Average Error: 16.5 → 8.1
Time: 20.3s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.4785650511945879 \cdot 10^{119} \lor \neg \left(t \le 5.62287539718969527 \cdot 10^{93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{t - z}{\sqrt[3]{a - t}}\right) + \left(x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.4785650511945879 \cdot 10^{119} \lor \neg \left(t \le 5.62287539718969527 \cdot 10^{93}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r759866 = x;
        double r759867 = y;
        double r759868 = r759866 + r759867;
        double r759869 = z;
        double r759870 = t;
        double r759871 = r759869 - r759870;
        double r759872 = r759871 * r759867;
        double r759873 = a;
        double r759874 = r759873 - r759870;
        double r759875 = r759872 / r759874;
        double r759876 = r759868 - r759875;
        return r759876;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r759877 = t;
        double r759878 = -1.4785650511945879e+119;
        bool r759879 = r759877 <= r759878;
        double r759880 = 5.622875397189695e+93;
        bool r759881 = r759877 <= r759880;
        double r759882 = !r759881;
        bool r759883 = r759879 || r759882;
        double r759884 = z;
        double r759885 = r759884 / r759877;
        double r759886 = y;
        double r759887 = x;
        double r759888 = fma(r759885, r759886, r759887);
        double r759889 = cbrt(r759886);
        double r759890 = r759889 * r759889;
        double r759891 = a;
        double r759892 = r759891 - r759877;
        double r759893 = cbrt(r759892);
        double r759894 = r759890 / r759893;
        double r759895 = r759889 / r759893;
        double r759896 = r759877 - r759884;
        double r759897 = r759896 / r759893;
        double r759898 = r759895 * r759897;
        double r759899 = r759894 * r759898;
        double r759900 = r759887 + r759886;
        double r759901 = r759899 + r759900;
        double r759902 = r759883 ? r759888 : r759901;
        return r759902;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.5
Target8.6
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.4785650511945879e+119 or 5.622875397189695e+93 < t

    1. Initial program 30.3

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

    2. Simplified20.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a - t}, y, x + y\right)}\]

    3. Taylor expanded around inf 17.5

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

    4. Simplified12.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}\]

    if -1.4785650511945879e+119 < t < 5.622875397189695e+93

    1. Initial program 8.8

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

    2. Simplified6.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a - t}, y, x + y\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt6.8

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}\right) \cdot \sqrt[3]{t - z}}}{a - t}, y, x + y\right)\]

    5. Applied associate-/l*6.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{a - t}{\sqrt[3]{t - z}}}}, y, x + y\right)\]
    6. Using strategy rm

    7. Applied fma-udef6.8

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

    8. Simplified6.7

      \[\leadsto \color{blue}{y \cdot \frac{t - z}{a - t}} + \left(x + y\right)\]
    9. Using strategy rm

    10. Applied add-cube-cbrt6.9

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

    11. Applied *-un-lft-identity6.9

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

    12. Applied times-frac6.9

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

    13. Applied associate-*r*6.7

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

    14. Simplified6.7

      \[\leadsto \color{blue}{\frac{y}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{t - z}{\sqrt[3]{a - t}} + \left(x + y\right)\]
    15. Using strategy rm

    16. Applied add-cube-cbrt6.7

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

    17. Applied times-frac6.7

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

    18. Applied associate-*l*5.7

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.4785650511945879 \cdot 10^{119} \lor \neg \left(t \le 5.62287539718969527 \cdot 10^{93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{t - z}{\sqrt[3]{a - t}}\right) + \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))