Average Error: 16.6 → 7.8
Time: 5.6s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.20925242721609934 \cdot 10^{153} \lor \neg \left(t \le 8.90353523668654675 \cdot 10^{157}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.20925242721609934 \cdot 10^{153} \lor \neg \left(t \le 8.90353523668654675 \cdot 10^{157}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r486644 = x;
        double r486645 = y;
        double r486646 = r486644 + r486645;
        double r486647 = z;
        double r486648 = t;
        double r486649 = r486647 - r486648;
        double r486650 = r486649 * r486645;
        double r486651 = a;
        double r486652 = r486651 - r486648;
        double r486653 = r486650 / r486652;
        double r486654 = r486646 - r486653;
        return r486654;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r486655 = t;
        double r486656 = -1.2092524272160993e+153;
        bool r486657 = r486655 <= r486656;
        double r486658 = 8.903535236686547e+157;
        bool r486659 = r486655 <= r486658;
        double r486660 = !r486659;
        bool r486661 = r486657 || r486660;
        double r486662 = z;
        double r486663 = r486662 / r486655;
        double r486664 = y;
        double r486665 = x;
        double r486666 = fma(r486663, r486664, r486665);
        double r486667 = r486655 - r486662;
        double r486668 = cbrt(r486667);
        double r486669 = r486668 * r486668;
        double r486670 = a;
        double r486671 = r486670 - r486655;
        double r486672 = cbrt(r486671);
        double r486673 = r486672 * r486672;
        double r486674 = cbrt(r486664);
        double r486675 = r486674 * r486674;
        double r486676 = r486673 / r486675;
        double r486677 = r486669 / r486676;
        double r486678 = r486672 / r486674;
        double r486679 = r486668 / r486678;
        double r486680 = r486665 + r486664;
        double r486681 = fma(r486677, r486679, r486680);
        double r486682 = r486661 ? r486666 : r486681;
        return r486682;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.6
Target8.5
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.2092524272160993e+153 or 8.903535236686547e+157 < t

    1. Initial program 33.3

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

    2. Simplified23.7

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

    4. Applied clear-num23.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
    5. Using strategy rm

    6. Applied fma-udef23.9

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

    7. Simplified23.9

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

    8. Taylor expanded around inf 17.0

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

    9. Simplified11.5

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

    if -1.2092524272160993e+153 < t < 8.903535236686547e+157

    1. Initial program 10.3

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

    2. Simplified7.6

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

    4. Applied clear-num7.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
    5. Using strategy rm

    6. Applied fma-udef7.7

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

    7. Simplified7.5

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

    9. Applied add-cube-cbrt7.7

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

    10. Applied add-cube-cbrt7.7

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

    11. Applied times-frac7.7

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

    12. Applied add-cube-cbrt7.7

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

    13. Applied times-frac6.4

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

    14. Applied fma-def6.4

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.20925242721609934 \cdot 10^{153} \lor \neg \left(t \le 8.90353523668654675 \cdot 10^{157}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))))