Average Error: 7.6 → 2.4
Time: 10.1s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.9656743812715667 \cdot 10^{-16} \lor \neg \left(z \le 7.8493791686123403 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.9656743812715667 \cdot 10^{-16} \lor \neg \left(z \le 7.8493791686123403 \cdot 10^{-10}\right):\\
\;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r575863 = x;
        double r575864 = y;
        double r575865 = r575863 + r575864;
        double r575866 = 1.0;
        double r575867 = z;
        double r575868 = r575864 / r575867;
        double r575869 = r575866 - r575868;
        double r575870 = r575865 / r575869;
        return r575870;
}
↓
double f(double x, double y, double z) {
        double r575871 = z;
        double r575872 = -2.9656743812715667e-16;
        bool r575873 = r575871 <= r575872;
        double r575874 = 7.84937916861234e-10;
        bool r575875 = r575871 <= r575874;
        double r575876 = !r575875;
        bool r575877 = r575873 || r575876;
        double r575878 = 1.0;
        double r575879 = 1.0;
        double r575880 = y;
        double r575881 = r575880 / r575871;
        double r575882 = r575879 - r575881;
        double r575883 = r575878 / r575882;
        double r575884 = x;
        double r575885 = r575884 + r575880;
        double r575886 = r575883 * r575885;
        double r575887 = r575879 / r575885;
        double r575888 = r575885 * r575871;
        double r575889 = r575880 / r575888;
        double r575890 = r575887 - r575889;
        double r575891 = r575878 / r575890;
        double r575892 = r575877 ? r575886 : r575891;
        return r575892;
}

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

Original7.6
Target4.0
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.9656743812715667e-16 or 7.84937916861234e-10 < z

    1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied clear-num0.2

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

    5. Applied div-inv0.2

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

    6. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(1 - \frac{y}{z}\right) \cdot \frac{1}{x + y}}\]

    7. Applied times-frac0.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1 - \frac{y}{z}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x + y}}}\]

    8. Simplified0.2

      \[\leadsto \color{blue}{\frac{1}{1 - \frac{y}{z}}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x + y}}\]

    9. Simplified0.1

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

    if -2.9656743812715667e-16 < z < 7.84937916861234e-10

    1. Initial program 15.6

      \[\frac{x + y}{1 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied clear-num15.7

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

    5. Applied div-sub15.7

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

    6. Simplified4.9

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.9656743812715667 \cdot 10^{-16} \lor \neg \left(z \le 7.8493791686123403 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1 (/ y z))))