Average Error: 6.5 → 1.6
Time: 20.5s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.584406202812830128924411129162326193187 \cdot 10^{-168} \lor \neg \left(t \le 150412601825096175811127520041088581632\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -4.584406202812830128924411129162326193187 \cdot 10^{-168} \lor \neg \left(t \le 150412601825096175811127520041088581632\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r242433 = x;
        double r242434 = y;
        double r242435 = z;
        double r242436 = r242435 - r242433;
        double r242437 = r242434 * r242436;
        double r242438 = t;
        double r242439 = r242437 / r242438;
        double r242440 = r242433 + r242439;
        return r242440;
}
↓
double f(double x, double y, double z, double t) {
        double r242441 = t;
        double r242442 = -4.58440620281283e-168;
        bool r242443 = r242441 <= r242442;
        double r242444 = 1.5041260182509618e+38;
        bool r242445 = r242441 <= r242444;
        double r242446 = !r242445;
        bool r242447 = r242443 || r242446;
        double r242448 = y;
        double r242449 = r242448 / r242441;
        double r242450 = z;
        double r242451 = x;
        double r242452 = r242450 - r242451;
        double r242453 = fma(r242449, r242452, r242451);
        double r242454 = r242448 * r242452;
        double r242455 = r242454 / r242441;
        double r242456 = r242451 + r242455;
        double r242457 = r242447 ? r242453 : r242456;
        return r242457;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.5
Target2.0
Herbie1.6
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if t < -4.58440620281283e-168 or 1.5041260182509618e+38 < t

    1. Initial program 8.4

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

    2. Simplified1.4

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

    if -4.58440620281283e-168 < t < 1.5041260182509618e+38

    1. Initial program 2.0

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.584406202812830128924411129162326193187 \cdot 10^{-168} \lor \neg \left(t \le 150412601825096175811127520041088581632\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))