Average Error: 6.2 → 0.3
Time: 11.6s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 1.4362507429651844 \cdot 10^{239}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
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 1.4362507429651844 \cdot 10^{239}\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 r391795 = x;
        double r391796 = y;
        double r391797 = z;
        double r391798 = t;
        double r391799 = r391797 - r391798;
        double r391800 = r391796 * r391799;
        double r391801 = a;
        double r391802 = r391800 / r391801;
        double r391803 = r391795 + r391802;
        return r391803;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r391804 = y;
        double r391805 = z;
        double r391806 = t;
        double r391807 = r391805 - r391806;
        double r391808 = r391804 * r391807;
        double r391809 = -inf.0;
        bool r391810 = r391808 <= r391809;
        double r391811 = 1.4362507429651844e+239;
        bool r391812 = r391808 <= r391811;
        double r391813 = !r391812;
        bool r391814 = r391810 || r391813;
        double r391815 = a;
        double r391816 = r391804 / r391815;
        double r391817 = x;
        double r391818 = fma(r391816, r391807, r391817);
        double r391819 = r391808 / r391815;
        double r391820 = r391817 + r391819;
        double r391821 = r391814 ? r391818 : r391820;
        return r391821;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -inf.0 or 1.4362507429651844e+239 < (* y (- z t))

    1. Initial program 47.6

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

    2. Simplified0.3

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

    if -inf.0 < (* y (- z t)) < 1.4362507429651844e+239

    1. Initial program 0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 1.4362507429651844 \cdot 10^{239}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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)))