Average Error: 15.4 → 2.7
Time: 4.4s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \left(x \cdot \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{z}\right)\right)}{z} \cdot \frac{\sqrt[3]{\sqrt[3]{y}}}{z + 1}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \left(x \cdot \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{z}\right)\right)}{z} \cdot \frac{\sqrt[3]{\sqrt[3]{y}}}{z + 1}
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) (((double) cbrt(y)) * ((double) (((double) cbrt(y)) * ((double) (x * (((double) cbrt(((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) / z))))))) / z) * (((double) cbrt(((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

Original15.4
Target4.2
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.4

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

  2. Simplified14.4

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

  4. Applied add-cube-cbrt14.8

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*3.3

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

  7. Simplified3.3

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

  9. Applied add-cube-cbrt3.3

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

  10. Applied cbrt-prod3.4

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

  11. Applied times-frac2.4

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

  12. Applied associate-*r*1.7

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

  13. Simplified2.8

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

  15. Applied associate-*r/4.2

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

  16. Applied associate-*r/4.2

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

  17. Applied associate-*r/3.4

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

  18. Simplified2.7

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

  19. Final simplification2.7

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

Reproduce

herbie shell --seed 2020198 
(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))))