Average Error: 6.2 → 0.5
Time: 9.5s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.286418287653091 \cdot 10^{194}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a}, y, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) = -\infty:\\
\;\;\;\;x - y \cdot \frac{z - t}{a}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 7.286418287653091 \cdot 10^{194}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a}, y, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r373614 = x;
        double r373615 = y;
        double r373616 = z;
        double r373617 = t;
        double r373618 = r373616 - r373617;
        double r373619 = r373615 * r373618;
        double r373620 = a;
        double r373621 = r373619 / r373620;
        double r373622 = r373614 - r373621;
        return r373622;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r373623 = y;
        double r373624 = z;
        double r373625 = t;
        double r373626 = r373624 - r373625;
        double r373627 = r373623 * r373626;
        double r373628 = -inf.0;
        bool r373629 = r373627 <= r373628;
        double r373630 = x;
        double r373631 = a;
        double r373632 = r373626 / r373631;
        double r373633 = r373623 * r373632;
        double r373634 = r373630 - r373633;
        double r373635 = 7.286418287653091e+194;
        bool r373636 = r373627 <= r373635;
        double r373637 = r373627 / r373631;
        double r373638 = r373630 - r373637;
        double r373639 = r373625 - r373624;
        double r373640 = 1.0;
        double r373641 = r373640 / r373631;
        double r373642 = r373639 * r373641;
        double r373643 = fma(r373642, r373623, r373630);
        double r373644 = r373636 ? r373638 : r373643;
        double r373645 = r373629 ? r373634 : r373644;
        return r373645;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -inf.0

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity64.0

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

    4. Applied times-frac0.2

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

    5. Simplified0.2

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

    if -inf.0 < (* y (- z t)) < 7.286418287653091e+194

    1. Initial program 0.4

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

    if 7.286418287653091e+194 < (* y (- z t))

    1. Initial program 27.7

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

    2. Simplified1.4

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

    4. Applied div-inv1.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.286418287653091 \cdot 10^{194}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a}, y, x\right)\\ \end{array}\]

Reproduce

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

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