Average Error: 16.8 → 8.3
Time: 22.2s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.272792786955583 \cdot 10^{-212}:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.83086 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(t - z\right), y, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.272792786955583 \cdot 10^{-212}:\\
\;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\

\mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.83086 \cdot 10^{-232}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(t - z\right), y, x + y\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r580828 = x;
        double r580829 = y;
        double r580830 = r580828 + r580829;
        double r580831 = z;
        double r580832 = t;
        double r580833 = r580831 - r580832;
        double r580834 = r580833 * r580829;
        double r580835 = a;
        double r580836 = r580835 - r580832;
        double r580837 = r580834 / r580836;
        double r580838 = r580830 - r580837;
        return r580838;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r580839 = x;
        double r580840 = y;
        double r580841 = r580839 + r580840;
        double r580842 = z;
        double r580843 = t;
        double r580844 = r580842 - r580843;
        double r580845 = r580844 * r580840;
        double r580846 = a;
        double r580847 = r580846 - r580843;
        double r580848 = r580845 / r580847;
        double r580849 = r580841 - r580848;
        double r580850 = -inf.0;
        bool r580851 = r580849 <= r580850;
        double r580852 = r580842 / r580843;
        double r580853 = fma(r580852, r580840, r580839);
        double r580854 = -1.272792786955583e-212;
        bool r580855 = r580849 <= r580854;
        double r580856 = 4.83086157719785e-232;
        bool r580857 = r580849 <= r580856;
        double r580858 = 1.0;
        double r580859 = r580858 / r580847;
        double r580860 = r580843 - r580842;
        double r580861 = r580859 * r580860;
        double r580862 = fma(r580861, r580840, r580841);
        double r580863 = r580857 ? r580853 : r580862;
        double r580864 = r580855 ? r580849 : r580863;
        double r580865 = r580851 ? r580853 : r580864;
        return r580865;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.8
Target8.6
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -inf.0 or -1.272792786955583e-212 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 4.83086157719785e-232

    1. Initial program 58.5

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

    2. Simplified41.6

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

    3. Taylor expanded around inf 29.7

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

    4. Simplified23.9

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

    if -inf.0 < (- (+ x y) (/ (* (- z t) y) (- a t))) < -1.272792786955583e-212

    1. Initial program 1.3

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

    if 4.83086157719785e-232 < (- (+ x y) (/ (* (- z t) y) (- a t)))

    1. Initial program 13.2

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

    2. Simplified7.9

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

    4. Applied div-inv7.9

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

    6. Applied *-un-lft-identity7.9

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

    7. Applied add-cube-cbrt7.9

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

    8. Applied times-frac7.9

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

    9. Simplified7.9

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

    10. Simplified7.9

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.272792786955583 \cdot 10^{-212}:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.83086 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(t - z\right), y, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))