Average Error: 16.5 → 8.1
Time: 22.9s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.4785650511945879 \cdot 10^{119} \lor \neg \left(t \le 5.62287539718969527 \cdot 10^{93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{t - z}{\sqrt[3]{a - t}}\right) + \left(x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.4785650511945879 \cdot 10^{119} \lor \neg \left(t \le 5.62287539718969527 \cdot 10^{93}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r733366 = x;
        double r733367 = y;
        double r733368 = r733366 + r733367;
        double r733369 = z;
        double r733370 = t;
        double r733371 = r733369 - r733370;
        double r733372 = r733371 * r733367;
        double r733373 = a;
        double r733374 = r733373 - r733370;
        double r733375 = r733372 / r733374;
        double r733376 = r733368 - r733375;
        return r733376;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r733377 = t;
        double r733378 = -1.4785650511945879e+119;
        bool r733379 = r733377 <= r733378;
        double r733380 = 5.622875397189695e+93;
        bool r733381 = r733377 <= r733380;
        double r733382 = !r733381;
        bool r733383 = r733379 || r733382;
        double r733384 = z;
        double r733385 = r733384 / r733377;
        double r733386 = y;
        double r733387 = x;
        double r733388 = fma(r733385, r733386, r733387);
        double r733389 = cbrt(r733386);
        double r733390 = r733389 * r733389;
        double r733391 = a;
        double r733392 = r733391 - r733377;
        double r733393 = cbrt(r733392);
        double r733394 = r733390 / r733393;
        double r733395 = r733389 / r733393;
        double r733396 = r733377 - r733384;
        double r733397 = r733396 / r733393;
        double r733398 = r733395 * r733397;
        double r733399 = r733394 * r733398;
        double r733400 = r733387 + r733386;
        double r733401 = r733399 + r733400;
        double r733402 = r733383 ? r733388 : r733401;
        return r733402;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.5
Target8.6
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.4785650511945879e+119 or 5.622875397189695e+93 < t

    1. Initial program 30.3

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

    2. Simplified20.9

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

    3. Taylor expanded around inf 17.5

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

    4. Simplified12.3

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

    if -1.4785650511945879e+119 < t < 5.622875397189695e+93

    1. Initial program 8.8

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

    2. Simplified6.7

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

    4. Applied add-cube-cbrt6.8

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

    5. Applied associate-/l*6.8

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

    7. Applied fma-udef6.8

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

    8. Simplified6.7

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

    10. Applied add-cube-cbrt6.9

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

    11. Applied *-un-lft-identity6.9

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

    12. Applied times-frac6.9

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

    13. Applied associate-*r*6.7

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

    14. Simplified6.7

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

    16. Applied add-cube-cbrt6.7

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

    17. Applied times-frac6.7

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

    18. Applied associate-*l*5.7

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.4785650511945879 \cdot 10^{119} \lor \neg \left(t \le 5.62287539718969527 \cdot 10^{93}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{t - z}{\sqrt[3]{a - t}}\right) + \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))