Average Error: 6.2 → 2.5
Time: 6.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;z \le 6.9390753974297484 \cdot 10^{-149}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138}:\\
\;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\

\mathbf{elif}\;z \le 6.9390753974297484 \cdot 10^{-149}:\\
\;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r336680 = x;
        double r336681 = y;
        double r336682 = z;
        double r336683 = t;
        double r336684 = r336682 - r336683;
        double r336685 = r336681 * r336684;
        double r336686 = a;
        double r336687 = r336685 / r336686;
        double r336688 = r336680 + r336687;
        return r336688;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r336689 = z;
        double r336690 = -1.8095903151115152e-138;
        bool r336691 = r336689 <= r336690;
        double r336692 = x;
        double r336693 = t;
        double r336694 = r336689 - r336693;
        double r336695 = y;
        double r336696 = a;
        double r336697 = r336695 / r336696;
        double r336698 = r336694 * r336697;
        double r336699 = r336692 + r336698;
        double r336700 = 6.939075397429748e-149;
        bool r336701 = r336689 <= r336700;
        double r336702 = r336696 / r336694;
        double r336703 = r336695 / r336702;
        double r336704 = r336692 + r336703;
        double r336705 = r336696 / r336695;
        double r336706 = r336694 / r336705;
        double r336707 = r336692 + r336706;
        double r336708 = r336701 ? r336704 : r336707;
        double r336709 = r336691 ? r336699 : r336708;
        return r336709;
}

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

Original6.2
Target0.7
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.8095903151115152e-138

    1. Initial program 7.4

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

    3. Applied associate-/l*6.1

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

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

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

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

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

    7. Applied times-frac6.1

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

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

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

    9. Applied times-frac6.1

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

    10. Simplified6.1

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

    11. Simplified2.1

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

    if -1.8095903151115152e-138 < z < 6.939075397429748e-149

    1. Initial program 4.3

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

    3. Applied associate-/l*3.3

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

    if 6.939075397429748e-149 < z

    1. Initial program 6.5

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

    2. Taylor expanded around 0 6.5

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

    3. Simplified2.2

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;z \le 6.9390753974297484 \cdot 10^{-149}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-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)))