Average Error: 14.8 → 1.6
Time: 7.8s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\sqrt[3]{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z}} \cdot \sqrt[3]{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z}} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\sqrt[3]{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z}} \cdot \sqrt[3]{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z}} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{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) (((double) cbrt((((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / z))) * ((double) cbrt((((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / z))))) * ((double) (((double) cbrt((((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / z))) * ((double) ((((double) cbrt(x)) / z) * (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
Target4.3
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.8

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

  3. Applied times-frac10.9

    \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt11.3

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

  6. Applied times-frac6.3

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

  7. Applied associate-*l*1.5

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

  9. Applied add-cube-cbrt1.6

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

  10. Applied associate-*l*1.6

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

  11. Final simplification1.6

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

Reproduce

herbie shell --seed 2020182 
(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))))