Average Error: 6.2 → 1.7
Time: 3.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.8744657462895403389224078436647417432 \cdot 10^{-286} \lor \neg \left(a \le 3146.272963083657941751880571246147155762\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.8744657462895403389224078436647417432 \cdot 10^{-286} \lor \neg \left(a \le 3146.272963083657941751880571246147155762\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 r308463 = x;
        double r308464 = y;
        double r308465 = z;
        double r308466 = t;
        double r308467 = r308465 - r308466;
        double r308468 = r308464 * r308467;
        double r308469 = a;
        double r308470 = r308468 / r308469;
        double r308471 = r308463 + r308470;
        return r308471;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r308472 = a;
        double r308473 = -1.8744657462895403e-286;
        bool r308474 = r308472 <= r308473;
        double r308475 = 3146.272963083658;
        bool r308476 = r308472 <= r308475;
        double r308477 = !r308476;
        bool r308478 = r308474 || r308477;
        double r308479 = y;
        double r308480 = r308479 / r308472;
        double r308481 = z;
        double r308482 = t;
        double r308483 = r308481 - r308482;
        double r308484 = x;
        double r308485 = fma(r308480, r308483, r308484);
        double r308486 = r308479 * r308483;
        double r308487 = r308486 / r308472;
        double r308488 = r308484 + r308487;
        double r308489 = r308478 ? r308485 : r308488;
        return r308489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.8744657462895403e-286 or 3146.272963083658 < a

    1. Initial program 7.6

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

    2. Simplified1.9

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

    if -1.8744657462895403e-286 < a < 3146.272963083658

    1. Initial program 1.0

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.8744657462895403389224078436647417432 \cdot 10^{-286} \lor \neg \left(a \le 3146.272963083657941751880571246147155762\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))