Average Error: 13.0 → 1.9
Time: 3.1s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.901646128522259257495615629314967561602 \cdot 10^{-252} \lor \neg \left(y \le 1.444583585396195924054622508482627471757 \cdot 10^{-125}\right):\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + x \cdot \left(-z\right)}{y}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.901646128522259257495615629314967561602 \cdot 10^{-252} \lor \neg \left(y \le 1.444583585396195924054622508482627471757 \cdot 10^{-125}\right):\\
\;\;\;\;\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r870599 = x;
        double r870600 = y;
        double r870601 = z;
        double r870602 = r870600 - r870601;
        double r870603 = r870599 * r870602;
        double r870604 = r870603 / r870600;
        return r870604;
}
↓
double f(double x, double y, double z) {
        double r870605 = y;
        double r870606 = -1.9016461285222593e-252;
        bool r870607 = r870605 <= r870606;
        double r870608 = 1.444583585396196e-125;
        bool r870609 = r870605 <= r870608;
        double r870610 = !r870609;
        bool r870611 = r870607 || r870610;
        double r870612 = x;
        double r870613 = z;
        double r870614 = r870605 - r870613;
        double r870615 = cbrt(r870614);
        double r870616 = r870615 * r870615;
        double r870617 = cbrt(r870605);
        double r870618 = r870617 * r870617;
        double r870619 = r870616 / r870618;
        double r870620 = r870612 * r870619;
        double r870621 = r870615 / r870617;
        double r870622 = r870620 * r870621;
        double r870623 = r870612 * r870605;
        double r870624 = -r870613;
        double r870625 = r870612 * r870624;
        double r870626 = r870623 + r870625;
        double r870627 = r870626 / r870605;
        double r870628 = r870611 ? r870622 : r870627;
        return r870628;
}

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

Original13.0
Target3.4
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.9016461285222593e-252 or 1.444583585396196e-125 < y

    1. Initial program 13.4

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

    3. Applied *-un-lft-identity13.4

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

    4. Applied times-frac1.8

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

    5. Simplified1.8

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

    7. Applied add-cube-cbrt3.1

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

    8. Applied add-cube-cbrt2.3

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

    9. Applied times-frac2.3

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

    10. Applied associate-*r*0.7

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

    if -1.9016461285222593e-252 < y < 1.444583585396196e-125

    1. Initial program 10.0

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

    3. Applied sub-neg10.0

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

    4. Applied distribute-lft-in10.0

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.901646128522259257495615629314967561602 \cdot 10^{-252} \lor \neg \left(y \le 1.444583585396195924054622508482627471757 \cdot 10^{-125}\right):\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + x \cdot \left(-z\right)}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))