Average Error: 15.7 → 1.4
Time: 10.7s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{z}} \cdot \sqrt[3]{\frac{\sqrt[3]{x}}{z}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{x}}{z}} \cdot \frac{y}{z + 1}\right)\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{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)
double f(double x, double y, double z) {
        double r355425 = x;
        double r355426 = y;
        double r355427 = r355425 * r355426;
        double r355428 = z;
        double r355429 = r355428 * r355428;
        double r355430 = 1.0;
        double r355431 = r355428 + r355430;
        double r355432 = r355429 * r355431;
        double r355433 = r355427 / r355432;
        return r355433;
}
↓
double f(double x, double y, double z) {
        double r355434 = x;
        double r355435 = cbrt(r355434);
        double r355436 = r355435 * r355435;
        double r355437 = z;
        double r355438 = r355436 / r355437;
        double r355439 = cbrt(r355435);
        double r355440 = cbrt(r355437);
        double r355441 = r355439 / r355440;
        double r355442 = r355435 / r355437;
        double r355443 = cbrt(r355442);
        double r355444 = r355441 * r355443;
        double r355445 = y;
        double r355446 = 1.0;
        double r355447 = r355437 + r355446;
        double r355448 = r355445 / r355447;
        double r355449 = r355443 * r355448;
        double r355450 = r355444 * r355449;
        double r355451 = r355438 * r355450;
        return r355451;
}

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 cbrt-div1.4

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

  13. Final simplification1.4

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

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))))