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

Average Error: 6.4 → 1.7

Time: 12.0s

Precision: binary64

Cost: 964

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 9 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot \frac{1}{z \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 9e+145)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (* (/ 1.0 (* z y)) (/ 1.0 (* 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 tmp;
	if ((z * z) <= 9e+145) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (z * y)) * (1.0 / (z * x));
	}
	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) <= 9d+145) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / (z * y)) * (1.0d0 / (z * x))
    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) <= 9e+145) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (z * y)) * (1.0 / (z * x));
	}
	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) <= 9e+145:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / (z * y)) * (1.0 / (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)
	tmp = 0.0
	if (Float64(z * z) <= 9e+145)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(z * y)) * Float64(1.0 / 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)
	tmp = 0.0;
	if ((z * z) <= 9e+145)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / (z * y)) * (1.0 / (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_] := If[LessEqual[N[(z * z), $MachinePrecision], 9e+145], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.4
Target	5.1
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 8.9999999999999996e145

   1. Initial program 0.9

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

   if 8.9999999999999996e145 < (*.f64 z z)

   1. Initial program 14.1

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

   2. Taylor expanded in z around inf 14.1

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

   3. Simplified 7.5

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

      [Start] 14.1	\[ \frac{\frac{1}{x}}{y \cdot {z}^{2}} \]
      *-commutative [=>] 14.1	\[ \frac{\frac{1}{x}}{\color{blue}{{z}^{2} \cdot y}} \]
      unpow2 [=>] 14.1	\[ \frac{\frac{1}{x}}{\color{blue}{\left(z \cdot z\right)} \cdot y} \]
      associate-*r* [<=] 7.5	\[ \frac{\frac{1}{x}}{\color{blue}{z \cdot \left(z \cdot y\right)}} \]

   4. Applied egg-rr 2.7

      \[\leadsto \color{blue}{\frac{1}{z \cdot y} \cdot \frac{1}{x \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 9 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot \frac{1}{z \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.7
Cost	13632

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

Alternative 2
Error	2.2
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{z \cdot x}\\ \mathbf{if}\;z \leq -0.86:\\ \;\;\;\;\frac{t_0}{z \cdot y}\\ \mathbf{elif}\;z \leq 0.88:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot t_0\\ \end{array} \]

Alternative 3
Error	4.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -12.5 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]

Alternative 4
Error	4.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -12.5 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]

Alternative 5
Error	4.2
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -0.86 \lor \neg \left(z \leq 0.88\right):\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \end{array} \]

Alternative 6
Error	4.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.05:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 7
Error	2.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.05:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternative 8
Error	2.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.05:\\ \;\;\;\;\frac{1 - z \cdot z}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{z \cdot y}\\ \end{array} \]

Alternative 9
Error	21.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -12.5 \lor \neg \left(z \leq 5.5 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{-1}{y \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]

Alternative 10
Error	29.0
Cost	320

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

Reproduce

herbie shell --seed 2023060 
(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)))))