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

Percentage Accurate: 90.9% → 97.3%

Time: 8.8s

Precision: binary64

Cost: 1736

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x \cdot z}}{z}\\ \end{array} \]

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 -2e+72)
     (/ 1.0 (* z (* y (* x z))))
     (if (<= t_0 4e+297) (/ (/ 1.0 x) t_0) (/ (/ (/ 1.0 y) (* x 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 t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -2e+72) {
		tmp = 1.0 / (z * (y * (x * z)));
	} else if (t_0 <= 4e+297) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / y) / (x * 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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= (-2d+72)) then
        tmp = 1.0d0 / (z * (y * (x * z)))
    else if (t_0 <= 4d+297) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = ((1.0d0 / y) / (x * 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 t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= -2e+72) {
		tmp = 1.0 / (z * (y * (x * z)));
	} else if (t_0 <= 4e+297) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / y) / (x * z)) / 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 <= -2e+72:
		tmp = 1.0 / (z * (y * (x * z)))
	elif t_0 <= 4e+297:
		tmp = (1.0 / x) / t_0
	else:
		tmp = ((1.0 / y) / (x * 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)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= -2e+72)
		tmp = Float64(1.0 / Float64(z * Float64(y * Float64(x * z))));
	elseif (t_0 <= 4e+297)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / y) / Float64(x * 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)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= -2e+72)
		tmp = 1.0 / (z * (y * (x * z)));
	elseif (t_0 <= 4e+297)
		tmp = (1.0 / x) / t_0;
	else
		tmp = ((1.0 / y) / (x * 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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+72], N[(1.0 / N[(z * N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+297], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	90.9%
Target	92.9%
Herbie	97.3%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < -1.99999999999999989e72

   1. Initial program 79.1%

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

   2. Simplified 79.1%

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

      [Start] 79.1%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 79.1%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 79.1%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 79.1%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Applied egg-rr 84.1%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}} \]
      Step-by-step derivation

      [Start] 79.1%	\[ \frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)} \]
      fma-udef [=>] 79.1%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      +-commutative [<=] 79.1%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(1 + z \cdot z\right)}\right)} \]
      associate-/r* [=>] 79.1%	\[ \color{blue}{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]
      associate-/r* [=>] 79.8%	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]
      add-sqr-sqrt [=>] 79.8%	\[ \frac{\frac{\frac{1}{x}}{y}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}} \]
      *-un-lft-identity [=>] 79.8%	\[ \frac{\color{blue}{1 \cdot \frac{\frac{1}{x}}{y}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}} \]
      times-frac [=>] 79.9%	\[ \color{blue}{\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}}} \]
      hypot-1-def [=>] 79.9%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}} \]
      associate-/l/ [=>] 79.8%	\[ \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\color{blue}{\frac{1}{y \cdot x}}}{\sqrt{1 + z \cdot z}} \]
      hypot-1-def [=>] 84.1%	\[ \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y \cdot x}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \]

   4. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}} \]
      Step-by-step derivation

      [Start] 84.1%	\[ \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-*l/ [=>] 84.1%	\[ \color{blue}{\frac{1 \cdot \frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}} \]
      *-lft-identity [=>] 84.1%	\[ \frac{\color{blue}{\frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-/r* [=>] 84.1%	\[ \frac{\frac{\color{blue}{\frac{\frac{1}{y}}{x}}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-/l/ [=>] 98.4%	\[ \frac{\color{blue}{\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right) \cdot x}}}{\mathsf{hypot}\left(1, z\right)} \]
      *-commutative [=>] 98.4%	\[ \frac{\frac{\frac{1}{y}}{\color{blue}{x \cdot \mathsf{hypot}\left(1, z\right)}}}{\mathsf{hypot}\left(1, z\right)} \]

   5. Taylor expanded in z around inf 57.0%

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

   6. Simplified 57.0%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(\left(z \cdot z\right) \cdot x\right)}} \]
      Step-by-step derivation

      [Start] 57.0%	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      unpow2 [=>] 57.0%	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)} \]

   7. Taylor expanded in y around 0 57.0%

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

   8. Simplified 76.4%

      \[\leadsto \frac{1}{\color{blue}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}} \]
      Step-by-step derivation

      [Start] 57.0%	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      *-commutative [=>] 57.0%	\[ \frac{1}{\color{blue}{\left({z}^{2} \cdot x\right) \cdot y}} \]
      unpow2 [=>] 57.0%	\[ \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot x\right) \cdot y} \]
      associate-*r* [<=] 60.2%	\[ \frac{1}{\color{blue}{\left(z \cdot z\right) \cdot \left(x \cdot y\right)}} \]
      *-commutative [<=] 60.2%	\[ \frac{1}{\left(z \cdot z\right) \cdot \color{blue}{\left(y \cdot x\right)}} \]
      associate-*l* [=>] 67.2%	\[ \frac{1}{\color{blue}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}} \]
      *-commutative [=>] 67.2%	\[ \frac{1}{z \cdot \left(z \cdot \color{blue}{\left(x \cdot y\right)}\right)} \]
      associate-*l* [<=] 76.4%	\[ \frac{1}{z \cdot \color{blue}{\left(\left(z \cdot x\right) \cdot y\right)}} \]
      *-commutative [<=] 76.4%	\[ \frac{1}{z \cdot \left(\color{blue}{\left(x \cdot z\right)} \cdot y\right)} \]
      *-commutative [=>] 76.4%	\[ \frac{1}{z \cdot \color{blue}{\left(y \cdot \left(x \cdot z\right)\right)}} \]
      *-commutative [=>] 76.4%	\[ \frac{1}{z \cdot \left(y \cdot \color{blue}{\left(z \cdot x\right)}\right)} \]

   if -1.99999999999999989e72 < (*.f64 y (+.f64 1 (*.f64 z z))) < 4.0000000000000001e297

   1. Initial program 99.7%

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

   if 4.0000000000000001e297 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 67.3%

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

   2. Simplified 67.3%

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

      [Start] 67.3%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 67.3%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 67.3%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 67.3%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Applied egg-rr 75.4%

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

      [Start] 67.3%	\[ \frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)} \]
      associate-/r* [=>] 67.3%	\[ \color{blue}{\frac{\frac{1}{x}}{y \cdot \mathsf{fma}\left(z, z, 1\right)}} \]
      *-un-lft-identity [=>] 67.3%	\[ \frac{\color{blue}{1 \cdot \frac{1}{x}}}{y \cdot \mathsf{fma}\left(z, z, 1\right)} \]
      fma-udef [=>] 67.3%	\[ \frac{1 \cdot \frac{1}{x}}{y \cdot \color{blue}{\left(z \cdot z + 1\right)}} \]
      +-commutative [<=] 67.3%	\[ \frac{1 \cdot \frac{1}{x}}{y \cdot \color{blue}{\left(1 + z \cdot z\right)}} \]
      times-frac [=>] 75.4%	\[ \color{blue}{\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}} \]
      +-commutative [=>] 75.4%	\[ \frac{1}{y} \cdot \frac{\frac{1}{x}}{\color{blue}{z \cdot z + 1}} \]
      fma-udef [<=] 75.4%	\[ \frac{1}{y} \cdot \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)}} \]

   4. Taylor expanded in z around inf 75.4%

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

   5. Simplified 83.7%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{1}{z \cdot \left(z \cdot x\right)}} \]
      Step-by-step derivation

      [Start] 75.4%	\[ \frac{1}{y} \cdot \frac{1}{{z}^{2} \cdot x} \]
      unpow2 [=>] 75.4%	\[ \frac{1}{y} \cdot \frac{1}{\color{blue}{\left(z \cdot z\right)} \cdot x} \]
      associate-*l* [=>] 83.7%	\[ \frac{1}{y} \cdot \frac{1}{\color{blue}{z \cdot \left(z \cdot x\right)}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{x \cdot z}}{z}} \]
      Step-by-step derivation

      [Start] 83.7%	\[ \frac{1}{y} \cdot \frac{1}{z \cdot \left(z \cdot x\right)} \]
      un-div-inv [=>] 83.7%	\[ \color{blue}{\frac{\frac{1}{y}}{z \cdot \left(z \cdot x\right)}} \]
      *-commutative [=>] 83.7%	\[ \frac{\frac{1}{y}}{\color{blue}{\left(z \cdot x\right) \cdot z}} \]
      associate-/r* [=>] 99.8%	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{z \cdot x}}{z}} \]
      *-commutative [=>] 99.8%	\[ \frac{\frac{\frac{1}{y}}{\color{blue}{x \cdot z}}}{z} \]

3. Recombined 3 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x \cdot z}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.3%
Cost	1736

\[\begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x \cdot z}}{z}\\ \end{array} \]

Alternative 2
Accuracy	97.7%
Cost	13504

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

Alternative 3
Accuracy	97.5%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	93.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{--1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	96.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{--1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	96.6%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{--1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x \cdot z}}{z}\\ \end{array} \]

Alternative 7
Accuracy	58.3%
Cost	384

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

Alternative 8
Accuracy	58.2%
Cost	320

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

Alternative 9
Accuracy	58.3%
Cost	320

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

Reproduce

herbie shell --seed 2023165 
(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)))))