Average Error: 15.7 → 1.4
Time: 10.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\frac{\sqrt[3]{x}}{z} \cdot \sqrt[3]{x}\right) \cdot \frac{y \cdot {\left(\sqrt[3]{\frac{\sqrt[3]{x}}{z}}\right)}^{3}}{z + 1}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\frac{\sqrt[3]{x}}{z} \cdot \sqrt[3]{x}\right) \cdot \frac{y \cdot {\left(\sqrt[3]{\frac{\sqrt[3]{x}}{z}}\right)}^{3}}{z + 1}
double f(double x, double y, double z) {
        double r366396 = x;
        double r366397 = y;
        double r366398 = r366396 * r366397;
        double r366399 = z;
        double r366400 = r366399 * r366399;
        double r366401 = 1.0;
        double r366402 = r366399 + r366401;
        double r366403 = r366400 * r366402;
        double r366404 = r366398 / r366403;
        return r366404;
}
↓
double f(double x, double y, double z) {
        double r366405 = x;
        double r366406 = cbrt(r366405);
        double r366407 = z;
        double r366408 = r366406 / r366407;
        double r366409 = r366408 * r366406;
        double r366410 = y;
        double r366411 = cbrt(r366408);
        double r366412 = 3.0;
        double r366413 = pow(r366411, r366412);
        double r366414 = r366410 * r366413;
        double r366415 = 1.0;
        double r366416 = r366407 + r366415;
        double r366417 = r366414 / r366416;
        double r366418 = r366409 * r366417;
        return r366418;
}

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

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 *-un-lft-identity1.5

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

  13. Applied associate-*l*1.5

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

  14. Simplified1.4

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

  15. Final simplification1.4

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

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))))