Average Error: 14.6 → 2.8
Time: 15.8s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.585967125230204065987521886435233291452 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{1 + z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;y \le 2.062958957082107387615416084806118908628 \cdot 10^{-214}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{1 + z}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(\frac{y}{1 + z} \cdot \frac{x}{z}\right)\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.585967125230204065987521886435233291452 \cdot 10^{-130}:\\
\;\;\;\;\frac{y}{1 + z} \cdot \frac{\frac{x}{z}}{z}\\

\mathbf{elif}\;y \le 2.062958957082107387615416084806118908628 \cdot 10^{-214}:\\
\;\;\;\;x \cdot \frac{\frac{\frac{y}{1 + z}}{z}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r14153405 = x;
        double r14153406 = y;
        double r14153407 = r14153405 * r14153406;
        double r14153408 = z;
        double r14153409 = r14153408 * r14153408;
        double r14153410 = 1.0;
        double r14153411 = r14153408 + r14153410;
        double r14153412 = r14153409 * r14153411;
        double r14153413 = r14153407 / r14153412;
        return r14153413;
}
↓
double f(double x, double y, double z) {
        double r14153414 = y;
        double r14153415 = -1.585967125230204e-130;
        bool r14153416 = r14153414 <= r14153415;
        double r14153417 = 1.0;
        double r14153418 = z;
        double r14153419 = r14153417 + r14153418;
        double r14153420 = r14153414 / r14153419;
        double r14153421 = x;
        double r14153422 = r14153421 / r14153418;
        double r14153423 = r14153422 / r14153418;
        double r14153424 = r14153420 * r14153423;
        double r14153425 = 2.0629589570821074e-214;
        bool r14153426 = r14153414 <= r14153425;
        double r14153427 = r14153420 / r14153418;
        double r14153428 = r14153427 / r14153418;
        double r14153429 = r14153421 * r14153428;
        double r14153430 = 1.0;
        double r14153431 = r14153430 / r14153418;
        double r14153432 = r14153420 * r14153422;
        double r14153433 = r14153431 * r14153432;
        double r14153434 = r14153426 ? r14153429 : r14153433;
        double r14153435 = r14153416 ? r14153424 : r14153434;
        return r14153435;
}

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
Target3.9
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.585967125230204e-130

    1. Initial program 15.3

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

    3. Applied times-frac9.0

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

    5. Applied associate-/r*3.2

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

    if -1.585967125230204e-130 < y < 2.0629589570821074e-214

    1. Initial program 14.3

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

    3. Applied times-frac15.5

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

    5. Applied div-inv15.8

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

    6. Applied associate-*l*13.6

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

    7. Simplified4.3

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

    if 2.0629589570821074e-214 < y

    1. Initial program 14.3

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

    3. Applied times-frac9.5

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

    5. Applied *-un-lft-identity9.5

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

    6. Applied times-frac4.4

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

    7. Applied associate-*l*1.5

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.585967125230204065987521886435233291452 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{1 + z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;y \le 2.062958957082107387615416084806118908628 \cdot 10^{-214}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{1 + z}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \left(\frac{y}{1 + z} \cdot \frac{x}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))