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

Percentage Accurate: 91.3% → 97.6%

Time: 11.0s

Precision: binary64

Cost: 964

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

Math

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

↓

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

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

↓

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+107)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ 1.0 (* y (* x z))) z)))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+107) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} 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

↓

NOTE: x and y should be sorted in increasing order before calling this 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 * z) <= 2d+107) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    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)));
}

↓

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+107) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} 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)))

↓

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+107:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	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

↓

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+107)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(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

↓

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+107)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	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]

↓

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+107], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	91.3%
Target	93.2%
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 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e107

   1. Initial program 99.6%

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

   if 1.9999999999999999e107 < (*.f64 z z)

   1. Initial program 77.9%

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

   2. Simplified 77.2%

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

      [Start] 77.9%	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 77.2%	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 77.2%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 77.2%	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Taylor expanded in x around 0 75.2%

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

   4. Simplified 71.5%

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

      [Start] 75.2%	\[ \frac{1}{\left({z}^{2} + 1\right) \cdot \left(y \cdot x\right)} \]
      unpow2 [=>] 75.2%	\[ \frac{1}{\left(\color{blue}{z \cdot z} + 1\right) \cdot \left(y \cdot x\right)} \]
      fma-udef [<=] 75.2%	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(y \cdot x\right)} \]
      *-commutative [<=] 75.2%	\[ \frac{1}{\color{blue}{\left(y \cdot x\right) \cdot \mathsf{fma}\left(z, z, 1\right)}} \]
      associate-*r* [<=] 71.5%	\[ \frac{1}{\color{blue}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]

   5. Taylor expanded in z around inf 71.5%

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

   6. Simplified 85.4%

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

      [Start] 71.5%	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      unpow2 [=>] 71.5%	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)} \]
      associate-*l* [=>] 85.4%	\[ \frac{1}{y \cdot \color{blue}{\left(z \cdot \left(z \cdot x\right)\right)}} \]

   7. Applied egg-rr 96.2%

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

      [Start] 85.4%	\[ \frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)} \]
      associate-*r* [=>] 71.5%	\[ \frac{1}{y \cdot \color{blue}{\left(\left(z \cdot z\right) \cdot x\right)}} \]
      associate-/r* [=>] 73.1%	\[ \color{blue}{\frac{\frac{1}{y}}{\left(z \cdot z\right) \cdot x}} \]
      *-un-lft-identity [=>] 73.1%	\[ \frac{\color{blue}{1 \cdot \frac{1}{y}}}{\left(z \cdot z\right) \cdot x} \]
      associate-*r* [<=] 87.2%	\[ \frac{1 \cdot \frac{1}{y}}{\color{blue}{z \cdot \left(z \cdot x\right)}} \]
      *-commutative [=>] 87.2%	\[ \frac{1 \cdot \frac{1}{y}}{\color{blue}{\left(z \cdot x\right) \cdot z}} \]
      times-frac [=>] 96.2%	\[ \color{blue}{\frac{1}{z \cdot x} \cdot \frac{\frac{1}{y}}{z}} \]
      *-commutative [=>] 96.2%	\[ \frac{1}{\color{blue}{x \cdot z}} \cdot \frac{\frac{1}{y}}{z} \]

   8. Applied egg-rr 97.2%

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

      [Start] 96.2%	\[ \frac{1}{x \cdot z} \cdot \frac{\frac{1}{y}}{z} \]
      associate-*r/ [=>] 97.2%	\[ \color{blue}{\frac{\frac{1}{x \cdot z} \cdot \frac{1}{y}}{z}} \]
      frac-times [=>] 97.2%	\[ \frac{\color{blue}{\frac{1 \cdot 1}{\left(x \cdot z\right) \cdot y}}}{z} \]
      metadata-eval [=>] 97.2%	\[ \frac{\frac{\color{blue}{1}}{\left(x \cdot z\right) \cdot y}}{z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]

Alternative 2
Accuracy	97.8%
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

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]

Alternative 4
Accuracy	97.6%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]

Alternative 5
Accuracy	93.8%
Cost	841

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

Alternative 6
Accuracy	93.4%
Cost	841

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

Alternative 7
Accuracy	96.8%
Cost	841

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

Alternative 8
Accuracy	96.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{y \cdot z}\\ \end{array} \]

Alternative 10
Accuracy	96.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0002:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]

Alternative 11
Accuracy	58.8%
Cost	320

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

Reproduce

herbie shell --seed 2023167 
(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)))))