Average Error: 16.5 → 8.3
Time: 14.9s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.22266 \cdot 10^{-261}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{3}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.246342234530279 \cdot 10^{-220}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\left(\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.22266 \cdot 10^{-261}:\\
\;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{3}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\

\mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.246342234530279 \cdot 10^{-220}:\\
\;\;\;\;\frac{z \cdot y}{t} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r743863 = x;
        double r743864 = y;
        double r743865 = r743863 + r743864;
        double r743866 = z;
        double r743867 = t;
        double r743868 = r743866 - r743867;
        double r743869 = r743868 * r743864;
        double r743870 = a;
        double r743871 = r743870 - r743867;
        double r743872 = r743869 / r743871;
        double r743873 = r743865 - r743872;
        return r743873;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r743874 = x;
        double r743875 = y;
        double r743876 = r743874 + r743875;
        double r743877 = z;
        double r743878 = t;
        double r743879 = r743877 - r743878;
        double r743880 = r743879 * r743875;
        double r743881 = a;
        double r743882 = r743881 - r743878;
        double r743883 = r743880 / r743882;
        double r743884 = r743876 - r743883;
        double r743885 = -1.222656639302079e-261;
        bool r743886 = r743884 <= r743885;
        double r743887 = cbrt(r743882);
        double r743888 = r743887 * r743887;
        double r743889 = r743879 / r743888;
        double r743890 = cbrt(r743875);
        double r743891 = r743890 * r743890;
        double r743892 = cbrt(r743888);
        double r743893 = r743891 / r743892;
        double r743894 = r743889 * r743893;
        double r743895 = 3.0;
        double r743896 = pow(r743890, r743895);
        double r743897 = cbrt(r743896);
        double r743898 = cbrt(r743887);
        double r743899 = r743897 / r743898;
        double r743900 = r743894 * r743899;
        double r743901 = r743876 - r743900;
        double r743902 = 2.2463422345302793e-220;
        bool r743903 = r743884 <= r743902;
        double r743904 = r743877 * r743875;
        double r743905 = r743904 / r743878;
        double r743906 = r743905 + r743874;
        double r743907 = r743898 * r743898;
        double r743908 = r743887 * r743907;
        double r743909 = r743908 * r743898;
        double r743910 = r743879 / r743909;
        double r743911 = r743875 / r743887;
        double r743912 = r743910 * r743911;
        double r743913 = r743876 - r743912;
        double r743914 = r743903 ? r743906 : r743913;
        double r743915 = r743886 ? r743901 : r743914;
        return r743915;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.5
Target8.5
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -1.222656639302079e-261

    1. Initial program 12.9

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

    3. Applied add-cube-cbrt13.1

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

    4. Applied times-frac7.5

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

    6. Applied add-cube-cbrt7.6

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

    7. Applied cbrt-prod7.6

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

    8. Applied add-cube-cbrt7.6

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

    9. Applied times-frac7.6

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

    10. Applied associate-*r*7.3

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

    12. Applied add-cbrt-cube7.3

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

    13. Simplified7.3

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

    if -1.222656639302079e-261 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 2.2463422345302793e-220

    1. Initial program 54.5

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

    2. Taylor expanded around inf 16.9

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

    if 2.2463422345302793e-220 < (- (+ x y) (/ (* (- z t) y) (- a t)))

    1. Initial program 12.9

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

    3. Applied add-cube-cbrt13.1

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

    4. Applied times-frac7.5

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

    6. Applied add-cube-cbrt7.6

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

    7. Applied associate-*r*7.6

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.22266 \cdot 10^{-261}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{3}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.246342234530279 \cdot 10^{-220}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\left(\sqrt[3]{a - t} \cdot \left(\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))