Average Error: 14.8 → 2.3
Time: 4.7s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{z}\right) \cdot x\right) \cdot \left(\frac{1}{z} \cdot \frac{\sqrt[3]{y}}{z + 1}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{z}\right) \cdot x\right) \cdot \left(\frac{1}{z} \cdot \frac{\sqrt[3]{y}}{z + 1}\right)
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(y)) * ((double) (((double) cbrt(y)) / z)))) * x)) * ((double) (((double) (1.0 / z)) * ((double) (((double) cbrt(y)) / ((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

Original14.8
Target4.2
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.8

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

  2. Simplified14.2

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

  4. Applied add-cube-cbrt14.5

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

  5. Applied times-frac9.2

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

  6. Applied associate-*r*3.1

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

  7. Simplified3.1

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

  9. Applied *-un-lft-identity3.1

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

  10. Applied cbrt-prod3.1

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

  11. Applied times-frac2.3

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

  12. Simplified2.3

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

  13. Final simplification2.3

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

Reproduce

herbie shell --seed 2020184 
(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))))