Average Error: 14.5 → 1.4
Time: 3.6s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{x}}{z} \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot \frac{\sqrt[3]{x}}{z}\right)\right)}{z + 1}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{\sqrt[3]{x}}{z} \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot \frac{\sqrt[3]{x}}{z}\right)\right)}{z + 1}
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (/ (* (/ (cbrt x) z) (* (cbrt x) (* y (/ (cbrt x) 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) cbrt(x)) / z) * ((double) (((double) cbrt(x)) * ((double) (y * (((double) cbrt(x)) / 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

Original14.5
Target4.2
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program Error: 14.5 bits

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

  3. Applied associate-/r*Error: 12.8 bits

    \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}}\]

  4. SimplifiedError: 11.1 bits

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

  6. Applied add-cube-cbrtError: 11.5 bits

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

  7. Applied times-fracError: 6.8 bits

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

  8. Applied associate-*r*Error: 1.4 bits

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

  9. SimplifiedError: 1.4 bits

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

  11. Applied *-un-lft-identityError: 1.4 bits

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

  12. Applied times-fracError: 1.5 bits

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

  13. Applied associate-*l*Error: 1.4 bits

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

  14. SimplifiedError: 1.4 bits

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

  15. Final simplificationError: 1.4 bits

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

Reproduce

herbie shell --seed 2020205 
(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))))