Average Error: 14.4 → 2.0
Time: 8.6s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{1}{z} \cdot \left(\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{1}{z} \cdot \left(\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)
double f(double x, double y, double z) {
        double r207715 = x;
        double r207716 = y;
        double r207717 = r207715 * r207716;
        double r207718 = z;
        double r207719 = r207718 * r207718;
        double r207720 = 1.0;
        double r207721 = r207718 + r207720;
        double r207722 = r207719 * r207721;
        double r207723 = r207717 / r207722;
        return r207723;
}
↓
double f(double x, double y, double z) {
        double r207724 = 1.0;
        double r207725 = z;
        double r207726 = r207724 / r207725;
        double r207727 = x;
        double r207728 = cbrt(r207727);
        double r207729 = r207728 / r207725;
        double r207730 = y;
        double r207731 = 1.0;
        double r207732 = r207725 + r207731;
        double r207733 = r207730 / r207732;
        double r207734 = r207729 * r207733;
        double r207735 = r207728 * r207728;
        double r207736 = r207734 * r207735;
        double r207737 = r207726 * r207736;
        return r207737;
}

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

Original14.4
Target4.1
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.4

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

  3. Applied times-frac11.0

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

  5. Applied *-un-lft-identity11.0

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

  6. Applied times-frac5.9

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

  7. Applied associate-*l*2.7

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

  9. Applied *-un-lft-identity2.7

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

  10. Applied add-cube-cbrt3.1

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

  11. Applied times-frac3.2

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

  12. Applied associate-*l*2.0

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

  13. Final simplification2.0

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

Reproduce

herbie shell --seed 2020046 
(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))))