Average Error: 15.7 → 1.5
Time: 4.6s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{x}}{z}} \cdot \sqrt[3]{\frac{\sqrt[3]{x}}{z}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{x}}{z}} \cdot \frac{y}{z + 1}\right)\right)\right)\]
\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(\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)
double f(double x, double y, double z) {
        double r275464 = x;
        double r275465 = y;
        double r275466 = r275464 * r275465;
        double r275467 = z;
        double r275468 = r275467 * r275467;
        double r275469 = 1.0;
        double r275470 = r275467 + r275469;
        double r275471 = r275468 * r275470;
        double r275472 = r275466 / r275471;
        return r275472;
}
↓
double f(double x, double y, double z) {
        double r275473 = x;
        double r275474 = cbrt(r275473);
        double r275475 = z;
        double r275476 = cbrt(r275475);
        double r275477 = r275476 * r275476;
        double r275478 = r275474 / r275477;
        double r275479 = r275474 / r275476;
        double r275480 = r275474 / r275475;
        double r275481 = cbrt(r275480);
        double r275482 = r275481 * r275481;
        double r275483 = y;
        double r275484 = 1.0;
        double r275485 = r275475 + r275484;
        double r275486 = r275483 / r275485;
        double r275487 = r275481 * r275486;
        double r275488 = r275482 * r275487;
        double r275489 = r275479 * r275488;
        double r275490 = r275478 * r275489;
        return r275490;
}

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

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

  10. Applied times-frac1.5

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

  11. Applied associate-*l*1.4

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

  13. Applied add-cube-cbrt1.5

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

  14. Applied associate-*l*1.5

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

  15. Final simplification1.5

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

Reproduce

herbie shell --seed 2020047 +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))))