Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 14.7 → 1.2

Time: 2.7m

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return ((double) (((double) (x * y)) / ((double) (((double) (z * z)) * ((double) (z + 1.0))))));
}

↓

double code(double x, double y, double z) {
	return ((double) (((double) (((double) (((double) cbrt(x)) / ((double) (1.0 / ((double) cbrt(1.0)))))) * ((double) (((double) (((double) (((double) cbrt(x)) / z)) * ((double) (y / ((double) (((double) cbrt(((double) (z + 1.0)))) * 1.0)))))) / ((double) cbrt(((double) (z + 1.0)))))))) / ((double) (((double) (z / ((double) cbrt(x)))) * ((double) cbrt(((double) (z + 1.0))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.7
Target	4.0
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

1. Initial program 14.7

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

3. Applied associate-*l* 14.7

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

4. Applied add-cube-cbrt 15.0

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

5. Applied associate-*l* 15.0

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

6. Applied times-frac 4.9

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

7. Simplified 4.9

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

8. Simplified 1.3

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

10. Applied add-cube-cbrt 1.4

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

11. Applied associate-/r* 1.4

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

12. Applied associate-*r/ 1.4

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

13. Applied *-un-lft-identity 1.4

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

14. Applied cbrt-prod 1.4

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

15. Applied *-un-lft-identity 1.4

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

16. Applied times-frac 1.4

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

17. Applied associate-/r* 1.4

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

18. Applied frac-times 1.2

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

20. Applied *-un-lft-identity 1.2

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

21. Applied associate-*r* 1.2

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

22. Applied associate-/r* 1.2

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

23. Applied associate-*r/ 1.2

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

24. Final simplification 1.2

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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