Average Error: 15.0 → 1.3
Time: 2.0m
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\frac{\left|z\right|}{\sqrt[3]{x}}} \cdot \left(\frac{\sqrt[3]{x}}{\left|z\right|} \cdot \frac{y}{z + 1}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x}}{\frac{\left|z\right|}{\sqrt[3]{x}}} \cdot \left(\frac{\sqrt[3]{x}}{\left|z\right|} \cdot \frac{y}{z + 1}\right)
double code(double x, double y, double z) {
	return ((x * y) / ((z * z) * (z + 1.0)));
}
↓
double code(double x, double y, double z) {
	return ((cbrt(x) / (fabs(z) / cbrt(x))) * ((cbrt(x) / fabs(z)) * (y / (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

Original15.0
Target4.2
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.0

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

  3. Applied add-sqr-sqrt15.0

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

  4. Applied associate-*l*15.0

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

  5. Applied add-cube-cbrt15.4

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

  6. Applied associate-*l*15.4

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

  7. Applied times-frac10.5

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

  8. Simplified10.5

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

  9. Simplified1.3

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

  10. Final simplification1.3

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))