Average Error: 14.8 → 2.7
Time: 15.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -58055444970433757472358400 \lor \neg \left(y \le 1.000178951499777507692981026699353331534 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{1 + z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(1 + z\right) \cdot z}{x \cdot \frac{y}{z}}}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -58055444970433757472358400 \lor \neg \left(y \le 1.000178951499777507692981026699353331534 \cdot 10^{-36}\right):\\
\;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{1 + z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r228682 = x;
        double r228683 = y;
        double r228684 = r228682 * r228683;
        double r228685 = z;
        double r228686 = r228685 * r228685;
        double r228687 = 1.0;
        double r228688 = r228685 + r228687;
        double r228689 = r228686 * r228688;
        double r228690 = r228684 / r228689;
        return r228690;
}
↓
double f(double x, double y, double z) {
        double r228691 = y;
        double r228692 = -5.805544497043376e+25;
        bool r228693 = r228691 <= r228692;
        double r228694 = 1.0001789514997775e-36;
        bool r228695 = r228691 <= r228694;
        double r228696 = !r228695;
        bool r228697 = r228693 || r228696;
        double r228698 = x;
        double r228699 = z;
        double r228700 = r228698 / r228699;
        double r228701 = r228700 / r228699;
        double r228702 = r228691 * r228701;
        double r228703 = 1.0;
        double r228704 = r228703 + r228699;
        double r228705 = r228702 / r228704;
        double r228706 = 1.0;
        double r228707 = r228704 * r228699;
        double r228708 = r228691 / r228699;
        double r228709 = r228698 * r228708;
        double r228710 = r228707 / r228709;
        double r228711 = r228706 / r228710;
        double r228712 = r228697 ? r228705 : r228711;
        return r228712;
}

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

Original14.8
Target3.8
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -5.805544497043376e+25 or 1.0001789514997775e-36 < y

    1. Initial program 17.4

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

    3. Applied associate-/r*14.0

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

    4. Simplified3.2

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

    6. Applied div-inv3.3

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

    7. Applied associate-*l*3.6

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

    8. Simplified3.6

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

    if -5.805544497043376e+25 < y < 1.0001789514997775e-36

    1. Initial program 12.8

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

    3. Applied associate-/r*11.8

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

    4. Simplified2.0

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

    6. Applied associate-*r/1.7

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

    7. Simplified1.7

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

    9. Applied clear-num1.9

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

    10. Simplified2.1

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(1 + z\right) \cdot z}{x \cdot \frac{y}{z}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -58055444970433757472358400 \lor \neg \left(y \le 1.000178951499777507692981026699353331534 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{1 + z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(1 + z\right) \cdot z}{x \cdot \frac{y}{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))))