Average Error: 12.4 → 2.9
Time: 18.4s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.699929332195363087572042602326430898574 \cdot 10^{-162} \lor \neg \left(y \le 2.724407101896751811416130978286132894297 \cdot 10^{-107}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\left(y - z\right) \cdot x}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.699929332195363087572042602326430898574 \cdot 10^{-162} \lor \neg \left(y \le 2.724407101896751811416130978286132894297 \cdot 10^{-107}\right):\\
\;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r483795 = x;
        double r483796 = y;
        double r483797 = z;
        double r483798 = r483796 - r483797;
        double r483799 = r483795 * r483798;
        double r483800 = r483799 / r483796;
        return r483800;
}
↓
double f(double x, double y, double z) {
        double r483801 = y;
        double r483802 = -7.699929332195363e-162;
        bool r483803 = r483801 <= r483802;
        double r483804 = 2.7244071018967518e-107;
        bool r483805 = r483801 <= r483804;
        double r483806 = !r483805;
        bool r483807 = r483803 || r483806;
        double r483808 = x;
        double r483809 = 1.0;
        double r483810 = z;
        double r483811 = r483810 / r483801;
        double r483812 = r483809 - r483811;
        double r483813 = r483808 * r483812;
        double r483814 = r483801 - r483810;
        double r483815 = r483814 * r483808;
        double r483816 = r483801 / r483815;
        double r483817 = r483809 / r483816;
        double r483818 = r483807 ? r483813 : r483817;
        return r483818;
}

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

Original12.4
Target3.2
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -7.699929332195363e-162 or 2.7244071018967518e-107 < y

    1. Initial program 13.2

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

    3. Applied associate-/l*0.9

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

    5. Applied div-inv1.2

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

    6. Simplified1.1

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

    if -7.699929332195363e-162 < y < 2.7244071018967518e-107

    1. Initial program 9.5

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

    3. Applied associate-/l*11.1

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

    5. Applied clear-num11.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{y - z}}{x}}}\]
    6. Using strategy rm

    7. Applied div-inv11.2

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

    8. Applied associate-/l*9.6

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

    9. Simplified9.5

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.699929332195363087572042602326430898574 \cdot 10^{-162} \lor \neg \left(y \le 2.724407101896751811416130978286132894297 \cdot 10^{-107}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\left(y - z\right) \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))