Average Error: 14.7 → 2.1
Time: 3.8s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{-1} \cdot \sqrt[3]{\frac{1}{\frac{-1}{y}}}}{z}\right) \cdot x\right) \cdot \frac{\frac{\sqrt[3]{y}}{z}}{z + 1}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{-1} \cdot \sqrt[3]{\frac{1}{\frac{-1}{y}}}}{z}\right) \cdot x\right) \cdot \frac{\frac{\sqrt[3]{y}}{z}}{z + 1}
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (* (* (cbrt y) (/ (* (cbrt -1.0) (cbrt (/ 1.0 (/ -1.0 y)))) z)) x)
  (/ (/ (cbrt y) z) (+ z 1.0))))
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) (((double) cbrt(y)) * (((double) (((double) cbrt(-1.0)) * ((double) cbrt((1.0 / (-1.0 / y)))))) / z))) * x)) * ((((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

Original14.7
Target4.0
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program Error: 14.7 bits

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

  2. SimplifiedError: 13.7 bits

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

  4. Applied add-cube-cbrtError: 14.0 bits

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

  5. Applied times-fracError: 8.8 bits

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

  6. Applied associate-*r*Error: 3.2 bits

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

  7. SimplifiedError: 3.2 bits

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

  9. Applied associate-/r*Error: 2.1 bits

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

  10. Taylor expanded around -inf Error: 34.1 bits

    \[\leadsto \left(\left(\sqrt[3]{y} \cdot \frac{\color{blue}{\sqrt[3]{-1} \cdot e^{0.3333333333333333 \cdot \left(\log 1 - \log \left(\frac{-1}{y}\right)\right)}}}{z}\right) \cdot x\right) \cdot \frac{\frac{\sqrt[3]{y}}{z}}{z + 1}\]

  11. SimplifiedError: 2.1 bits

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

  12. Final simplificationError: 2.1 bits

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

Reproduce

herbie shell --seed 2020203 
(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))))