Average Error: 6.2 → 0.9
Time: 25.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.294102167235718175854547572327975906299 \cdot 10^{56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.218276811061773599006994335448883110359 \cdot 10^{300}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -3.294102167235718175854547572327975906299 \cdot 10^{56}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.218276811061773599006994335448883110359 \cdot 10^{300}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r45078785 = x;
        double r45078786 = y;
        double r45078787 = z;
        double r45078788 = t;
        double r45078789 = r45078787 - r45078788;
        double r45078790 = r45078786 * r45078789;
        double r45078791 = a;
        double r45078792 = r45078790 / r45078791;
        double r45078793 = r45078785 - r45078792;
        return r45078793;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r45078794 = y;
        double r45078795 = z;
        double r45078796 = t;
        double r45078797 = r45078795 - r45078796;
        double r45078798 = r45078794 * r45078797;
        double r45078799 = -3.294102167235718e+56;
        bool r45078800 = r45078798 <= r45078799;
        double r45078801 = a;
        double r45078802 = r45078794 / r45078801;
        double r45078803 = r45078796 - r45078795;
        double r45078804 = x;
        double r45078805 = fma(r45078802, r45078803, r45078804);
        double r45078806 = 2.2182768110617736e+300;
        bool r45078807 = r45078798 <= r45078806;
        double r45078808 = r45078798 / r45078801;
        double r45078809 = r45078804 - r45078808;
        double r45078810 = r45078807 ? r45078809 : r45078805;
        double r45078811 = r45078800 ? r45078805 : r45078810;
        return r45078811;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -3.294102167235718e+56 or 2.2182768110617736e+300 < (* y (- z t))

    1. Initial program 22.6

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

    2. Simplified2.0

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

    if -3.294102167235718e+56 < (* y (- z t)) < 2.2182768110617736e+300

    1. Initial program 0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.294102167235718175854547572327975906299 \cdot 10^{56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.218276811061773599006994335448883110359 \cdot 10^{300}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(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)))