Average Error: 6.2 → 0.4
Time: 9.9s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.4829459359694524 \cdot 10^{214} \lor \neg \left(y \cdot \left(z - t\right) \le 1.7601478476406493 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -1.4829459359694524 \cdot 10^{214} \lor \neg \left(y \cdot \left(z - t\right) \le 1.7601478476406493 \cdot 10^{275}\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 r321772 = x;
        double r321773 = y;
        double r321774 = z;
        double r321775 = t;
        double r321776 = r321774 - r321775;
        double r321777 = r321773 * r321776;
        double r321778 = a;
        double r321779 = r321777 / r321778;
        double r321780 = r321772 + r321779;
        return r321780;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r321781 = y;
        double r321782 = z;
        double r321783 = t;
        double r321784 = r321782 - r321783;
        double r321785 = r321781 * r321784;
        double r321786 = -1.4829459359694524e+214;
        bool r321787 = r321785 <= r321786;
        double r321788 = 1.7601478476406493e+275;
        bool r321789 = r321785 <= r321788;
        double r321790 = !r321789;
        bool r321791 = r321787 || r321790;
        double r321792 = a;
        double r321793 = r321781 / r321792;
        double r321794 = x;
        double r321795 = fma(r321793, r321784, r321794);
        double r321796 = r321785 / r321792;
        double r321797 = r321794 + r321796;
        double r321798 = r321791 ? r321795 : r321797;
        return r321798;
}

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
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)) < -1.4829459359694524e+214 or 1.7601478476406493e+275 < (* y (- z t))

    1. Initial program 38.3

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

    2. Simplified0.5

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

    if -1.4829459359694524e+214 < (* y (- z t)) < 1.7601478476406493e+275

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.4829459359694524 \cdot 10^{214} \lor \neg \left(y \cdot \left(z - t\right) \le 1.7601478476406493 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))