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

Average Error: 6.4 → 0.7

Time: 11.9s

Precision: binary64

Cost: 7492

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot x\right)} \cdot \frac{1}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 1e+305)
   (/ (/ 1.0 x) (fma (* y z) z y))
   (* (/ 1.0 (* y (* z x))) (/ 1.0 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 ((y * (1.0 + (z * z))) <= 1e+305) {
		tmp = (1.0 / x) / fma((y * z), z, y);
	} else {
		tmp = (1.0 / (y * (z * x))) * (1.0 / 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(y * Float64(1.0 + Float64(z * z))) <= 1e+305)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(z * x))) * Float64(1.0 / z));
	end
	return 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[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.4
Target	5.1
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 9.9999999999999994e304

   1. Initial program 2.0

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

   2. Taylor expanded in y around 0 2.0

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

   3. Simplified 0.5

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

      [Start] 2.0	\[ \frac{\frac{1}{x}}{\left({z}^{2} + 1\right) \cdot y} \]
      unpow2 [=>] 2.0	\[ \frac{\frac{1}{x}}{\left(\color{blue}{z \cdot z} + 1\right) \cdot y} \]
      distribute-lft1-in [<=] 2.0	\[ \frac{\frac{1}{x}}{\color{blue}{\left(z \cdot z\right) \cdot y + y}} \]
      *-commutative [<=] 2.0	\[ \frac{\frac{1}{x}}{\color{blue}{y \cdot \left(z \cdot z\right)} + y} \]
      associate-*r* [=>] 0.5	\[ \frac{\frac{1}{x}}{\color{blue}{\left(y \cdot z\right) \cdot z} + y} \]
      fma-def [=>] 0.5	\[ \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(y \cdot z, z, y\right)}} \]

   if 9.9999999999999994e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 18.6

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

   2. Simplified 13.8

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

      [Start] 18.6	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 13.8	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   3. Taylor expanded in z around inf 14.1

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

   4. Simplified 5.1

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

      [Start] 14.1	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      associate-/r* [=>] 14.0	\[ \color{blue}{\frac{\frac{1}{y}}{{z}^{2} \cdot x}} \]
      unpow2 [=>] 14.0	\[ \frac{\frac{1}{y}}{\color{blue}{\left(z \cdot z\right)} \cdot x} \]
      associate-*l* [=>] 5.1	\[ \frac{\frac{1}{y}}{\color{blue}{z \cdot \left(z \cdot x\right)}} \]

   5. Applied egg-rr 1.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot x\right)} \cdot \frac{1}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -750000000000:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \mathbf{elif}\;z \leq 0.86:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot x\right)} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 2
Error	1.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \mathbf{elif}\;z \leq 19500000:\\ \;\;\;\;\frac{1}{y \cdot \left(x + \left(z \cdot z\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot x\right)} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 3
Error	1.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\frac{y \cdot z}{\frac{1}{z \cdot x}}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot x\right)} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 4
Error	4.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -750000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \end{array} \]

Alternative 5
Error	4.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -750000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \end{array} \]

Alternative 6
Error	2.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -750000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \end{array} \]

Alternative 7
Error	2.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 8
Error	28.7
Cost	320

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

Reproduce

herbie shell --seed 2023066 
(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)))))