Average Error: 16.0 → 9.3
Time: 24.9s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -5.277016346357952993745277225912104768213 \cdot 10^{-4} \lor \neg \left(a \le 4.889529153193783306974873529274732222518 \cdot 10^{-116}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -5.277016346357952993745277225912104768213 \cdot 10^{-4} \lor \neg \left(a \le 4.889529153193783306974873529274732222518 \cdot 10^{-116}\right):\\
\;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}, x + y\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 r360458 = x;
        double r360459 = y;
        double r360460 = r360458 + r360459;
        double r360461 = z;
        double r360462 = t;
        double r360463 = r360461 - r360462;
        double r360464 = r360463 * r360459;
        double r360465 = a;
        double r360466 = r360465 - r360462;
        double r360467 = r360464 / r360466;
        double r360468 = r360460 - r360467;
        return r360468;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r360469 = a;
        double r360470 = -0.0005277016346357953;
        bool r360471 = r360469 <= r360470;
        double r360472 = 4.8895291531937833e-116;
        bool r360473 = r360469 <= r360472;
        double r360474 = !r360473;
        bool r360475 = r360471 || r360474;
        double r360476 = t;
        double r360477 = z;
        double r360478 = r360476 - r360477;
        double r360479 = y;
        double r360480 = cbrt(r360479);
        double r360481 = r360480 * r360480;
        double r360482 = r360469 - r360476;
        double r360483 = cbrt(r360482);
        double r360484 = r360483 * r360483;
        double r360485 = r360481 / r360484;
        double r360486 = r360478 * r360485;
        double r360487 = r360480 / r360483;
        double r360488 = x;
        double r360489 = r360488 + r360479;
        double r360490 = fma(r360486, r360487, r360489);
        double r360491 = r360477 / r360476;
        double r360492 = fma(r360491, r360479, r360488);
        double r360493 = r360475 ? r360490 : r360492;
        return r360493;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.0
Target8.1
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -0.0005277016346357953 or 4.8895291531937833e-116 < a

    1. Initial program 14.3

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

    2. Simplified7.1

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

    4. Applied fma-udef7.1

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

    6. Applied div-inv7.1

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

    7. Applied associate-*l*7.8

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

    8. Simplified7.8

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

    10. Applied add-cube-cbrt7.9

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

    11. Applied add-cube-cbrt7.9

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

    12. Applied times-frac7.9

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

    13. Applied associate-*r*7.2

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

    15. Applied fma-def7.2

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

    if -0.0005277016346357953 < a < 4.8895291531937833e-116

    1. Initial program 18.7

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

    2. Simplified17.8

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

    3. Taylor expanded around inf 14.4

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

    4. Simplified13.0

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.277016346357952993745277225912104768213 \cdot 10^{-4} \lor \neg \left(a \le 4.889529153193783306974873529274732222518 \cdot 10^{-116}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))