Average Error: 6.1 → 1.6
Time: 8.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z - t \le -8.381032267864286848111598473235001808927 \cdot 10^{-13} \lor \neg \left(z - t \le 2.979324857395300632013474840610844551682 \cdot 10^{-117}\right):\\ \;\;\;\;x - \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\sqrt[3]{y}}{a} \cdot \left(z - t\right)\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\\ \end{array}\]
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(\frac{\sqrt[3]{y}}{a} \cdot \left(z - t\right)\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r250419 = x;
        double r250420 = y;
        double r250421 = z;
        double r250422 = t;
        double r250423 = r250421 - r250422;
        double r250424 = r250420 * r250423;
        double r250425 = a;
        double r250426 = r250424 / r250425;
        double r250427 = r250419 - r250426;
        return r250427;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r250428 = z;
        double r250429 = t;
        double r250430 = r250428 - r250429;
        double r250431 = -8.381032267864287e-13;
        bool r250432 = r250430 <= r250431;
        double r250433 = 2.9793248573953006e-117;
        bool r250434 = r250430 <= r250433;
        double r250435 = !r250434;
        bool r250436 = r250432 || r250435;
        double r250437 = x;
        double r250438 = y;
        double r250439 = a;
        double r250440 = r250438 / r250439;
        double r250441 = r250440 * r250430;
        double r250442 = r250437 - r250441;
        double r250443 = cbrt(r250438);
        double r250444 = r250443 / r250439;
        double r250445 = r250444 * r250430;
        double r250446 = r250443 * r250443;
        double r250447 = r250445 * r250446;
        double r250448 = r250437 - r250447;
        double r250449 = r250436 ? r250442 : r250448;
        return r250449;
}

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 associate-/r/5.6

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

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

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

    8. Applied add-cube-cbrt5.8

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

    9. Applied times-frac5.8

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

    10. Applied associate-*l*1.6

      \[\leadsto x - \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \left(z - t\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

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

Reproduce

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