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

Average Accuracy: 89.4% → 97.7%

Time: 12.0s

Precision: binary64

Cost: 7300

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1500000000:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{-1}{z}}{x}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -1500000000.0)
   (* (/ (/ 1.0 y) (hypot 1.0 z)) (/ (/ -1.0 z) x))
   (if (<= z 1.5e+19)
     (/ (/ (/ 1.0 y) x) (fma z z 1.0))
     (/ (/ (/ (/ 1.0 z) x) 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 <= -1500000000.0) {
		tmp = ((1.0 / y) / hypot(1.0, z)) * ((-1.0 / z) / x);
	} else if (z <= 1.5e+19) {
		tmp = ((1.0 / y) / x) / fma(z, z, 1.0);
	} else {
		tmp = (((1.0 / z) / x) / 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 (z <= -1500000000.0)
		tmp = Float64(Float64(Float64(1.0 / y) / hypot(1.0, z)) * Float64(Float64(-1.0 / z) / x));
	elseif (z <= 1.5e+19)
		tmp = Float64(Float64(Float64(1.0 / y) / x) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / z) / x) / y) / 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[z, -1500000000.0], N[(N[(N[(1.0 / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+19], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	89.4%
Target	91.6%
Herbie	97.7%

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

Derivation

1. Split input into 3 regimes

2. if z < -1.5e9

   1. Initial program 79.6%

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

   2. Simplified 79.4%

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

      [Start] 79.6	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [=>] 79.4	\[ \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}} \]

   3. Applied egg-rr 95.9%

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

      [Start] 79.4	\[ \frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z} \]
      div-inv [=>] 79.4	\[ \frac{\color{blue}{\frac{1}{x} \cdot \frac{1}{y}}}{1 + z \cdot z} \]
      add-sqr-sqrt [=>] 79.4	\[ \frac{\frac{1}{x} \cdot \frac{1}{y}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}} \]
      times-frac [=>] 81.2	\[ \color{blue}{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}} \]
      hypot-1-def [=>] 81.2	\[ \frac{\frac{1}{x}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \cdot \frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}} \]
      hypot-1-def [=>] 95.9	\[ \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y}}{\color{blue}{\mathsf{hypot}\left(1, z\right)}} \]

   4. Taylor expanded in z around -inf 95.8%

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

   5. Simplified 95.9%

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

      [Start] 95.8	\[ \frac{-1}{z \cdot x} \cdot \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \]
      associate-/r* [=>] 95.9	\[ \color{blue}{\frac{\frac{-1}{z}}{x}} \cdot \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \]

   if -1.5e9 < z < 1.5e19

   1. Initial program 99.6%

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

   2. Simplified 99.1%

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

      [Start] 99.6	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 99.1	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 99.1	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 99.1	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Taylor expanded in x around 0 99.1%

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

   4. Simplified 99.6%

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

      [Start] 99.1	\[ \frac{1}{y \cdot \left(\left({z}^{2} + 1\right) \cdot x\right)} \]
      associate-/r* [=>] 99.6	\[ \color{blue}{\frac{\frac{1}{y}}{\left({z}^{2} + 1\right) \cdot x}} \]
      *-commutative [=>] 99.6	\[ \frac{\frac{1}{y}}{\color{blue}{x \cdot \left({z}^{2} + 1\right)}} \]
      associate-/r* [=>] 99.6	\[ \color{blue}{\frac{\frac{\frac{1}{y}}{x}}{{z}^{2} + 1}} \]
      unpow2 [=>] 99.6	\[ \frac{\frac{\frac{1}{y}}{x}}{\color{blue}{z \cdot z} + 1} \]
      fma-udef [<=] 99.6	\[ \frac{\frac{\frac{1}{y}}{x}}{\color{blue}{\mathsf{fma}\left(z, z, 1\right)}} \]

   if 1.5e19 < z

   1. Initial program 79.0%

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

   2. Simplified 78.7%

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

      [Start] 79.0	\[ \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
      associate-/r* [<=] 78.7	\[ \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]
      +-commutative [=>] 78.7	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]
      fma-def [=>] 78.7	\[ \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Taylor expanded in z around inf 79.4%

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

   4. Simplified 78.5%

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

      [Start] 79.4	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      unpow2 [=>] 79.4	\[ \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)} \]
      *-commutative [=>] 79.4	\[ \frac{1}{y \cdot \color{blue}{\left(x \cdot \left(z \cdot z\right)\right)}} \]
      associate-*l* [<=] 78.5	\[ \frac{1}{\color{blue}{\left(y \cdot x\right) \cdot \left(z \cdot z\right)}} \]

   5. Taylor expanded in y around 0 79.4%

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

   6. Simplified 95.9%

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

      [Start] 79.4	\[ \frac{1}{y \cdot \left({z}^{2} \cdot x\right)} \]
      *-commutative [=>] 79.4	\[ \frac{1}{\color{blue}{\left({z}^{2} \cdot x\right) \cdot y}} \]
      unpow2 [=>] 79.4	\[ \frac{1}{\left(\color{blue}{\left(z \cdot z\right)} \cdot x\right) \cdot y} \]
      *-commutative [=>] 79.4	\[ \frac{1}{\color{blue}{\left(x \cdot \left(z \cdot z\right)\right)} \cdot y} \]
      associate-*r* [=>] 89.3	\[ \frac{1}{\color{blue}{\left(\left(x \cdot z\right) \cdot z\right)} \cdot y} \]
      associate-*l* [=>] 95.1	\[ \frac{1}{\color{blue}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}} \]
      associate-/r* [=>] 95.6	\[ \color{blue}{\frac{\frac{1}{x \cdot z}}{z \cdot y}} \]
      associate-/l/ [<=] 95.5	\[ \color{blue}{\frac{\frac{\frac{1}{x \cdot z}}{y}}{z}} \]
      associate-/l/ [<=] 95.9	\[ \frac{\frac{\color{blue}{\frac{\frac{1}{z}}{x}}}{y}}{z} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1500000000:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{-1}{z}}{x}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	13632

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

Alternative 2
Accuracy	97.7%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{-1}{z}}{y}}{x \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

Alternative 3
Accuracy	97.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\frac{-1}{z}}{y}}{x \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 1050000000:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

Alternative 4
Accuracy	97.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\frac{-1}{z}}{y}}{x \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -3.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{-1}{z}}{y}}{x \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 96000000:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

Alternative 6
Accuracy	96.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	96.8%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

Alternative 8
Accuracy	96.9%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15:\\ \;\;\;\;\frac{\frac{\frac{-1}{z}}{y}}{x \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{y}}{z}\\ \end{array} \]

Alternative 9
Accuracy	88.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	67.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15 \lor \neg \left(z \leq 1.38 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1}{y \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \end{array} \]

Alternative 12
Accuracy	67.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.15:\\ \;\;\;\;\frac{-1}{y \cdot \left(x \cdot z\right)}\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{z}}{y}}{x}\\ \end{array} \]

Alternative 13
Accuracy	55.2%
Cost	320

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

Reproduce

herbie shell --seed 2023147 
(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)))))