Average Error: 6.1 → 1.1
Time: 21.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -3.273801318257331161706233360861922033076 \cdot 10^{273}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le -8.544258254814858046435073701156046376542 \cdot 10^{-245}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\sqrt[3]{a}} \cdot \left(\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}}{\sqrt[3]{a}}\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\left(z - t\right) \cdot y \le -3.273801318257331161706233360861922033076 \cdot 10^{273}:\\
\;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;\left(z - t\right) \cdot y \le -8.544258254814858046435073701156046376542 \cdot 10^{-245}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13625901 = x;
        double r13625902 = y;
        double r13625903 = z;
        double r13625904 = t;
        double r13625905 = r13625903 - r13625904;
        double r13625906 = r13625902 * r13625905;
        double r13625907 = a;
        double r13625908 = r13625906 / r13625907;
        double r13625909 = r13625901 + r13625908;
        return r13625909;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13625910 = z;
        double r13625911 = t;
        double r13625912 = r13625910 - r13625911;
        double r13625913 = y;
        double r13625914 = r13625912 * r13625913;
        double r13625915 = -3.273801318257331e+273;
        bool r13625916 = r13625914 <= r13625915;
        double r13625917 = x;
        double r13625918 = a;
        double r13625919 = r13625918 / r13625912;
        double r13625920 = r13625913 / r13625919;
        double r13625921 = r13625917 + r13625920;
        double r13625922 = -8.544258254814858e-245;
        bool r13625923 = r13625914 <= r13625922;
        double r13625924 = r13625914 / r13625918;
        double r13625925 = r13625924 + r13625917;
        double r13625926 = 1.0;
        double r13625927 = cbrt(r13625918);
        double r13625928 = r13625926 / r13625927;
        double r13625929 = cbrt(r13625913);
        double r13625930 = r13625929 / r13625927;
        double r13625931 = r13625912 * r13625930;
        double r13625932 = r13625929 * r13625929;
        double r13625933 = r13625929 * r13625932;
        double r13625934 = cbrt(r13625933);
        double r13625935 = r13625929 * r13625934;
        double r13625936 = r13625935 / r13625927;
        double r13625937 = r13625931 * r13625936;
        double r13625938 = r13625928 * r13625937;
        double r13625939 = r13625917 + r13625938;
        double r13625940 = r13625923 ? r13625925 : r13625939;
        double r13625941 = r13625916 ? r13625921 : r13625940;
        return r13625941;
}

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

Original6.1
Target0.6
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -3.273801318257331e+273

    1. Initial program 48.2

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

    3. Applied associate-/l*0.2

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

    if -3.273801318257331e+273 < (* y (- z t)) < -8.544258254814858e-245

    1. Initial program 0.1

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

    if -8.544258254814858e-245 < (* y (- z t))

    1. Initial program 5.8

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

    3. Applied add-cube-cbrt6.3

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

    4. Applied times-frac3.0

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

    6. Applied div-inv3.0

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

    7. Applied associate-*r*2.7

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

    9. Applied add-cube-cbrt2.9

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

    10. Applied times-frac2.9

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

    11. Applied associate-*l*1.9

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

    13. Applied add-cbrt-cube1.9

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -3.273801318257331161706233360861922033076 \cdot 10^{273}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le -8.544258254814858046435073701156046376542 \cdot 10^{-245}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\sqrt[3]{a}} \cdot \left(\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}}{\sqrt[3]{a}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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