Average Error: 6.3 → 0.8
Time: 14.7s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.506169073250434556576682563271829432785 \cdot 10^{60} \lor \neg \left(y \le 5122316469.390033721923828125\right):\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t \cdot y}{a} - \frac{z \cdot y}{a}\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.506169073250434556576682563271829432785 \cdot 10^{60} \lor \neg \left(y \le 5122316469.390033721923828125\right):\\
\;\;\;\;x + y \cdot \frac{t - z}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r319579 = x;
        double r319580 = y;
        double r319581 = z;
        double r319582 = t;
        double r319583 = r319581 - r319582;
        double r319584 = r319580 * r319583;
        double r319585 = a;
        double r319586 = r319584 / r319585;
        double r319587 = r319579 - r319586;
        return r319587;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r319588 = y;
        double r319589 = -3.5061690732504346e+60;
        bool r319590 = r319588 <= r319589;
        double r319591 = 5122316469.390034;
        bool r319592 = r319588 <= r319591;
        double r319593 = !r319592;
        bool r319594 = r319590 || r319593;
        double r319595 = x;
        double r319596 = t;
        double r319597 = z;
        double r319598 = r319596 - r319597;
        double r319599 = a;
        double r319600 = r319598 / r319599;
        double r319601 = r319588 * r319600;
        double r319602 = r319595 + r319601;
        double r319603 = r319596 * r319588;
        double r319604 = r319603 / r319599;
        double r319605 = r319597 * r319588;
        double r319606 = r319605 / r319599;
        double r319607 = r319604 - r319606;
        double r319608 = r319595 + r319607;
        double r319609 = r319594 ? r319602 : r319608;
        return r319609;
}

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

Original6.3
Target0.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.5061690732504346e+60 or 5122316469.390034 < y

    1. Initial program 18.0

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

    2. Simplified4.5

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

    4. Applied div-inv4.6

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

    5. Applied associate-*l*1.0

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

    6. Simplified0.9

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

    if -3.5061690732504346e+60 < y < 5122316469.390034

    1. Initial program 0.8

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

    2. Simplified1.7

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

    4. Applied add-cube-cbrt2.2

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

    5. Applied add-cube-cbrt2.3

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

    6. Applied times-frac2.3

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

    7. Applied associate-*l*0.8

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

    8. Taylor expanded around 0 0.8

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

    9. Simplified0.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.506169073250434556576682563271829432785 \cdot 10^{60} \lor \neg \left(y \le 5122316469.390033721923828125\right):\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t \cdot y}{a} - \frac{z \cdot y}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))