmp = (1.0 / (y * (z * x))) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y * ((z * z) + 1.0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 1.0 / ((z * x) * (z * y))
	elif t_0 <= 1e+301:
		tmp = (1.0 / x) / (y + ((z * z) * y))
	else:
		tmp = (1.0 / (y * (z * x))) / 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)
	t_0 = Float64(y * Float64(Float64(z * z) + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(z * x) * Float64(z * y)));
	elseif (t_0 <= 1e+301)
		tmp = Float64(Float64(1.0 / x) / Float64(y + Float64(Float64(z * z) * y)));
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(z * x))) / 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)
	t_0 = y * ((z * z) + 1.0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 1.0 / ((z * x) * (z * y));
	elseif (t_0 <= 1e+301)
		tmp = (1.0 / x) / (y + ((z * z) * y));
	else
		tmp = (1.0 / (y * (z * x))) / 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_] := Block[{t$95$0 = N[(y * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(1.0 / N[(N[(z * x), $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], N[(N[(1.0 / x), $MachinePrecision] / N[(y + N[(N[(z * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	10.09%
Target	7.92%
Herbie	1.14%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < -inf.0

   1. Initial program 27.54

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

   2. Simplified 27.54

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

      [Start] 27.54	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/l/ [=>] 27.54	\[ \color{blue}{\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}} \]
      associate-*l* [=>] 27.54	\[ \frac{1}{\color{blue}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}} \]
      +-commutative [=>] 27.54	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z + 1\right)} \cdot x\right)} \]
      fma-def [=>] 27.54	\[ \frac{1}{y \cdot \left(\color{blue}{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)} \]

   3. Applied egg-rr 27.54

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

   4. Simplified 2.75

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

      [Start] 27.54	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\frac{\frac{1}{y}}{x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}} \]
      associate-/l/ [=>] 27.54	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\color{blue}{\frac{1}{\left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot y}}}} \]
      associate-/r* [=>] 27.54	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\color{blue}{\frac{\frac{1}{x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}}}} \]
      associate-/r* [=>] 27.54	\[ \frac{1}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\frac{\color{blue}{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{y}}} \]
      associate-/l* [<=] 27.54	\[ \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot y}{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}} \]
      *-commutative [<=] 27.54	\[ \frac{1}{\frac{\color{blue}{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}} \]
      associate-/r* [<=] 27.54	\[ \frac{1}{\frac{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}{\color{blue}{\frac{1}{x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}} \]
      associate-/r/ [=>] 27.54	\[ \frac{1}{\color{blue}{\frac{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}{1} \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)}} \]
      /-rgt-identity [=>] 27.54	\[ \frac{1}{\color{blue}{\left(y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      fma-udef [=>] 27.54	\[ \frac{1}{\left(y \cdot \sqrt{\color{blue}{z \cdot z + 1}}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      +-commutative [<=] 27.54	\[ \frac{1}{\left(y \cdot \sqrt{\color{blue}{1 + z \cdot z}}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      hypot-1-def [=>] 27.54	\[ \frac{1}{\left(y \cdot \color{blue}{\mathsf{hypot}\left(1, z\right)}\right) \cdot \left(x \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}\right)} \]
      fma-udef [=>] 27.54	\[ \frac{1}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \sqrt{\color{blue}{z \cdot z + 1}}\right)} \]
      +-commutative [<=] 27.54	\[ \frac{1}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \sqrt{\color{blue}{1 + z \cdot z}}\right)} \]
      hypot-1-def [=>] 2.75	\[ \frac{1}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \color{blue}{\mathsf{hypot}\left(1, z\right)}\right)} \]

   5. Applied egg-rr 27.54

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

   6. Taylor expanded in z around inf 27.54

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

   7. Simplified 2.75

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

      [Start] 27.54	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      *-commutative [=>] 27.54	\[ \frac{1}{\color{blue}{\left({z}^{2} \cdot x\right) \cdot y}} \]
      *-commutative [=>] 27.54	\[ \frac{1}{\color{blue}{\left(x \cdot {z}^{2}\right)} \cdot y} \]
      unpow2 [=>] 27.54	\[ \frac{1}{\left(x \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot y} \]
      associate-*r* [=>] 21.66	\[ \frac{1}{\color{blue}{\left(\left(x \cdot z\right) \cdot z\right)} \cdot y} \]
      associate-*l* [=>] 2.75	\[ \frac{1}{\color{blue}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}} \]
      *-commutative [=>] 2.75	\[ \frac{1}{\color{blue}{\left(z \cdot x\right)} \cdot \left(z \cdot y\right)} \]

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   1. Initial program 0.41

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

   2. Applied egg-rr 0.41

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 28.28

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

   2. Simplified 20.76

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

      [Start] 28.28	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 20.76	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   3. Applied egg-rr 84.01

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\left(1 - {z}^{4}\right) \cdot y} \cdot 1 + \frac{\frac{1}{x}}{\left(1 - {z}^{4}\right) \cdot y} \cdot \left(z \cdot \left(-z\right)\right)} \]

   4. Simplified 86.46

      \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \left(1 - z \cdot z\right)}{y \cdot \left(1 - {z}^{4}\right)}} \]
      Proof

      [Start] 84.01	\[ \frac{\frac{1}{x}}{\left(1 - {z}^{4}\right) \cdot y} \cdot 1 + \frac{\frac{1}{x}}{\left(1 - {z}^{4}\right) \cdot y} \cdot \left(z \cdot \left(-z\right)\right) \]
      distribute-lft-out [=>] 84.01	\[ \color{blue}{\frac{\frac{1}{x}}{\left(1 - {z}^{4}\right) \cdot y} \cdot \left(1 + z \cdot \left(-z\right)\right)} \]
      *-commutative [=>] 84.01	\[ \frac{\frac{1}{x}}{\left(1 - {z}^{4}\right) \cdot y} \cdot \left(1 + \color{blue}{\left(-z\right) \cdot z}\right) \]
      cancel-sign-sub-inv [<=] 84.01	\[ \frac{\frac{1}{x}}{\left(1 - {z}^{4}\right) \cdot y} \cdot \color{blue}{\left(1 - z \cdot z\right)} \]
      associate-*l/ [=>] 86.46	\[ \color{blue}{\frac{\frac{1}{x} \cdot \left(1 - z \cdot z\right)}{\left(1 - {z}^{4}\right) \cdot y}} \]
      *-commutative [=>] 86.46	\[ \frac{\frac{1}{x} \cdot \left(1 - z \cdot z\right)}{\color{blue}{y \cdot \left(1 - {z}^{4}\right)}} \]

   5. Taylor expanded in z around inf 21.95

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

   6. Simplified 21.89

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

      [Start] 21.95	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 21.68	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      *-commutative [=>] 21.68	\[ \frac{\frac{1}{y}}{\color{blue}{x \cdot {z}^{2}}} \]
      unpow2 [=>] 21.68	\[ \frac{\frac{1}{y}}{x \cdot \color{blue}{\left(z \cdot z\right)}} \]
      associate-/r* [=>] 21.89	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{x}}{z \cdot z}} \]
      associate-/r* [<=] 21.89	\[ \frac{\color{blue}{\frac{1}{y \cdot x}}}{z \cdot z} \]

   7. Taylor expanded in y around 0 21.95

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

   8. Simplified 2.45

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

      [Start] 21.95	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 21.68	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      associate-/l/ [<=] 21.89	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{x}}{{z}^{2}}} \]
      associate-/r* [<=] 21.89	\[ \frac{\color{blue}{\frac{1}{y \cdot x}}}{{z}^{2}} \]
      unpow2 [=>] 21.89	\[ \frac{\frac{1}{y \cdot x}}{\color{blue}{z \cdot z}} \]
      associate-/r* [=>] 9.75	\[ \color{blue}{\frac{\frac{\frac{1}{y \cdot x}}{z}}{z}} \]
      associate-/r* [<=] 9.19	\[ \frac{\color{blue}{\frac{1}{\left(y \cdot x\right) \cdot z}}}{z} \]
      associate-/l/ [<=] 9.19	\[ \frac{\color{blue}{\frac{\frac{1}{z}}{y \cdot x}}}{z} \]
      *-lft-identity [<=] 9.19	\[ \frac{\frac{\color{blue}{1 \cdot \frac{1}{z}}}{y \cdot x}}{z} \]
      unpow-1 [<=] 9.19	\[ \frac{\frac{1 \cdot \color{blue}{{z}^{-1}}}{y \cdot x}}{z} \]
      metadata-eval [<=] 9.19	\[ \frac{\frac{1 \cdot {z}^{\color{blue}{\left(\frac{-2}{2}\right)}}}{y \cdot x}}{z} \]
      associate-/l* [=>] 9.21	\[ \frac{\color{blue}{\frac{1}{\frac{y \cdot x}{{z}^{\left(\frac{-2}{2}\right)}}}}}{z} \]
      metadata-eval [=>] 9.21	\[ \frac{\frac{1}{\frac{y \cdot x}{{z}^{\color{blue}{-1}}}}}{z} \]
      unpow-1 [=>] 9.21	\[ \frac{\frac{1}{\frac{y \cdot x}{\color{blue}{\frac{1}{z}}}}}{z} \]
      associate-/r/ [=>] 9.19	\[ \frac{\frac{1}{\color{blue}{\frac{y \cdot x}{1} \cdot z}}}{z} \]
      /-rgt-identity [=>] 9.19	\[ \frac{\frac{1}{\color{blue}{\left(y \cdot x\right)} \cdot z}}{z} \]
      associate-*l* [=>] 2.45	\[ \frac{\frac{1}{\color{blue}{y \cdot \left(x \cdot z\right)}}}{z} \]
      *-commutative [=>] 2.45	\[ \frac{\frac{1}{y \cdot \color{blue}{\left(z \cdot x\right)}}}{z} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z \cdot z + 1\right) \leq -\infty:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;y \cdot \left(z \cdot z + 1\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + \left(z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.54%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{\frac{-1}{y}}{-x}}{z \cdot z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(x \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \end{array} \]

Alternative 2
Error	1.15%
Cost	1736

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot z + 1\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;t_0 \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternative 3
Error	6.76%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.8\right):\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \end{array} \]

Alternative 4
Error	3.28%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -0.88 \lor \neg \left(z \leq 0.8\right):\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \end{array} \]

Alternative 5
Error	6.72%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{elif}\;z \leq 0.8:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 6
Error	6.64%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\\ \mathbf{elif}\;z \leq 0.8:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 7
Error	3.57%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.005:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 8
Error	3.63%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.005:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot x\right) \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 9
Error	45.16%
Cost	320

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

Reproduce

herbie shell --seed 2023089 
(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)))))