Average Error: 6.0 → 0.7
Time: 7.1s
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 r1508 = x;
        double r1509 = y;
        double r1510 = z;
        double r1511 = t;
        double r1512 = r1510 - r1511;
        double r1513 = r1509 * r1512;
        double r1514 = a;
        double r1515 = r1513 / r1514;
        double r1516 = r1508 + r1515;
        return r1516;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1517 = y;
        double r1518 = z;
        double r1519 = t;
        double r1520 = r1518 - r1519;
        double r1521 = r1517 * r1520;
        double r1522 = -3.2688106303761396e+233;
        bool r1523 = r1521 <= r1522;
        double r1524 = 1.0086260599490307e+83;
        bool r1525 = r1521 <= r1524;
        double r1526 = !r1525;
        bool r1527 = r1523 || r1526;
        double r1528 = a;
        double r1529 = r1517 / r1528;
        double r1530 = x;
        double r1531 = fma(r1529, r1520, r1530);
        double r1532 = r1521 / r1528;
        double r1533 = r1530 + r1532;
        double r1534 = r1527 ? r1531 : r1533;
        return r1534;
}

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)))