Average Error: 14.6 → 2.3
Time: 3.8s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{z}\right)\right) \cdot \left(\sqrt[3]{\frac{\frac{\sqrt[3]{y}}{z}}{z + 1}} \cdot \left(\sqrt[3]{\frac{\frac{\sqrt[3]{y}}{z}}{z + 1}} \cdot \sqrt[3]{\frac{\frac{\sqrt[3]{y}}{z}}{z + 1}}\right)\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{z}\right)\right) \cdot \left(\sqrt[3]{\frac{\frac{\sqrt[3]{y}}{z}}{z + 1}} \cdot \left(\sqrt[3]{\frac{\frac{\sqrt[3]{y}}{z}}{z + 1}} \cdot \sqrt[3]{\frac{\frac{\sqrt[3]{y}}{z}}{z + 1}}\right)\right)
double code(double x, double y, double z) {
	return (((double) (x * y)) / ((double) (((double) (z * z)) * ((double) (z + 1.0)))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (x * ((double) (((double) cbrt(y)) * (((double) cbrt(y)) / z))))) * ((double) (((double) cbrt(((((double) cbrt(y)) / z) / ((double) (z + 1.0))))) * ((double) (((double) cbrt(((((double) cbrt(y)) / z) / ((double) (z + 1.0))))) * ((double) cbrt(((((double) cbrt(y)) / z) / ((double) (z + 1.0)))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.6
Target3.8
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.6

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

  2. Simplified13.4

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

  4. Applied add-cube-cbrt13.7

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

  5. Applied times-frac8.6

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

  6. Applied associate-*r*3.0

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

  7. Simplified3.0

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

  9. Applied add-cube-cbrt3.2

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

  10. Applied associate-/l*3.2

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

  11. Simplified3.0

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

  13. Applied add-cube-cbrt3.1

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

  14. Simplified2.9

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

  15. Simplified2.3

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

  16. Final simplification2.3

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

Reproduce

herbie shell --seed 2020196 
(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) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))