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

Average Error: 6.4 → 1.8

Time: 12.4s

Precision: binary64

Cost: 1028

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{\frac{-1}{y}}{-x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+185)
   (/ (/ (/ -1.0 y) (- x)) (+ 1.0 (* z z)))
   (/ (/ (/ 1.0 x) (* z y)) 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 tmp;
	if ((z * z) <= 2e+185) {
		tmp = ((-1.0 / y) / -x) / (1.0 + (z * z));
	} else {
		tmp = ((1.0 / x) / (z * y)) / 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) :: tmp
    if ((z * z) <= 2d+185) then
        tmp = (((-1.0d0) / y) / -x) / (1.0d0 + (z * z))
    else
        tmp = ((1.0d0 / x) / (z * y)) / 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 tmp;
	if ((z * z) <= 2e+185) {
		tmp = ((-1.0 / y) / -x) / (1.0 + (z * z));
	} else {
		tmp = ((1.0 / x) / (z * y)) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+185:
		tmp = ((-1.0 / y) / -x) / (1.0 + (z * z))
	else:
		tmp = ((1.0 / x) / (z * y)) / 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)
	tmp = 0.0
	if (Float64(z * z) <= 2e+185)
		tmp = Float64(Float64(Float64(-1.0 / y) / Float64(-x)) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(z * y)) / 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)
	tmp = 0.0;
	if ((z * z) <= 2e+185)
		tmp = ((-1.0 / y) / -x) / (1.0 + (z * z));
	else
		tmp = ((1.0 / x) / (z * y)) / 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_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+185], N[(N[(N[(-1.0 / y), $MachinePrecision] / (-x)), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	6.4
Target	5.0
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e185

   1. Initial program 1.1

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

   2. Simplified 1.2

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

      [Start] 1.1	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 1.2	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   3. Applied egg-rr 1.2

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

   if 2e185 < (*.f64 z z)

   1. Initial program 15.4

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

   2. Simplified 15.3

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

      [Start] 15.4	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 15.3	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   3. Applied egg-rr 2.7

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

   4. Applied egg-rr 2.3

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

   5. Taylor expanded in z around inf 15.4

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

   6. Simplified 7.9

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

      [Start] 15.4	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      unpow2 [=>] 15.4	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)} \]
      associate-*l* [=>] 7.9	\[ \frac{1}{y \cdot \color{blue}{\left(z \cdot \left(z \cdot x\right)\right)}} \]

   7. Taylor expanded in y around 0 15.4

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

   8. Simplified 2.7

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

      [Start] 15.4	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 15.2	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      unpow2 [=>] 15.2	\[ \frac{\frac{1}{y}}{\color{blue}{\left(z \cdot z\right)} \cdot x} \]
      associate-*r* [<=] 7.7	\[ \frac{\frac{1}{y}}{\color{blue}{z \cdot \left(z \cdot x\right)}} \]
      associate-/l/ [<=] 2.8	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{z \cdot x}}{z}} \]
      associate-/l/ [=>] 2.8	\[ \frac{\color{blue}{\frac{1}{\left(z \cdot x\right) \cdot y}}}{z} \]
      associate-/r* [=>] 2.8	\[ \frac{\color{blue}{\frac{\frac{1}{z \cdot x}}{y}}}{z} \]
      associate-/l/ [<=] 2.7	\[ \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{z}}}{y}}{z} \]
      associate-/l/ [=>] 2.7	\[ \frac{\color{blue}{\frac{\frac{1}{x}}{y \cdot z}}}{z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{\frac{-1}{y}}{-x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.6
Cost	13632

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

Alternative 2
Error	1.4
Cost	13504

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

Alternative 3
Error	1.8
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+37} \lor \neg \left(z \leq 6.6 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+45} \lor \neg \left(z \leq 92000000\right):\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{z \cdot y}\\ \end{array} \]

Alternative 6
Error	4.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -62000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \end{array} \]

Alternative 7
Error	2.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -62000000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \end{array} \]

Alternative 8
Error	4.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -62000000000000:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 9
Error	2.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -62000000000000:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot y\right) \cdot \left(z \cdot x\right)}\\ \end{array} \]

Alternative 10
Error	2.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{z \cdot y}\\ \end{array} \]

Alternative 11
Error	2.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \]

Alternative 12
Error	28.3
Cost	320

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

Reproduce

herbie shell --seed 2023057 
(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)))))