Average Error: 16.5 → 6.8
Time: 19.0s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.4270769705098634 \cdot 10^{-178} \lor \neg \left(a \le 4.1955295958532206 \cdot 10^{-120}\right):\\ \;\;\;\;x + \left(\sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.4270769705098634 \cdot 10^{-178} \lor \neg \left(a \le 4.1955295958532206 \cdot 10^{-120}\right):\\
\;\;\;\;x + \left(\sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r687866 = x;
        double r687867 = y;
        double r687868 = r687866 + r687867;
        double r687869 = z;
        double r687870 = t;
        double r687871 = r687869 - r687870;
        double r687872 = r687871 * r687867;
        double r687873 = a;
        double r687874 = r687873 - r687870;
        double r687875 = r687872 / r687874;
        double r687876 = r687868 - r687875;
        return r687876;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r687877 = a;
        double r687878 = -2.4270769705098634e-178;
        bool r687879 = r687877 <= r687878;
        double r687880 = 4.1955295958532206e-120;
        bool r687881 = r687877 <= r687880;
        double r687882 = !r687881;
        bool r687883 = r687879 || r687882;
        double r687884 = x;
        double r687885 = t;
        double r687886 = z;
        double r687887 = r687885 - r687886;
        double r687888 = r687877 - r687885;
        double r687889 = r687887 / r687888;
        double r687890 = y;
        double r687891 = fma(r687889, r687890, r687890);
        double r687892 = cbrt(r687891);
        double r687893 = r687892 * r687892;
        double r687894 = r687893 * r687892;
        double r687895 = r687884 + r687894;
        double r687896 = r687886 / r687885;
        double r687897 = fma(r687896, r687890, r687884);
        double r687898 = r687883 ? r687895 : r687897;
        return r687898;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.5
Target8.5
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.4270769705098634e-178 or 4.1955295958532206e-120 < a

    1. Initial program 15.1

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

    2. Simplified5.8

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

    4. Applied add-cube-cbrt6.2

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

    if -2.4270769705098634e-178 < a < 4.1955295958532206e-120

    1. Initial program 21.0

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

    2. Simplified11.3

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

    3. Taylor expanded around inf 9.6

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

    4. Simplified8.4

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.4270769705098634 \cdot 10^{-178} \lor \neg \left(a \le 4.1955295958532206 \cdot 10^{-120}\right):\\ \;\;\;\;x + \left(\sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{t - z}{a - t}, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))