Average Error: 14.6 → 1.1
Time: 18.3s
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 r271367 = x;
        double r271368 = y;
        double r271369 = r271367 * r271368;
        double r271370 = z;
        double r271371 = r271370 * r271370;
        double r271372 = 1.0;
        double r271373 = r271370 + r271372;
        double r271374 = r271371 * r271373;
        double r271375 = r271369 / r271374;
        return r271375;
}
↓
double f(double x, double y, double z) {
        double r271376 = x;
        double r271377 = y;
        double r271378 = r271376 * r271377;
        double r271379 = 1.1961511694853e-315;
        bool r271380 = r271378 <= r271379;
        double r271381 = cbrt(r271376);
        double r271382 = cbrt(r271381);
        double r271383 = r271382 * r271382;
        double r271384 = r271383 * r271382;
        double r271385 = r271381 * r271384;
        double r271386 = z;
        double r271387 = r271385 / r271386;
        double r271388 = r271381 / r271386;
        double r271389 = 1.0;
        double r271390 = r271386 + r271389;
        double r271391 = r271377 / r271390;
        double r271392 = r271388 * r271391;
        double r271393 = r271387 * r271392;
        double r271394 = 7.94789564876583e+212;
        bool r271395 = r271378 <= r271394;
        double r271396 = r271378 / r271386;
        double r271397 = r271386 * r271390;
        double r271398 = r271396 / r271397;
        double r271399 = r271381 * r271381;
        double r271400 = r271399 / r271386;
        double r271401 = r271400 * r271388;
        double r271402 = r271401 * r271391;
        double r271403 = r271395 ? r271398 : r271402;
        double r271404 = r271380 ? r271393 : r271403;
        return r271404;
}

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))))