Average Error: 6.5 → 0.4
Time: 2.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 1.4616697667384146 \cdot 10^{247}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 1.4616697667384146 \cdot 10^{247}\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 r295621 = x;
        double r295622 = y;
        double r295623 = z;
        double r295624 = t;
        double r295625 = r295623 - r295624;
        double r295626 = r295622 * r295625;
        double r295627 = a;
        double r295628 = r295626 / r295627;
        double r295629 = r295621 + r295628;
        return r295629;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r295630 = y;
        double r295631 = z;
        double r295632 = t;
        double r295633 = r295631 - r295632;
        double r295634 = r295630 * r295633;
        double r295635 = -inf.0;
        bool r295636 = r295634 <= r295635;
        double r295637 = 1.4616697667384146e+247;
        bool r295638 = r295634 <= r295637;
        double r295639 = !r295638;
        bool r295640 = r295636 || r295639;
        double r295641 = a;
        double r295642 = r295630 / r295641;
        double r295643 = x;
        double r295644 = fma(r295642, r295633, r295643);
        double r295645 = r295634 / r295641;
        double r295646 = r295643 + r295645;
        double r295647 = r295640 ? r295644 : r295646;
        return r295647;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.5
Target0.8
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)) < -inf.0 or 1.4616697667384146e+247 < (* y (- z t))

    1. Initial program 48.9

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

    2. Simplified0.2

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

    if -inf.0 < (* y (- z t)) < 1.4616697667384146e+247

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 1.4616697667384146 \cdot 10^{247}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))