Average Error: 6.0 → 0.7
Time: 3.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.2688106303761396 \cdot 10^{233} \lor \neg \left(y \cdot \left(z - t\right) \le 1.0086260599490307 \cdot 10^{83}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -3.2688106303761396 \cdot 10^{233} \lor \neg \left(y \cdot \left(z - t\right) \le 1.0086260599490307 \cdot 10^{83}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, 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 r237465 = x;
        double r237466 = y;
        double r237467 = z;
        double r237468 = t;
        double r237469 = r237467 - r237468;
        double r237470 = r237466 * r237469;
        double r237471 = a;
        double r237472 = r237470 / r237471;
        double r237473 = r237465 + r237472;
        return r237473;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r237474 = y;
        double r237475 = z;
        double r237476 = t;
        double r237477 = r237475 - r237476;
        double r237478 = r237474 * r237477;
        double r237479 = -3.2688106303761396e+233;
        bool r237480 = r237478 <= r237479;
        double r237481 = 1.0086260599490307e+83;
        bool r237482 = r237478 <= r237481;
        double r237483 = !r237482;
        bool r237484 = r237480 || r237483;
        double r237485 = a;
        double r237486 = r237474 / r237485;
        double r237487 = x;
        double r237488 = fma(r237486, r237477, r237487);
        double r237489 = r237478 / r237485;
        double r237490 = r237487 + r237489;
        double r237491 = r237484 ? r237488 : r237490;
        return r237491;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -3.2688106303761396e+233 or 1.0086260599490307e+83 < (* y (- z t))

    1. Initial program 20.5

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

    2. Simplified1.7

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

    if -3.2688106303761396e+233 < (* y (- z t)) < 1.0086260599490307e+83

    1. Initial program 0.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.2688106303761396 \cdot 10^{233} \lor \neg \left(y \cdot \left(z - t\right) \le 1.0086260599490307 \cdot 10^{83}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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)))