Average Error: 6.1 → 1.6
Time: 11.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z - t \le -8.381032267864286848111598473235001808927 \cdot 10^{-13} \lor \neg \left(z - t \le 2.979324857395300632013474840610844551682 \cdot 10^{-117}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \left(z - t\right)\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z - t \le -8.381032267864286848111598473235001808927 \cdot 10^{-13} \lor \neg \left(z - t \le 2.979324857395300632013474840610844551682 \cdot 10^{-117}\right):\\
\;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r242808 = x;
        double r242809 = y;
        double r242810 = z;
        double r242811 = t;
        double r242812 = r242810 - r242811;
        double r242813 = r242809 * r242812;
        double r242814 = a;
        double r242815 = r242813 / r242814;
        double r242816 = r242808 + r242815;
        return r242816;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r242817 = z;
        double r242818 = t;
        double r242819 = r242817 - r242818;
        double r242820 = -8.381032267864287e-13;
        bool r242821 = r242819 <= r242820;
        double r242822 = 2.9793248573953006e-117;
        bool r242823 = r242819 <= r242822;
        double r242824 = !r242823;
        bool r242825 = r242821 || r242824;
        double r242826 = x;
        double r242827 = y;
        double r242828 = a;
        double r242829 = r242827 / r242828;
        double r242830 = r242829 * r242819;
        double r242831 = r242826 + r242830;
        double r242832 = cbrt(r242827);
        double r242833 = r242832 * r242832;
        double r242834 = r242832 / r242828;
        double r242835 = r242834 * r242819;
        double r242836 = r242833 * r242835;
        double r242837 = r242826 + r242836;
        double r242838 = r242825 ? r242831 : r242837;
        return r242838;
}

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
Target0.7
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- z t) < -8.381032267864287e-13 or 2.9793248573953006e-117 < (- z t)

    1. Initial program 7.4

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

    3. Applied associate-/l*7.2

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

    5. Applied associate-/r/1.5

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

    if -8.381032267864287e-13 < (- z t) < 2.9793248573953006e-117

    1. Initial program 1.2

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

    3. Applied associate-/l*1.1

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

    5. Applied div-inv1.1

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

    6. Applied associate-/r*5.6

      \[\leadsto x + \color{blue}{\frac{\frac{y}{a}}{\frac{1}{z - t}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity5.6

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

    9. Applied add-cube-cbrt5.6

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

    10. Applied times-frac5.6

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

    11. Applied *-un-lft-identity5.6

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

    12. Applied add-cube-cbrt5.8

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

    13. Applied times-frac5.8

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

    14. Applied times-frac1.6

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

    15. Simplified1.6

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

    16. Simplified1.6

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \le -8.381032267864286848111598473235001808927 \cdot 10^{-13} \lor \neg \left(z - t \le 2.979324857395300632013474840610844551682 \cdot 10^{-117}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \left(z - t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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