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

Average Error: 6.0 → 5.8

Time: 12.2s

Precision: binary64

Math

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

↓

\[\frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \left(1 + z \cdot z\right)}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (*
  (/ (* (cbrt (/ 1.0 y)) (cbrt (/ 1.0 y))) (cbrt x))
  (/ (/ (cbrt (/ 1.0 y)) (cbrt x)) (* (cbrt x) (+ 1.0 (* z 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) {
	return ((cbrt(1.0 / y) * cbrt(1.0 / y)) / cbrt(x)) * ((cbrt(1.0 / y) / cbrt(x)) / (cbrt(x) * (1.0 + (z * z))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	5.3
Herbie	5.8

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

Derivation

1. Initial program 6.0

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
2. Using strategy rm

3. Applied add-cube-cbrt_binary64_1829 6.6

   \[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{y \cdot \left(1 + z \cdot z\right)}\]

4. Applied *-un-lft-identity_binary64_1858 6.6

   \[\leadsto \frac{\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{y \cdot \left(1 + z \cdot z\right)}\]

5. Applied times-frac_binary64_1853 6.6

   \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}}}{y \cdot \left(1 + z \cdot z\right)}\]

6. Applied times-frac_binary64_1853 6.4

   \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}}\]

7. Simplified 6.4

   \[\leadsto \color{blue}{\frac{\frac{1}{y}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}\]
8. Using strategy rm

9. Applied add-cube-cbrt_binary64_1829 6.5

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}\right) \cdot \sqrt[3]{\frac{1}{y}}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}\]

10. Applied times-frac_binary64_1853 6.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}}\right)} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}\]

11. Applied associate-*l*_binary64_1918 5.7

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}\right)}\]

12. Simplified 5.8

    \[\leadsto \frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \left(1 + z \cdot z\right)}}\]

13. Final simplification 5.8

    \[\leadsto \frac{\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{y}}}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \left(1 + z \cdot z\right)}\]

Reproduce

herbie shell --seed 2020231 
(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)))))