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

Average Accuracy: 90.0% → 97.6%

Time: 12.0s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;t_0 \leq 10^{+287}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

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 (+ 1.0 (* z z)))))
   (if (<= t_0 (- INFINITY))
     (* (/ 1.0 (* y z)) (/ 1.0 (* z x)))
     (if (<= t_0 1e+287) (/ (/ 1.0 x) t_0) (/ 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 t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 / (y * z)) * (1.0 / (z * x));
	} else if (t_0 <= 1e+287) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = 1.0 / (z * (y * (z * x)));
	}
	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 * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (y * z)) * (1.0 / (z * x));
	} else if (t_0 <= 1e+287) {
		tmp = (1.0 / x) / t_0;
	} 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):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (1.0 / (y * z)) * (1.0 / (z * x))
	elif t_0 <= 1e+287:
		tmp = (1.0 / x) / t_0
	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)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(1.0 / Float64(z * x)));
	elseif (t_0 <= 1e+287)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(1.0 / Float64(z * Float64(y * 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)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (1.0 / (y * z)) * (1.0 / (z * x));
	elseif (t_0 <= 1e+287)
		tmp = (1.0 / x) / t_0;
	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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+287], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(z * N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	90.0%
Target	92.1%
Herbie	97.6%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.1%

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

   2. Taylor expanded in z around inf 76.1%

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

   3. Simplified 76.1%

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

      [Start] 76.1	\[ \frac{\frac{1}{x}}{y \cdot {z}^{2}} \]
      unpow2 [=>] 76.1	\[ \frac{\frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z\right)}} \]

   4. Applied egg-rr 97.9%

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

      [Start] 76.1	\[ \frac{\frac{1}{x}}{y \cdot \left(z \cdot z\right)} \]
      *-un-lft-identity [=>] 76.1	\[ \frac{\color{blue}{1 \cdot \frac{1}{x}}}{y \cdot \left(z \cdot z\right)} \]
      associate-*r* [=>] 96.2	\[ \frac{1 \cdot \frac{1}{x}}{\color{blue}{\left(y \cdot z\right) \cdot z}} \]
      times-frac [=>] 98.4	\[ \color{blue}{\frac{1}{y \cdot z} \cdot \frac{\frac{1}{x}}{z}} \]
      associate-/l/ [=>] 97.9	\[ \frac{1}{y \cdot z} \cdot \color{blue}{\frac{1}{z \cdot x}} \]
      *-commutative [=>] 97.9	\[ \frac{1}{y \cdot z} \cdot \frac{1}{\color{blue}{x \cdot z}} \]

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 1.0000000000000001e287

   1. Initial program 99.5%

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

   if 1.0000000000000001e287 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 73.0%

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

   2. Simplified 80.0%

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

      [Start] 73.0	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/l/ [=>] 73.0	\[ \color{blue}{\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}} \]
      associate-*l* [=>] 80.0	\[ \frac{1}{\color{blue}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}} \]
      +-commutative [=>] 80.0	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z + 1\right)} \cdot x\right)} \]
      fma-def [=>] 80.0	\[ \frac{1}{y \cdot \left(\color{blue}{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)} \]

   3. Taylor expanded in z around inf 74.4%

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

   4. Simplified 87.4%

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

      [Start] 74.4	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      *-commutative [=>] 74.4	\[ \frac{1}{y \cdot \color{blue}{\left(x \cdot {z}^{2}\right)}} \]
      unpow2 [=>] 74.4	\[ \frac{1}{y \cdot \left(x \cdot \color{blue}{\left(z \cdot z\right)}\right)} \]
      associate-*r* [=>] 75.0	\[ \frac{1}{\color{blue}{\left(y \cdot x\right) \cdot \left(z \cdot z\right)}} \]
      *-commutative [=>] 75.0	\[ \frac{1}{\color{blue}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}} \]
      associate-*l* [=>] 87.4	\[ \frac{1}{\color{blue}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}} \]

   5. Taylor expanded in z around 0 93.2%

      \[\leadsto \frac{1}{z \cdot \color{blue}{\left(y \cdot \left(z \cdot x\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \frac{1}{z \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+287}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;y \leq 10000000:\\ \;\;\;\;\frac{1}{\frac{y \cdot \mathsf{hypot}\left(1, z\right)}{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot x\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2
Accuracy	97.1%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+287}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	97.4%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+287}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	97.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -10000000000 \lor \neg \left(z \leq 1.36 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	93.3%
Cost	841

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

Alternative 6
Accuracy	96.5%
Cost	841

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

Alternative 7
Accuracy	96.5%
Cost	841

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

Alternative 8
Accuracy	96.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0001:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	54.7%
Cost	320

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

Reproduce

herbie shell --seed 2023137 
(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)))))