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

Average Error: 6.6 → 2.0

Time: 10.5s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -10000000000000:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(z \cdot x\right)}{\frac{1}{z}}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{x}}{z \cdot y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -10000000000000.0)
   (/ 1.0 (/ (* y (* z x)) (/ 1.0 z)))
   (if (<= z 5e+152)
     (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
     (/ (/ (/ 1.0 z) x) (* z y)))))

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 <= -10000000000000.0) {
		tmp = 1.0 / ((y * (z * x)) / (1.0 / z));
	} else if (z <= 5e+152) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = ((1.0 / z) / x) / (z * y);
	}
	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 <= (-10000000000000.0d0)) then
        tmp = 1.0d0 / ((y * (z * x)) / (1.0d0 / z))
    else if (z <= 5d+152) then
        tmp = ((1.0d0 / x) / y) / (1.0d0 + (z * z))
    else
        tmp = ((1.0d0 / z) / x) / (z * y)
    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 <= -10000000000000.0) {
		tmp = 1.0 / ((y * (z * x)) / (1.0 / z));
	} else if (z <= 5e+152) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = ((1.0 / z) / x) / (z * y);
	}
	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 <= -10000000000000.0:
		tmp = 1.0 / ((y * (z * x)) / (1.0 / z))
	elif z <= 5e+152:
		tmp = ((1.0 / x) / y) / (1.0 + (z * z))
	else:
		tmp = ((1.0 / z) / x) / (z * y)
	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 (z <= -10000000000000.0)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(z * x)) / Float64(1.0 / z)));
	elseif (z <= 5e+152)
		tmp = Float64(Float64(Float64(1.0 / x) / y) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(1.0 / z) / x) / Float64(z * y));
	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 <= -10000000000000.0)
		tmp = 1.0 / ((y * (z * x)) / (1.0 / z));
	elseif (z <= 5e+152)
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	else
		tmp = ((1.0 / z) / x) / (z * y);
	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[z, -10000000000000.0], N[(1.0 / N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+152], N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.6
Target	5.0
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if z < -1e13

   1. Initial program 13.0

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

   2. Simplified 12.5

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

      [Start] 13.0	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/l/ [=>] 13.1	\[ \color{blue}{\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}} \]
      associate-*l* [=>] 12.5	\[ \frac{1}{\color{blue}{y \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}} \]
      +-commutative [=>] 12.5	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z + 1\right)} \cdot x\right)} \]
      fma-def [=>] 12.5	\[ \frac{1}{y \cdot \left(\color{blue}{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)} \]

   3. Taylor expanded in z around inf 12.5

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

   4. Simplified 12.6

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

      [Start] 12.5	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      *-commutative [=>] 12.5	\[ \frac{1}{y \cdot \color{blue}{\left(x \cdot {z}^{2}\right)}} \]
      unpow2 [=>] 12.5	\[ \frac{1}{y \cdot \left(x \cdot \color{blue}{\left(z \cdot z\right)}\right)} \]
      associate-*r* [=>] 12.6	\[ \frac{1}{\color{blue}{\left(y \cdot x\right) \cdot \left(z \cdot z\right)}} \]

   5. Applied egg-rr 3.2

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

   if -1e13 < z < 5e152

   1. Initial program 1.4

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

   2. Simplified 1.5

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

      [Start] 1.4	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 1.5	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   if 5e152 < z

   1. Initial program 18.5

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

   2. Taylor expanded in z around inf 18.5

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

   3. Simplified 18.5

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

      [Start] 18.5	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 18.5	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      *-commutative [=>] 18.5	\[ \frac{\frac{1}{y}}{\color{blue}{x \cdot {z}^{2}}} \]
      unpow2 [=>] 18.5	\[ \frac{\frac{1}{y}}{x \cdot \color{blue}{\left(z \cdot z\right)}} \]

   4. Taylor expanded in y around 0 18.5

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

   5. Simplified 7.2

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

      [Start] 18.5	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      *-commutative [=>] 18.5	\[ \frac{1}{\color{blue}{\left({z}^{2} \cdot x\right) \cdot y}} \]
      *-commutative [=>] 18.5	\[ \frac{1}{\color{blue}{\left(x \cdot {z}^{2}\right)} \cdot y} \]
      unpow2 [=>] 18.5	\[ \frac{1}{\left(x \cdot \color{blue}{\left(z \cdot z\right)}\right) \cdot y} \]
      associate-/r* [=>] 18.5	\[ \color{blue}{\frac{\frac{1}{x \cdot \left(z \cdot z\right)}}{y}} \]
      associate-/r* [=>] 18.5	\[ \frac{\color{blue}{\frac{\frac{1}{x}}{z \cdot z}}}{y} \]
      associate-/r* [=>] 7.2	\[ \frac{\color{blue}{\frac{\frac{\frac{1}{x}}{z}}{z}}}{y} \]

   6. Applied egg-rr 1.7

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10000000000000:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(z \cdot x\right)}{\frac{1}{z}}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{x}}{z \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	13504

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

Alternative 2
Error	1.8
Cost	968

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{z \cdot t_0}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\frac{1}{z}}}\\ \end{array} \]

Alternative 3
Error	1.8
Cost	968

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{z \cdot t_0}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\frac{1}{z}}}\\ \end{array} \]

Alternative 4
Error	2.4
Cost	964

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

Alternative 5
Error	4.4
Cost	841

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

Alternative 6
Error	2.4
Cost	836

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

Alternative 7
Error	28.7
Cost	320

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

Alternative 8
Error	28.7
Cost	320

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

Reproduce

herbie shell --seed 2023031 
(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)))))