Average Error: 6.2 → 2.0
Time: 13.3s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.89067874723543516820715729027922883886 \cdot 10^{-54}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;y \le 4.721761730699554045903946521897173928976 \cdot 10^{-150}:\\ \;\;\;\;x + \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t - z}{\frac{\sqrt[3]{a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{t}{\frac{a}{y}}\right) - \frac{y}{\frac{a}{z}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.89067874723543516820715729027922883886 \cdot 10^{-54}:\\
\;\;\;\;x + y \cdot \frac{t - z}{a}\\

\mathbf{elif}\;y \le 4.721761730699554045903946521897173928976 \cdot 10^{-150}:\\
\;\;\;\;x + \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t - z}{\frac{\sqrt[3]{a}}{y}}\\

\mathbf{else}:\\
\;\;\;\;\left(x + \frac{t}{\frac{a}{y}}\right) - \frac{y}{\frac{a}{z}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r253852 = x;
        double r253853 = y;
        double r253854 = z;
        double r253855 = t;
        double r253856 = r253854 - r253855;
        double r253857 = r253853 * r253856;
        double r253858 = a;
        double r253859 = r253857 / r253858;
        double r253860 = r253852 - r253859;
        return r253860;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r253861 = y;
        double r253862 = -3.890678747235435e-54;
        bool r253863 = r253861 <= r253862;
        double r253864 = x;
        double r253865 = t;
        double r253866 = z;
        double r253867 = r253865 - r253866;
        double r253868 = a;
        double r253869 = r253867 / r253868;
        double r253870 = r253861 * r253869;
        double r253871 = r253864 + r253870;
        double r253872 = 4.721761730699554e-150;
        bool r253873 = r253861 <= r253872;
        double r253874 = 1.0;
        double r253875 = cbrt(r253868);
        double r253876 = r253875 * r253875;
        double r253877 = r253874 / r253876;
        double r253878 = r253875 / r253861;
        double r253879 = r253867 / r253878;
        double r253880 = r253877 * r253879;
        double r253881 = r253864 + r253880;
        double r253882 = r253868 / r253861;
        double r253883 = r253865 / r253882;
        double r253884 = r253864 + r253883;
        double r253885 = r253868 / r253866;
        double r253886 = r253861 / r253885;
        double r253887 = r253884 - r253886;
        double r253888 = r253873 ? r253881 : r253887;
        double r253889 = r253863 ? r253871 : r253888;
        return r253889;
}

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

Original6.2
Target0.8
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -3.890678747235435e-54

    1. Initial program 12.7

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

    2. Simplified3.0

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

    4. Applied div-inv3.1

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

    5. Applied associate-*l*1.4

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

    6. Simplified1.3

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

    if -3.890678747235435e-54 < y < 4.721761730699554e-150

    1. Initial program 0.5

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

    2. Simplified2.1

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

    4. Applied add-cube-cbrt2.5

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

    5. Applied *-un-lft-identity2.5

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

    6. Applied times-frac2.5

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

    7. Applied associate-*l*1.0

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

    8. Simplified1.1

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

    if 4.721761730699554e-150 < y

    1. Initial program 9.1

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

    2. Simplified2.5

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

    3. Taylor expanded around 0 9.1

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

    4. Simplified3.6

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.89067874723543516820715729027922883886 \cdot 10^{-54}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;y \le 4.721761730699554045903946521897173928976 \cdot 10^{-150}:\\ \;\;\;\;x + \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t - z}{\frac{\sqrt[3]{a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{t}{\frac{a}{y}}\right) - \frac{y}{\frac{a}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :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)))