Average Error: 14.6 → 1.1
Time: 19.6s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le 1.196151169485302342273884046453796637229 \cdot 10^{-315}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\\ \mathbf{elif}\;x \cdot y \le 7.947895648765829788288380988796806250372 \cdot 10^{212}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right) \cdot \frac{y}{z + 1}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le 1.196151169485302342273884046453796637229 \cdot 10^{-315}:\\
\;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\\

\mathbf{elif}\;x \cdot y \le 7.947895648765829788288380988796806250372 \cdot 10^{212}:\\
\;\;\;\;\frac{\frac{x \cdot y}{z}}{z \cdot \left(z + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right) \cdot \frac{y}{z + 1}\\

\end{array}
double f(double x, double y, double z) {
        double r192532 = x;
        double r192533 = y;
        double r192534 = r192532 * r192533;
        double r192535 = z;
        double r192536 = r192535 * r192535;
        double r192537 = 1.0;
        double r192538 = r192535 + r192537;
        double r192539 = r192536 * r192538;
        double r192540 = r192534 / r192539;
        return r192540;
}
↓
double f(double x, double y, double z) {
        double r192541 = x;
        double r192542 = y;
        double r192543 = r192541 * r192542;
        double r192544 = 1.1961511694853e-315;
        bool r192545 = r192543 <= r192544;
        double r192546 = cbrt(r192541);
        double r192547 = cbrt(r192546);
        double r192548 = r192547 * r192547;
        double r192549 = r192548 * r192547;
        double r192550 = r192546 * r192549;
        double r192551 = z;
        double r192552 = r192550 / r192551;
        double r192553 = r192546 / r192551;
        double r192554 = 1.0;
        double r192555 = r192551 + r192554;
        double r192556 = r192542 / r192555;
        double r192557 = r192553 * r192556;
        double r192558 = r192552 * r192557;
        double r192559 = 7.94789564876583e+212;
        bool r192560 = r192543 <= r192559;
        double r192561 = r192543 / r192551;
        double r192562 = r192551 * r192555;
        double r192563 = r192561 / r192562;
        double r192564 = r192546 * r192546;
        double r192565 = r192564 / r192551;
        double r192566 = r192565 * r192553;
        double r192567 = r192566 * r192556;
        double r192568 = r192560 ? r192563 : r192567;
        double r192569 = r192545 ? r192558 : r192568;
        return r192569;
}

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

Original14.6
Target4.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < 1.1961511694853e-315

    1. Initial program 16.0

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

    3. Applied times-frac11.6

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

    5. Applied add-cube-cbrt11.9

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

    6. Applied times-frac6.4

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

    7. Applied associate-*l*1.2

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

    9. Applied add-cube-cbrt1.4

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

    if 1.1961511694853e-315 < (* x y) < 7.94789564876583e+212

    1. Initial program 6.9

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

    3. Applied times-frac9.7

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

    5. Applied add-cube-cbrt10.1

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

    6. Applied times-frac7.6

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

    7. Applied associate-*l*1.5

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

    9. Applied associate-*r/1.4

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

    10. Applied frac-times0.9

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

    11. Simplified0.4

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

    if 7.94789564876583e+212 < (* x y)

    1. Initial program 41.0

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

    3. Applied times-frac13.3

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

    5. Applied add-cube-cbrt13.7

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

    6. Applied times-frac1.7

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \frac{y}{z + 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le 1.196151169485302342273884046453796637229 \cdot 10^{-315}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\\ \mathbf{elif}\;x \cdot y \le 7.947895648765829788288380988796806250372 \cdot 10^{212}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z}}{z \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right) \cdot \frac{y}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))