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

Average Error: 6.6 → 6.1

Time: 10.8s

Precision: 64

Math

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

↓

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

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 (((1.0 / (cbrt(y) * cbrt(y))) / (fabs(cbrt((1.0 + (z * z)))) * (cbrt(x) * cbrt(x)))) * ((((1.0 / cbrt(y)) / cbrt(x)) / sqrt(cbrt((1.0 + (z * z))))) / sqrt((1.0 + (z * z)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.6
Target	6.0
Herbie	6.1

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Initial program 6.6

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

3. Applied associate-/r* 6.7

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

4. Simplified 6.7

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

6. Applied add-sqr-sqrt 6.7

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

7. Applied associate-/r* 6.7

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

9. Applied *-un-lft-identity 6.7

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

10. Applied add-cube-cbrt 6.8

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

11. Applied sqrt-prod 6.8

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

12. Applied add-cube-cbrt 7.3

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

13. Applied add-cube-cbrt 7.5

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

14. Applied *-un-lft-identity 7.5

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

15. Applied times-frac 7.5

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

16. Applied times-frac 7.5

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

17. Applied times-frac 6.6

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

18. Applied times-frac 6.2

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

19. Simplified 6.1

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

20. Final simplification 6.1

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

Reproduce

herbie shell --seed 2020057 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))