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

Average Error: 10.01% → 0.99%

Time: 11.2s

Precision: binary64

Cost: 1736

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{y}}{z}}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{-1}{y \cdot \left(-1 - z \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

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 (+ 1.0 (* z z)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ (/ (/ 1.0 y) z) z) x)
     (if (<= t_0 1e+307)
       (/ (/ -1.0 (* y (- -1.0 (* z z)))) x)
       (/ (/ 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 * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((1.0 / y) / z) / z) / x;
	} else if (t_0 <= 1e+307) {
		tmp = (-1.0 / (y * (-1.0 - (z * z)))) / x;
	} 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 * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (((1.0 / y) / z) / z) / x;
	} else if (t_0 <= 1e+307) {
		tmp = (-1.0 / (y * (-1.0 - (z * z)))) / x;
	} 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 * (1.0 + (z * z))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (((1.0 / y) / z) / z) / x
	elif t_0 <= 1e+307:
		tmp = (-1.0 / (y * (-1.0 - (z * z)))) / x
	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(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(1.0 / y) / z) / z) / x);
	elseif (t_0 <= 1e+307)
		tmp = Float64(Float64(-1.0 / Float64(y * Float64(-1.0 - Float64(z * z)))) / x);
	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 * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (((1.0 / y) / z) / z) / x;
	elseif (t_0 <= 1e+307)
		tmp = (-1.0 / (y * (-1.0 - (z * z)))) / x;
	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[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e+307], N[(N[(-1.0 / N[(y * N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	10.01%
Target	7.83%
Herbie	0.99%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 27.05

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

   2. Applied egg-rr 27.05

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

   3. Simplified 27.05

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

      [Start] 27.05	\[ \frac{-1}{x} \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)} \]
      associate-*l/ [=>] 27.05	\[ \color{blue}{\frac{-1 \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)}}{x}} \]
      associate-*r/ [=>] 27.05	\[ \frac{\color{blue}{\frac{-1 \cdot 1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)}}}{x} \]
      metadata-eval [=>] 27.05	\[ \frac{\frac{\color{blue}{-1}}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)}}{x} \]
      distribute-rgt-neg-out [=>] 27.05	\[ \frac{\frac{-1}{\color{blue}{-\mathsf{fma}\left(z, z, 1\right) \cdot y}}}{x} \]
      *-commutative [<=] 27.05	\[ \frac{\frac{-1}{-\color{blue}{y \cdot \mathsf{fma}\left(z, z, 1\right)}}}{x} \]
      distribute-rgt-neg-in [=>] 27.05	\[ \frac{\frac{-1}{\color{blue}{y \cdot \left(-\mathsf{fma}\left(z, z, 1\right)\right)}}}{x} \]
      fma-udef [=>] 27.05	\[ \frac{\frac{-1}{y \cdot \left(-\color{blue}{\left(z \cdot z + 1\right)}\right)}}{x} \]
      +-commutative [<=] 27.05	\[ \frac{\frac{-1}{y \cdot \left(-\color{blue}{\left(1 + z \cdot z\right)}\right)}}{x} \]
      distribute-neg-in [=>] 27.05	\[ \frac{\frac{-1}{y \cdot \color{blue}{\left(\left(-1\right) + \left(-z \cdot z\right)\right)}}}{x} \]
      metadata-eval [=>] 27.05	\[ \frac{\frac{-1}{y \cdot \left(\color{blue}{-1} + \left(-z \cdot z\right)\right)}}{x} \]
      neg-sub0 [=>] 27.05	\[ \frac{\frac{-1}{y \cdot \left(-1 + \color{blue}{\left(0 - z \cdot z\right)}\right)}}{x} \]
      associate-+r- [=>] 27.05	\[ \frac{\frac{-1}{y \cdot \color{blue}{\left(\left(-1 + 0\right) - z \cdot z\right)}}}{x} \]
      metadata-eval [=>] 27.05	\[ \frac{\frac{-1}{y \cdot \left(\color{blue}{-1} - z \cdot z\right)}}{x} \]

   4. Taylor expanded in z around inf 27.05

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

   5. Simplified 2.4

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

      [Start] 27.05	\[ \frac{\frac{1}{y \cdot {z}^{2}}}{x} \]
      associate-/r* [=>] 27.05	\[ \frac{\color{blue}{\frac{\frac{1}{y}}{{z}^{2}}}}{x} \]
      unpow2 [=>] 27.05	\[ \frac{\frac{\frac{1}{y}}{\color{blue}{z \cdot z}}}{x} \]
      associate-/r* [=>] 2.4	\[ \frac{\color{blue}{\frac{\frac{\frac{1}{y}}{z}}{z}}}{x} \]

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 9.99999999999999986e306

   1. Initial program 0.43

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

   2. Applied egg-rr 0.54

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

   3. Simplified 0.41

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

      [Start] 0.54	\[ \frac{-1}{x} \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)} \]
      associate-*l/ [=>] 0.42	\[ \color{blue}{\frac{-1 \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)}}{x}} \]
      associate-*r/ [=>] 0.42	\[ \frac{\color{blue}{\frac{-1 \cdot 1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)}}}{x} \]
      metadata-eval [=>] 0.42	\[ \frac{\frac{\color{blue}{-1}}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(-y\right)}}{x} \]
      distribute-rgt-neg-out [=>] 0.42	\[ \frac{\frac{-1}{\color{blue}{-\mathsf{fma}\left(z, z, 1\right) \cdot y}}}{x} \]
      *-commutative [<=] 0.42	\[ \frac{\frac{-1}{-\color{blue}{y \cdot \mathsf{fma}\left(z, z, 1\right)}}}{x} \]
      distribute-rgt-neg-in [=>] 0.42	\[ \frac{\frac{-1}{\color{blue}{y \cdot \left(-\mathsf{fma}\left(z, z, 1\right)\right)}}}{x} \]
      fma-udef [=>] 0.41	\[ \frac{\frac{-1}{y \cdot \left(-\color{blue}{\left(z \cdot z + 1\right)}\right)}}{x} \]
      +-commutative [<=] 0.41	\[ \frac{\frac{-1}{y \cdot \left(-\color{blue}{\left(1 + z \cdot z\right)}\right)}}{x} \]
      distribute-neg-in [=>] 0.41	\[ \frac{\frac{-1}{y \cdot \color{blue}{\left(\left(-1\right) + \left(-z \cdot z\right)\right)}}}{x} \]
      metadata-eval [=>] 0.41	\[ \frac{\frac{-1}{y \cdot \left(\color{blue}{-1} + \left(-z \cdot z\right)\right)}}{x} \]
      neg-sub0 [=>] 0.41	\[ \frac{\frac{-1}{y \cdot \left(-1 + \color{blue}{\left(0 - z \cdot z\right)}\right)}}{x} \]
      associate-+r- [=>] 0.41	\[ \frac{\frac{-1}{y \cdot \color{blue}{\left(\left(-1 + 0\right) - z \cdot z\right)}}}{x} \]
      metadata-eval [=>] 0.41	\[ \frac{\frac{-1}{y \cdot \left(\color{blue}{-1} - z \cdot z\right)}}{x} \]

   if 9.99999999999999986e306 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 28.67

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

   2. Simplified 20.94

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

      [Start] 28.67	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 20.94	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   3. Taylor expanded in z around inf 21.37

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

   4. Simplified 21.37

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

      [Start] 21.37	\[ \frac{\frac{\frac{1}{x}}{y}}{{z}^{2}} \]
      unpow2 [=>] 21.37	\[ \frac{\frac{\frac{1}{x}}{y}}{\color{blue}{z \cdot z}} \]

   5. Applied egg-rr 5.5

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

   6. Taylor expanded in x around 0 21.28

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

   7. Simplified 2.02

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

      [Start] 21.28	\[ -\frac{-1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-*r* [=>] 29.12	\[ -\frac{-1}{\color{blue}{\left(y \cdot {z}^{2}\right) \cdot x}} \]
      unpow2 [=>] 29.12	\[ -\frac{-1}{\left(y \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot x} \]
      +-rgt-identity [<=] 29.12	\[ -\frac{-1}{\left(y \cdot \color{blue}{\left(z \cdot z + 0\right)}\right) \cdot x} \]
      *-commutative [=>] 29.12	\[ -\frac{-1}{\color{blue}{\left(\left(z \cdot z + 0\right) \cdot y\right)} \cdot x} \]
      associate-*r* [<=] 21.56	\[ -\frac{-1}{\color{blue}{\left(z \cdot z + 0\right) \cdot \left(y \cdot x\right)}} \]
      +-rgt-identity [=>] 21.56	\[ -\frac{-1}{\color{blue}{\left(z \cdot z\right)} \cdot \left(y \cdot x\right)} \]
      associate-*r* [<=] 10.01	\[ -\frac{-1}{\color{blue}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}} \]
      *-commutative [=>] 10.01	\[ -\frac{-1}{\color{blue}{\left(z \cdot \left(y \cdot x\right)\right) \cdot z}} \]
      associate-/r* [=>] 9.13	\[ -\color{blue}{\frac{\frac{-1}{z \cdot \left(y \cdot x\right)}}{z}} \]
      associate-*r* [=>] 5.49	\[ -\frac{\frac{-1}{\color{blue}{\left(z \cdot y\right) \cdot x}}}{z} \]
      *-commutative [<=] 5.49	\[ -\frac{\frac{-1}{\color{blue}{\left(y \cdot z\right)} \cdot x}}{z} \]
      associate-*r* [<=] 2.02	\[ -\frac{\frac{-1}{\color{blue}{y \cdot \left(z \cdot x\right)}}}{z} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.99

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{y}}{z}}{z}}{x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{-1}{y \cdot \left(-1 - z \cdot z\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.05%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + z \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternative 2
Error	3.54%
Cost	900

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{y \cdot z}}{-z}\\ \end{array} \]

Alternative 3
Error	6.7%
Cost	840

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

Alternative 4
Error	6.61%
Cost	840

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

Alternative 5
Error	6.33%
Cost	840

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

Alternative 6
Error	3.29%
Cost	840

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

Alternative 7
Error	6.7%
Cost	836

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

Alternative 8
Error	3.45%
Cost	836

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

Alternative 9
Error	45.29%
Cost	320

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

Alternative 10
Error	45.34%
Cost	320

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

Alternative 11
Error	45.34%
Cost	320

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

Reproduce

herbie shell --seed 2023115 
(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)))))