Average Error: 14.5 → 2.0
Time: 3.6s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{z}\right)\right) \cdot \frac{\sqrt[3]{y}}{z + 1}}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{z}\right)\right) \cdot \frac{\sqrt[3]{y}}{z + 1}}{z}
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) cbrt(y)) / ((double) (z + 1.0))))) / z);
}

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

Original14.5
Target3.9
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.5

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

  2. Simplified13.2

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

  4. Applied add-cube-cbrt13.6

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

  5. Applied times-frac8.7

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

  6. Applied associate-*r*3.2

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

  7. Simplified3.2

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

  9. Applied associate-*r/3.2

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

  10. Applied associate-*r/7.2

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

  11. Applied associate-*l/7.3

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

  12. Simplified4.7

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

  14. Applied add-cube-cbrt5.2

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

  15. Applied times-frac3.5

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

  16. Applied associate-*r*1.9

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

  17. Simplified2.0

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

  18. Final simplification2.0

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

Reproduce

herbie shell --seed 2020199 
(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))))