Average Error: 15.9 → 7.6
Time: 27.9s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.8666316459197392 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 3.963652201301505 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \left(t - z\right), \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.8666316459197392 \cdot 10^{+129}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;t \le 3.963652201301505 \cdot 10^{+142}:\\
\;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \left(t - z\right), \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}, y + x\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 r23488603 = x;
        double r23488604 = y;
        double r23488605 = r23488603 + r23488604;
        double r23488606 = z;
        double r23488607 = t;
        double r23488608 = r23488606 - r23488607;
        double r23488609 = r23488608 * r23488604;
        double r23488610 = a;
        double r23488611 = r23488610 - r23488607;
        double r23488612 = r23488609 / r23488611;
        double r23488613 = r23488605 - r23488612;
        return r23488613;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r23488614 = t;
        double r23488615 = -1.8666316459197392e+129;
        bool r23488616 = r23488614 <= r23488615;
        double r23488617 = z;
        double r23488618 = r23488617 / r23488614;
        double r23488619 = y;
        double r23488620 = x;
        double r23488621 = fma(r23488618, r23488619, r23488620);
        double r23488622 = 3.963652201301505e+142;
        bool r23488623 = r23488614 <= r23488622;
        double r23488624 = cbrt(r23488619);
        double r23488625 = a;
        double r23488626 = r23488625 - r23488614;
        double r23488627 = cbrt(r23488626);
        double r23488628 = r23488624 / r23488627;
        double r23488629 = r23488628 * r23488628;
        double r23488630 = r23488614 - r23488617;
        double r23488631 = r23488629 * r23488630;
        double r23488632 = r23488627 * r23488627;
        double r23488633 = cbrt(r23488632);
        double r23488634 = cbrt(r23488627);
        double r23488635 = r23488633 * r23488634;
        double r23488636 = r23488624 / r23488635;
        double r23488637 = r23488619 + r23488620;
        double r23488638 = fma(r23488631, r23488636, r23488637);
        double r23488639 = r23488623 ? r23488638 : r23488621;
        double r23488640 = r23488616 ? r23488621 : r23488639;
        return r23488640;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original15.9
Target8.6
Herbie7.6
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.8666316459197392e+129 or 3.963652201301505e+142 < t

    1. Initial program 31.4

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

    2. Simplified23.6

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

    3. Taylor expanded around inf 16.7

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

    4. Simplified11.5

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

    if -1.8666316459197392e+129 < t < 3.963652201301505e+142

    1. Initial program 9.3

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

    2. Simplified7.1

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

    4. Applied fma-udef7.1

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

    6. Applied add-cube-cbrt7.3

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

    7. Applied add-cube-cbrt7.4

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

    8. Applied times-frac7.4

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

    9. Applied associate-*r*5.9

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

    10. Simplified5.9

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

    12. Applied add-cube-cbrt5.9

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

    13. Applied cbrt-prod5.9

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

    15. Applied fma-def5.9

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.8666316459197392 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 3.963652201301505 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \left(t - z\right), \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :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))))