Average Error: 16.6 → 10.6
Time: 7.2s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.50992464194075363 \cdot 10^{216}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(t - z\right), x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.50992464194075363 \cdot 10^{216}:\\
\;\;\;\;x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r575231 = x;
        double r575232 = y;
        double r575233 = r575231 + r575232;
        double r575234 = z;
        double r575235 = t;
        double r575236 = r575234 - r575235;
        double r575237 = r575236 * r575232;
        double r575238 = a;
        double r575239 = r575238 - r575235;
        double r575240 = r575237 / r575239;
        double r575241 = r575233 - r575240;
        return r575241;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r575242 = t;
        double r575243 = -1.5099246419407536e+216;
        bool r575244 = r575242 <= r575243;
        double r575245 = x;
        double r575246 = y;
        double r575247 = cbrt(r575246);
        double r575248 = r575247 * r575247;
        double r575249 = cbrt(r575248);
        double r575250 = cbrt(r575247);
        double r575251 = r575249 * r575250;
        double r575252 = r575251 * r575247;
        double r575253 = a;
        double r575254 = r575253 - r575242;
        double r575255 = cbrt(r575254);
        double r575256 = r575255 * r575255;
        double r575257 = r575252 / r575256;
        double r575258 = r575247 / r575255;
        double r575259 = z;
        double r575260 = r575242 - r575259;
        double r575261 = r575258 * r575260;
        double r575262 = r575245 + r575246;
        double r575263 = fma(r575257, r575261, r575262);
        double r575264 = r575244 ? r575245 : r575263;
        return r575264;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.6
Target8.2
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.5099246419407536e+216

    1. Initial program 36.5

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

    2. Simplified26.6

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

    3. Taylor expanded around 0 20.5

      \[\leadsto \color{blue}{x}\]

    if -1.5099246419407536e+216 < t

    1. Initial program 14.9

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

    2. Simplified10.6

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

    4. Applied fma-udef10.7

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

    6. Applied add-cube-cbrt10.9

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

    7. Applied add-cube-cbrt10.9

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

    8. Applied times-frac10.9

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

    9. Applied associate-*l*9.7

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

    11. Applied fma-def9.7

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

    13. Applied add-cube-cbrt9.7

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

    14. Applied cbrt-prod9.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.50992464194075363 \cdot 10^{216}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(t - z\right), x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))