Average Error: 16.3 → 7.6
Time: 6.5s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.89442403115317649 \cdot 10^{171} \lor \neg \left(t \le 2.00789644304095614 \cdot 10^{115}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.89442403115317649 \cdot 10^{171} \lor \neg \left(t \le 2.00789644304095614 \cdot 10^{115}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{a - t}}{\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 r583862 = x;
        double r583863 = y;
        double r583864 = r583862 + r583863;
        double r583865 = z;
        double r583866 = t;
        double r583867 = r583865 - r583866;
        double r583868 = r583867 * r583863;
        double r583869 = a;
        double r583870 = r583869 - r583866;
        double r583871 = r583868 / r583870;
        double r583872 = r583864 - r583871;
        return r583872;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r583873 = t;
        double r583874 = -1.8944240311531765e+171;
        bool r583875 = r583873 <= r583874;
        double r583876 = 2.007896443040956e+115;
        bool r583877 = r583873 <= r583876;
        double r583878 = !r583877;
        bool r583879 = r583875 || r583878;
        double r583880 = z;
        double r583881 = r583880 / r583873;
        double r583882 = y;
        double r583883 = x;
        double r583884 = fma(r583881, r583882, r583883);
        double r583885 = r583873 - r583880;
        double r583886 = cbrt(r583885);
        double r583887 = r583886 * r583886;
        double r583888 = a;
        double r583889 = r583888 - r583873;
        double r583890 = cbrt(r583889);
        double r583891 = cbrt(r583882);
        double r583892 = r583890 / r583891;
        double r583893 = r583887 / r583892;
        double r583894 = r583893 / r583892;
        double r583895 = r583886 / r583892;
        double r583896 = r583883 + r583882;
        double r583897 = fma(r583894, r583895, r583896);
        double r583898 = r583879 ? r583884 : r583897;
        return r583898;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.3
Target8.3
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.8944240311531765e+171 or 2.007896443040956e+115 < t

    1. Initial program 32.2

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

    2. Simplified23.2

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

    4. Applied clear-num23.2

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

    6. Applied fma-udef23.2

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

    7. Simplified23.3

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

    8. Taylor expanded around inf 17.2

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

    9. Simplified11.6

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

    if -1.8944240311531765e+171 < t < 2.007896443040956e+115

    1. Initial program 9.9

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

    2. Simplified7.5

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

    4. Applied clear-num7.6

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

    6. Applied fma-udef7.6

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

    7. Simplified7.4

      \[\leadsto \color{blue}{\frac{t - z}{\frac{a - t}{y}}} + \left(x + y\right)\]
    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-frac6.1

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

    14. Applied fma-def6.1

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

    16. Applied times-frac6.1

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

    17. Applied associate-/r*6.0

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.89442403115317649 \cdot 10^{171} \lor \neg \left(t \le 2.00789644304095614 \cdot 10^{115}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, \frac{\sqrt[3]{t - z}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))