Average Error: 15.7 → 1.5
Time: 11.3s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\left({\left(\sqrt[3]{\frac{\sqrt[3]{x}}{z}}\right)}^{3} \cdot \frac{y}{z + 1}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)\]
\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({\left(\sqrt[3]{\frac{\sqrt[3]{x}}{z}}\right)}^{3} \cdot \frac{y}{z + 1}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)
double f(double x, double y, double z) {
        double r388512 = x;
        double r388513 = y;
        double r388514 = r388512 * r388513;
        double r388515 = z;
        double r388516 = r388515 * r388515;
        double r388517 = 1.0;
        double r388518 = r388515 + r388517;
        double r388519 = r388516 * r388518;
        double r388520 = r388514 / r388519;
        return r388520;
}
↓
double f(double x, double y, double z) {
        double r388521 = x;
        double r388522 = cbrt(r388521);
        double r388523 = z;
        double r388524 = cbrt(r388523);
        double r388525 = r388524 * r388524;
        double r388526 = r388522 / r388525;
        double r388527 = r388522 / r388523;
        double r388528 = cbrt(r388527);
        double r388529 = 3.0;
        double r388530 = pow(r388528, r388529);
        double r388531 = y;
        double r388532 = 1.0;
        double r388533 = r388523 + r388532;
        double r388534 = r388531 / r388533;
        double r388535 = r388530 * r388534;
        double r388536 = r388522 / r388524;
        double r388537 = r388535 * r388536;
        double r388538 = r388526 * r388537;
        return r388538;
}

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

Original15.7
Target4.4
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.7

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

  3. Applied times-frac11.8

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

  5. Applied add-cube-cbrt12.1

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

  6. Applied times-frac7.0

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

  7. Applied associate-*l*1.3

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

  9. Applied add-cube-cbrt1.5

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

  10. Applied associate-*l*1.5

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

  12. Applied add-cube-cbrt1.6

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

  13. Applied times-frac1.6

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

  14. Applied associate-*l*1.5

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

  15. Simplified1.5

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

  16. Final simplification1.5

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

Reproduce

herbie shell --seed 2020047 
(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))))