Average Error: 16.3 → 8.7
Time: 23.3s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.76278196464351678 \cdot 10^{56} \lor \neg \left(t \le 3.70931520882170438 \cdot 10^{143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}, \left(\frac{\sqrt[3]{t - z}}{\sqrt[3]{a - t}} \cdot y\right) \cdot \frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -4.76278196464351678 \cdot 10^{56} \lor \neg \left(t \le 3.70931520882170438 \cdot 10^{143}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}, \left(\frac{\sqrt[3]{t - z}}{\sqrt[3]{a - t}} \cdot y\right) \cdot \frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, x + y\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r445389 = x;
        double r445390 = y;
        double r445391 = r445389 + r445390;
        double r445392 = z;
        double r445393 = t;
        double r445394 = r445392 - r445393;
        double r445395 = r445394 * r445390;
        double r445396 = a;
        double r445397 = r445396 - r445393;
        double r445398 = r445395 / r445397;
        double r445399 = r445391 - r445398;
        return r445399;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r445400 = t;
        double r445401 = -4.762781964643517e+56;
        bool r445402 = r445400 <= r445401;
        double r445403 = 3.7093152088217044e+143;
        bool r445404 = r445400 <= r445403;
        double r445405 = !r445404;
        bool r445406 = r445402 || r445405;
        double r445407 = z;
        double r445408 = r445407 / r445400;
        double r445409 = y;
        double r445410 = x;
        double r445411 = fma(r445408, r445409, r445410);
        double r445412 = r445400 - r445407;
        double r445413 = cbrt(r445412);
        double r445414 = r445413 * r445413;
        double r445415 = a;
        double r445416 = r445415 - r445400;
        double r445417 = cbrt(r445416);
        double r445418 = r445413 / r445417;
        double r445419 = r445418 * r445409;
        double r445420 = 1.0;
        double r445421 = r445417 * r445417;
        double r445422 = r445420 / r445421;
        double r445423 = r445419 * r445422;
        double r445424 = r445410 + r445409;
        double r445425 = fma(r445414, r445423, r445424);
        double r445426 = r445406 ? r445411 : r445425;
        return r445426;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

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

Derivation

  1. Split input into 2 regimes

  2. if t < -4.762781964643517e+56 or 3.7093152088217044e+143 < t

    1. Initial program 29.5

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

    2. Simplified19.3

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

    3. Taylor expanded around inf 17.6

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

    4. Simplified12.5

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

    if -4.762781964643517e+56 < t < 3.7093152088217044e+143

    1. Initial program 8.7

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

    2. Simplified6.9

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

    4. Applied fma-udef6.9

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

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

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

    7. Applied add-cube-cbrt7.1

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

    8. Applied times-frac7.1

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

    9. Applied associate-*l*6.5

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

    11. Applied add-cube-cbrt6.6

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

    12. Applied *-un-lft-identity6.6

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

    13. Applied cbrt-prod6.6

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

    14. Applied times-frac6.6

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

    15. Applied associate-*l*6.5

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

    17. Applied fma-def6.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}{1}, \frac{\sqrt[3]{1}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{t - z}}{\sqrt[3]{a - t}} \cdot y\right), x + y\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.76278196464351678 \cdot 10^{56} \lor \neg \left(t \le 3.70931520882170438 \cdot 10^{143}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}, \left(\frac{\sqrt[3]{t - z}}{\sqrt[3]{a - t}} \cdot y\right) \cdot \frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +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) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))