Average Error: 5.5 → 0.6
Time: 19.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.1358477187951435 \cdot 10^{+146}:\\ \;\;\;\;x + \left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right)\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 4.602631262379337 \cdot 10^{+178}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\left(z - t\right) \cdot y \le -1.1358477187951435 \cdot 10^{+146}:\\
\;\;\;\;x + \left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right)\\

\mathbf{elif}\;\left(z - t\right) \cdot y \le 4.602631262379337 \cdot 10^{+178}:\\
\;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r15461788 = x;
        double r15461789 = y;
        double r15461790 = z;
        double r15461791 = t;
        double r15461792 = r15461790 - r15461791;
        double r15461793 = r15461789 * r15461792;
        double r15461794 = a;
        double r15461795 = r15461793 / r15461794;
        double r15461796 = r15461788 + r15461795;
        return r15461796;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r15461797 = z;
        double r15461798 = t;
        double r15461799 = r15461797 - r15461798;
        double r15461800 = y;
        double r15461801 = r15461799 * r15461800;
        double r15461802 = -1.1358477187951435e+146;
        bool r15461803 = r15461801 <= r15461802;
        double r15461804 = x;
        double r15461805 = a;
        double r15461806 = r15461805 / r15461800;
        double r15461807 = r15461797 / r15461806;
        double r15461808 = r15461798 / r15461806;
        double r15461809 = r15461807 - r15461808;
        double r15461810 = r15461804 + r15461809;
        double r15461811 = 4.602631262379337e+178;
        bool r15461812 = r15461801 <= r15461811;
        double r15461813 = r15461801 / r15461805;
        double r15461814 = r15461804 + r15461813;
        double r15461815 = r15461800 / r15461805;
        double r15461816 = r15461799 * r15461815;
        double r15461817 = r15461804 + r15461816;
        double r15461818 = r15461812 ? r15461814 : r15461817;
        double r15461819 = r15461803 ? r15461810 : r15461818;
        return r15461819;
}

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

Original5.5
Target0.7
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -1.1358477187951435e+146

    1. Initial program 19.1

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

    2. Taylor expanded around 0 19.2

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

    3. Simplified1.2

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

    5. Applied div-sub1.2

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

    if -1.1358477187951435e+146 < (* y (- z t)) < 4.602631262379337e+178

    1. Initial program 0.5

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

    if 4.602631262379337e+178 < (* y (- z t))

    1. Initial program 23.3

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

    2. Taylor expanded around 0 23.3

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

    3. Simplified0.9

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

    5. Applied div-inv1.0

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

    6. Simplified0.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.1358477187951435 \cdot 10^{+146}:\\ \;\;\;\;x + \left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right)\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 4.602631262379337 \cdot 10^{+178}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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