Average Error: 6.1 → 0.4
Time: 14.6s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.358413248954256348213035766917854469977 \cdot 10^{175}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
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 2.358413248954256348213035766917854469977 \cdot 10^{175}\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 r215802 = x;
        double r215803 = y;
        double r215804 = z;
        double r215805 = t;
        double r215806 = r215804 - r215805;
        double r215807 = r215803 * r215806;
        double r215808 = a;
        double r215809 = r215807 / r215808;
        double r215810 = r215802 + r215809;
        return r215810;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r215811 = y;
        double r215812 = z;
        double r215813 = t;
        double r215814 = r215812 - r215813;
        double r215815 = r215811 * r215814;
        double r215816 = -inf.0;
        bool r215817 = r215815 <= r215816;
        double r215818 = 2.3584132489542563e+175;
        bool r215819 = r215815 <= r215818;
        double r215820 = !r215819;
        bool r215821 = r215817 || r215820;
        double r215822 = a;
        double r215823 = r215811 / r215822;
        double r215824 = x;
        double r215825 = fma(r215823, r215814, r215824);
        double r215826 = r215815 / r215822;
        double r215827 = r215824 + r215826;
        double r215828 = r215821 ? r215825 : r215827;
        return r215828;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -inf.0 or 2.3584132489542563e+175 < (* y (- z t))

    1. Initial program 35.2

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

    2. Simplified0.6

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

    if -inf.0 < (* y (- z t)) < 2.3584132489542563e+175

    1. Initial program 0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty \lor \neg \left(y \cdot \left(z - t\right) \le 2.358413248954256348213035766917854469977 \cdot 10^{175}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))