Average Error: 6.2 → 0.9
Time: 3.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -7.9616902482949608 \cdot 10^{133} \lor \neg \left(y \cdot \left(z - t\right) \le 9.8153332846651028 \cdot 10^{112}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -7.9616902482949608 \cdot 10^{133} \lor \neg \left(y \cdot \left(z - t\right) \le 9.8153332846651028 \cdot 10^{112}\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 r285136 = x;
        double r285137 = y;
        double r285138 = z;
        double r285139 = t;
        double r285140 = r285138 - r285139;
        double r285141 = r285137 * r285140;
        double r285142 = a;
        double r285143 = r285141 / r285142;
        double r285144 = r285136 + r285143;
        return r285144;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r285145 = y;
        double r285146 = z;
        double r285147 = t;
        double r285148 = r285146 - r285147;
        double r285149 = r285145 * r285148;
        double r285150 = -7.961690248294961e+133;
        bool r285151 = r285149 <= r285150;
        double r285152 = 9.815333284665103e+112;
        bool r285153 = r285149 <= r285152;
        double r285154 = !r285153;
        bool r285155 = r285151 || r285154;
        double r285156 = a;
        double r285157 = r285145 / r285156;
        double r285158 = x;
        double r285159 = fma(r285157, r285148, r285158);
        double r285160 = r285149 / r285156;
        double r285161 = r285158 + r285160;
        double r285162 = r285155 ? r285159 : r285161;
        return r285162;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.8
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -7.961690248294961e+133 or 9.815333284665103e+112 < (* y (- z t))

    1. Initial program 18.0

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

    2. Simplified1.6

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

    if -7.961690248294961e+133 < (* y (- z t)) < 9.815333284665103e+112

    1. Initial program 0.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -7.9616902482949608 \cdot 10^{133} \lor \neg \left(y \cdot \left(z - t\right) \le 9.8153332846651028 \cdot 10^{112}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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)))