Average Error: 15.3 → 1.3
Time: 27.9s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)
double f(double x, double y, double z) {
        double r233481 = x;
        double r233482 = y;
        double r233483 = r233481 * r233482;
        double r233484 = z;
        double r233485 = r233484 * r233484;
        double r233486 = 1.0;
        double r233487 = r233484 + r233486;
        double r233488 = r233485 * r233487;
        double r233489 = r233483 / r233488;
        return r233489;
}
↓
double f(double x, double y, double z) {
        double r233490 = x;
        double r233491 = cbrt(r233490);
        double r233492 = z;
        double r233493 = cbrt(r233492);
        double r233494 = r233493 * r233493;
        double r233495 = r233491 / r233494;
        double r233496 = r233491 / r233492;
        double r233497 = y;
        double r233498 = 1.0;
        double r233499 = r233492 + r233498;
        double r233500 = r233497 / r233499;
        double r233501 = r233496 * r233500;
        double r233502 = r233491 / r233493;
        double r233503 = r233501 * r233502;
        double r233504 = r233495 * r233503;
        return r233504;
}

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

Original15.3
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 15.3

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

  3. Applied times-frac11.2

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

  4. Simplified11.2

    \[\leadsto \color{blue}{\frac{x}{{z}^{2}}} \cdot \frac{y}{z + 1}\]
  5. Using strategy rm

  6. Applied sqr-pow11.2

    \[\leadsto \frac{x}{\color{blue}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}}} \cdot \frac{y}{z + 1}\]

  7. Applied add-cube-cbrt11.6

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}} \cdot \frac{y}{z + 1}\]

  8. Applied times-frac6.2

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \frac{\sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}}\right)} \cdot \frac{y}{z + 1}\]

  9. Applied associate-*l*1.3

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \frac{y}{z + 1}\right)}\]

  10. Simplified1.3

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{z}^{\left(\frac{2}{2}\right)}} \cdot \color{blue}{\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt1.4

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\left(\sqrt[3]{{z}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{z}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt[3]{{z}^{\left(\frac{2}{2}\right)}}}} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\]

  13. Applied times-frac1.4

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{{z}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{z}^{\left(\frac{2}{2}\right)}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{{z}^{\left(\frac{2}{2}\right)}}}\right)} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\]

  14. Applied associate-*l*1.3

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{\sqrt[3]{{z}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{z}^{\left(\frac{2}{2}\right)}}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{{z}^{\left(\frac{2}{2}\right)}}} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\right)}\]

  15. Simplified1.3

    \[\leadsto \frac{\sqrt[3]{x}}{\sqrt[3]{{z}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{z}^{\left(\frac{2}{2}\right)}}} \cdot \color{blue}{\left(\left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)}\]

  16. Final simplification1.3

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(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))))