Average Error: 15.4 → 1.6
Time: 43.3s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
↓
\[\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}{z} \cdot \left(\frac{y}{1.0 + z} \cdot \frac{\sqrt[3]{x}}{z}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}
↓
\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}{z} \cdot \left(\frac{y}{1.0 + z} \cdot \frac{\sqrt[3]{x}}{z}\right)
double f(double x, double y, double z) {
        double r20334492 = x;
        double r20334493 = y;
        double r20334494 = r20334492 * r20334493;
        double r20334495 = z;
        double r20334496 = r20334495 * r20334495;
        double r20334497 = 1.0;
        double r20334498 = r20334495 + r20334497;
        double r20334499 = r20334496 * r20334498;
        double r20334500 = r20334494 / r20334499;
        return r20334500;
}
↓
double f(double x, double y, double z) {
        double r20334501 = x;
        double r20334502 = cbrt(r20334501);
        double r20334503 = cbrt(r20334502);
        double r20334504 = r20334503 * r20334503;
        double r20334505 = r20334502 * r20334502;
        double r20334506 = cbrt(r20334505);
        double r20334507 = r20334506 * r20334506;
        double r20334508 = r20334504 * r20334507;
        double r20334509 = z;
        double r20334510 = r20334508 / r20334509;
        double r20334511 = y;
        double r20334512 = 1.0;
        double r20334513 = r20334512 + r20334509;
        double r20334514 = r20334511 / r20334513;
        double r20334515 = r20334502 / r20334509;
        double r20334516 = r20334514 * r20334515;
        double r20334517 = r20334510 * r20334516;
        return r20334517;
}

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

Original15.4
Target4.3
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1.0 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.4

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac11.2

    \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1.0}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt11.6

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{z \cdot z} \cdot \frac{y}{z + 1.0}\]

  6. Applied times-frac6.5

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \frac{y}{z + 1.0}\]

  7. Applied associate-*l*1.3

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)\]

  10. Applied cbrt-prod1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)\]

  11. Applied add-cube-cbrt1.5

    \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)\]

  12. Applied cbrt-prod1.6

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)\]

  13. Applied swap-sqr1.6

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)\]

  14. Final simplification1.6

    \[\leadsto \frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}{z} \cdot \left(\frac{y}{1.0 + z} \cdot \frac{\sqrt[3]{x}}{z}\right)\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :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))))