Average Error: 6.0 → 0.6
Time: 2.6s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.6927132490810346 \cdot 10^{197}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 8.20385041309340159 \cdot 10^{120}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -5.6927132490810346 \cdot 10^{197}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 8.20385041309340159 \cdot 10^{120}:\\
\;\;\;\;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 r325826 = x;
        double r325827 = y;
        double r325828 = z;
        double r325829 = t;
        double r325830 = r325828 - r325829;
        double r325831 = r325827 * r325830;
        double r325832 = a;
        double r325833 = r325831 / r325832;
        double r325834 = r325826 - r325833;
        return r325834;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r325835 = y;
        double r325836 = z;
        double r325837 = t;
        double r325838 = r325836 - r325837;
        double r325839 = r325835 * r325838;
        double r325840 = -5.6927132490810346e+197;
        bool r325841 = r325839 <= r325840;
        double r325842 = x;
        double r325843 = a;
        double r325844 = r325843 / r325838;
        double r325845 = r325835 / r325844;
        double r325846 = r325842 - r325845;
        double r325847 = 8.203850413093402e+120;
        bool r325848 = r325839 <= r325847;
        double r325849 = r325839 / r325843;
        double r325850 = r325842 - r325849;
        double r325851 = r325835 / r325843;
        double r325852 = r325837 - r325836;
        double r325853 = fma(r325851, r325852, r325842);
        double r325854 = r325848 ? r325850 : r325853;
        double r325855 = r325841 ? r325846 : r325854;
        return r325855;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.7
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -5.6927132490810346e+197

    1. Initial program 27.5

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

    3. Applied associate-/l*0.9

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

    if -5.6927132490810346e+197 < (* y (- z t)) < 8.203850413093402e+120

    1. Initial program 0.3

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

    if 8.203850413093402e+120 < (* y (- z t))

    1. Initial program 17.4

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

    2. Simplified1.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -5.6927132490810346 \cdot 10^{197}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 8.20385041309340159 \cdot 10^{120}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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)))