Average Error: 14.9 → 1.1
Time: 5.2s
Precision: 64
\[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.819752026637393519506049512320603510099 \cdot 10^{-42} \lor \neg \left(y \le 3.147408210159048214381836933498385687136 \cdot 10^{-101}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y}}}\\ \end{array}\]
\frac{\left(x \cdot 2\right) \cdot y}{x - y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.819752026637393519506049512320603510099 \cdot 10^{-42} \lor \neg \left(y \le 3.147408210159048214381836933498385687136 \cdot 10^{-101}\right):\\
\;\;\;\;\left(x \cdot 2\right) \cdot \frac{y}{x - y}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y}}}\\

\end{array}
double f(double x, double y) {
        double r660551 = x;
        double r660552 = 2.0;
        double r660553 = r660551 * r660552;
        double r660554 = y;
        double r660555 = r660553 * r660554;
        double r660556 = r660551 - r660554;
        double r660557 = r660555 / r660556;
        return r660557;
}
↓
double f(double x, double y) {
        double r660558 = y;
        double r660559 = -6.819752026637394e-42;
        bool r660560 = r660558 <= r660559;
        double r660561 = 3.147408210159048e-101;
        bool r660562 = r660558 <= r660561;
        double r660563 = !r660562;
        bool r660564 = r660560 || r660563;
        double r660565 = x;
        double r660566 = 2.0;
        double r660567 = r660565 * r660566;
        double r660568 = r660565 - r660558;
        double r660569 = r660558 / r660568;
        double r660570 = r660567 * r660569;
        double r660571 = cbrt(r660568);
        double r660572 = r660571 * r660571;
        double r660573 = r660567 / r660572;
        double r660574 = cbrt(r660558);
        double r660575 = r660574 * r660574;
        double r660576 = cbrt(r660572);
        double r660577 = r660575 / r660576;
        double r660578 = r660573 * r660577;
        double r660579 = cbrt(r660571);
        double r660580 = r660574 / r660579;
        double r660581 = r660578 * r660580;
        double r660582 = r660564 ? r660570 : r660581;
        return r660582;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.3
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.721044263414944729490876394165887012892 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x \lt 83645045635564432:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -6.819752026637394e-42 or 3.147408210159048e-101 < y

    1. Initial program 13.3

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

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

      \[\leadsto \frac{\left(x \cdot 2\right) \cdot y}{\color{blue}{1 \cdot \left(x - y\right)}}\]

    4. Applied times-frac0.7

      \[\leadsto \color{blue}{\frac{x \cdot 2}{1} \cdot \frac{y}{x - y}}\]

    5. Simplified0.7

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \frac{y}{x - y}\]

    if -6.819752026637394e-42 < y < 3.147408210159048e-101

    1. Initial program 17.5

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

    3. Applied add-cube-cbrt18.4

      \[\leadsto \frac{\left(x \cdot 2\right) \cdot y}{\color{blue}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \sqrt[3]{x - y}}}\]

    4. Applied times-frac6.8

      \[\leadsto \color{blue}{\frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{y}{\sqrt[3]{x - y}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt6.9

      \[\leadsto \frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{y}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \sqrt[3]{x - y}}}}\]

    7. Applied cbrt-prod7.0

      \[\leadsto \frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{y}{\color{blue}{\sqrt[3]{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \sqrt[3]{\sqrt[3]{x - y}}}}\]

    8. Applied add-cube-cbrt7.0

      \[\leadsto \frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \sqrt[3]{\sqrt[3]{x - y}}}\]

    9. Applied times-frac7.0

      \[\leadsto \frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y}}}\right)}\]

    10. Applied associate-*r*1.8

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.819752026637393519506049512320603510099 \cdot 10^{-42} \lor \neg \left(y \le 3.147408210159048214381836933498385687136 \cdot 10^{-101}\right):\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot 2}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{x - y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564432) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y)))

  (/ (* (* x 2) y) (- x y)))