Average Error: 6.1 → 1.4
Time: 12.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -5.562195680964302490149328614550000366002 \cdot 10^{178} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 1.989157763035480554275360497772684535848 \cdot 10^{111}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -5.562195680964302490149328614550000366002 \cdot 10^{178} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 1.989157763035480554275360497772684535848 \cdot 10^{111}\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 r185471 = x;
        double r185472 = y;
        double r185473 = z;
        double r185474 = t;
        double r185475 = r185473 - r185474;
        double r185476 = r185472 * r185475;
        double r185477 = a;
        double r185478 = r185476 / r185477;
        double r185479 = r185471 + r185478;
        return r185479;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r185480 = y;
        double r185481 = z;
        double r185482 = t;
        double r185483 = r185481 - r185482;
        double r185484 = r185480 * r185483;
        double r185485 = a;
        double r185486 = r185484 / r185485;
        double r185487 = -5.5621956809643025e+178;
        bool r185488 = r185486 <= r185487;
        double r185489 = 1.9891577630354806e+111;
        bool r185490 = r185486 <= r185489;
        double r185491 = !r185490;
        bool r185492 = r185488 || r185491;
        double r185493 = r185480 / r185485;
        double r185494 = x;
        double r185495 = fma(r185493, r185483, r185494);
        double r185496 = r185494 + r185486;
        double r185497 = r185492 ? r185495 : r185496;
        return r185497;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* y (- z t)) a) < -5.5621956809643025e+178 or 1.9891577630354806e+111 < (/ (* y (- z t)) a)

    1. Initial program 19.3

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

    2. Simplified3.9

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

    if -5.5621956809643025e+178 < (/ (* y (- z t)) a) < 1.9891577630354806e+111

    1. Initial program 0.4

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -5.562195680964302490149328614550000366002 \cdot 10^{178} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 1.989157763035480554275360497772684535848 \cdot 10^{111}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))