Average Error: 6.0 → 1.4
Time: 13.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2984.114765121556956728454679250717163086:\\ \;\;\;\;x - y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;y \le 1.030424506076636929976550508445270348949 \cdot 10^{-83}:\\ \;\;\;\;\left(x - \frac{t \cdot y}{a}\right) - \left(-\frac{z \cdot y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(t \cdot \frac{y}{a} + \frac{y}{\frac{a}{-z}}\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2984.114765121556956728454679250717163086:\\
\;\;\;\;x - y \cdot \frac{t - z}{a}\\

\mathbf{elif}\;y \le 1.030424506076636929976550508445270348949 \cdot 10^{-83}:\\
\;\;\;\;\left(x - \frac{t \cdot y}{a}\right) - \left(-\frac{z \cdot y}{a}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r200663 = x;
        double r200664 = y;
        double r200665 = z;
        double r200666 = t;
        double r200667 = r200665 - r200666;
        double r200668 = r200664 * r200667;
        double r200669 = a;
        double r200670 = r200668 / r200669;
        double r200671 = r200663 + r200670;
        return r200671;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r200672 = y;
        double r200673 = -2984.114765121557;
        bool r200674 = r200672 <= r200673;
        double r200675 = x;
        double r200676 = t;
        double r200677 = z;
        double r200678 = r200676 - r200677;
        double r200679 = a;
        double r200680 = r200678 / r200679;
        double r200681 = r200672 * r200680;
        double r200682 = r200675 - r200681;
        double r200683 = 1.030424506076637e-83;
        bool r200684 = r200672 <= r200683;
        double r200685 = r200676 * r200672;
        double r200686 = r200685 / r200679;
        double r200687 = r200675 - r200686;
        double r200688 = r200677 * r200672;
        double r200689 = r200688 / r200679;
        double r200690 = -r200689;
        double r200691 = r200687 - r200690;
        double r200692 = r200672 / r200679;
        double r200693 = r200676 * r200692;
        double r200694 = -r200677;
        double r200695 = r200679 / r200694;
        double r200696 = r200672 / r200695;
        double r200697 = r200693 + r200696;
        double r200698 = r200675 - r200697;
        double r200699 = r200684 ? r200691 : r200698;
        double r200700 = r200674 ? r200682 : r200699;
        return r200700;
}

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

Original6.0
Target0.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2984.114765121557

    1. Initial program 15.2

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

    2. Simplified3.4

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

    4. Applied div-inv3.5

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

    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 -2984.114765121557 < y < 1.030424506076637e-83

    1. Initial program 0.6

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

    2. Simplified1.8

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

    4. Applied sub-neg1.8

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

    5. Applied distribute-lft-in1.8

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

    6. Simplified6.1

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

    8. Applied *-un-lft-identity6.1

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

    9. Applied *-un-lft-identity6.1

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

    10. Applied distribute-lft-out--6.1

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

    11. Simplified0.6

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

    if 1.030424506076637e-83 < y

    1. Initial program 10.8

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

    2. Simplified2.8

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

    4. Applied sub-neg2.8

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

    5. Applied distribute-lft-in2.8

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

    6. Simplified3.3

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2984.114765121556956728454679250717163086:\\ \;\;\;\;x - y \cdot \frac{t - z}{a}\\ \mathbf{elif}\;y \le 1.030424506076636929976550508445270348949 \cdot 10^{-83}:\\ \;\;\;\;\left(x - \frac{t \cdot y}{a}\right) - \left(-\frac{z \cdot y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(t \cdot \frac{y}{a} + \frac{y}{\frac{a}{-z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))