Average Error: 6.2 → 0.4
Time: 12.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -7.977929768354411 \cdot 10^{224} \lor \neg \left(y \cdot \left(z - t\right) \le 1.3695700649275306 \cdot 10^{232}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -7.977929768354411 \cdot 10^{224} \lor \neg \left(y \cdot \left(z - t\right) \le 1.3695700649275306 \cdot 10^{232}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r310585 = x;
        double r310586 = y;
        double r310587 = z;
        double r310588 = t;
        double r310589 = r310587 - r310588;
        double r310590 = r310586 * r310589;
        double r310591 = a;
        double r310592 = r310590 / r310591;
        double r310593 = r310585 - r310592;
        return r310593;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r310594 = y;
        double r310595 = z;
        double r310596 = t;
        double r310597 = r310595 - r310596;
        double r310598 = r310594 * r310597;
        double r310599 = -7.977929768354411e+224;
        bool r310600 = r310598 <= r310599;
        double r310601 = 1.3695700649275306e+232;
        bool r310602 = r310598 <= r310601;
        double r310603 = !r310602;
        bool r310604 = r310600 || r310603;
        double r310605 = r310596 - r310595;
        double r310606 = a;
        double r310607 = r310605 / r310606;
        double r310608 = x;
        double r310609 = fma(r310607, r310594, r310608);
        double r310610 = r310598 / r310606;
        double r310611 = r310608 - r310610;
        double r310612 = r310604 ? r310609 : r310611;
        return r310612;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -7.977929768354411e+224 or 1.3695700649275306e+232 < (* y (- z t))

    1. Initial program 34.2

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

    2. Simplified0.5

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

    if -7.977929768354411e+224 < (* y (- z t)) < 1.3695700649275306e+232

    1. Initial program 0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -7.977929768354411 \cdot 10^{224} \lor \neg \left(y \cdot \left(z - t\right) \le 1.3695700649275306 \cdot 10^{232}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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)))