Average Error: 16.2 → 8.5
Time: 5.5s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -14925789107353.796875 \lor \neg \left(t \le 3.005206846125476302796241908980101272797 \cdot 10^{191}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} + \left(x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -14925789107353.796875 \lor \neg \left(t \le 3.005206846125476302796241908980101272797 \cdot 10^{191}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r464383 = x;
        double r464384 = y;
        double r464385 = r464383 + r464384;
        double r464386 = z;
        double r464387 = t;
        double r464388 = r464386 - r464387;
        double r464389 = r464388 * r464384;
        double r464390 = a;
        double r464391 = r464390 - r464387;
        double r464392 = r464389 / r464391;
        double r464393 = r464385 - r464392;
        return r464393;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r464394 = t;
        double r464395 = -14925789107353.797;
        bool r464396 = r464394 <= r464395;
        double r464397 = 3.005206846125476e+191;
        bool r464398 = r464394 <= r464397;
        double r464399 = !r464398;
        bool r464400 = r464396 || r464399;
        double r464401 = z;
        double r464402 = r464401 / r464394;
        double r464403 = y;
        double r464404 = x;
        double r464405 = fma(r464402, r464403, r464404);
        double r464406 = r464394 - r464401;
        double r464407 = cbrt(r464403);
        double r464408 = r464407 * r464407;
        double r464409 = a;
        double r464410 = r464409 - r464394;
        double r464411 = cbrt(r464410);
        double r464412 = r464411 * r464411;
        double r464413 = r464408 / r464412;
        double r464414 = r464406 * r464413;
        double r464415 = r464407 / r464411;
        double r464416 = r464414 * r464415;
        double r464417 = r464404 + r464403;
        double r464418 = r464416 + r464417;
        double r464419 = r464400 ? r464405 : r464418;
        return r464419;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.2
Target8.0
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -14925789107353.797 or 3.005206846125476e+191 < t

    1. Initial program 27.7

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

    2. Simplified19.7

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

    4. Applied div-inv19.7

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

    6. Applied fma-udef19.8

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

    7. Simplified19.7

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

    8. Taylor expanded around inf 18.1

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

    9. Simplified13.4

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

    if -14925789107353.797 < t < 3.005206846125476e+191

    1. Initial program 9.8

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

    2. Simplified6.9

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

    4. Applied div-inv7.0

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

    6. Applied fma-udef7.0

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

    7. Simplified7.0

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

    9. Applied add-cube-cbrt7.2

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

    10. Applied add-cube-cbrt7.2

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

    11. Applied times-frac7.2

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

    12. Applied associate-*r*5.8

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -14925789107353.796875 \lor \neg \left(t \le 3.005206846125476302796241908980101272797 \cdot 10^{191}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} + \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))))