Average Error: 6.1 → 1.4
Time: 13.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -5.562195680964302490149328614550000366002 \cdot 10^{178} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 1.989157763035480554275360497772684535848 \cdot 10^{111}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -5.562195680964302490149328614550000366002 \cdot 10^{178} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 1.989157763035480554275360497772684535848 \cdot 10^{111}\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 r205991 = x;
        double r205992 = y;
        double r205993 = z;
        double r205994 = t;
        double r205995 = r205993 - r205994;
        double r205996 = r205992 * r205995;
        double r205997 = a;
        double r205998 = r205996 / r205997;
        double r205999 = r205991 + r205998;
        return r205999;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r206000 = y;
        double r206001 = z;
        double r206002 = t;
        double r206003 = r206001 - r206002;
        double r206004 = r206000 * r206003;
        double r206005 = a;
        double r206006 = r206004 / r206005;
        double r206007 = -5.5621956809643025e+178;
        bool r206008 = r206006 <= r206007;
        double r206009 = 1.9891577630354806e+111;
        bool r206010 = r206006 <= r206009;
        double r206011 = !r206010;
        bool r206012 = r206008 || r206011;
        double r206013 = r206000 / r206005;
        double r206014 = x;
        double r206015 = fma(r206013, r206003, r206014);
        double r206016 = r206014 + r206006;
        double r206017 = r206012 ? r206015 : r206016;
        return r206017;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* y (- z t)) a) < -5.5621956809643025e+178 or 1.9891577630354806e+111 < (/ (* y (- z t)) a)

    1. Initial program 19.3

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

    2. Simplified3.9

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

    if -5.5621956809643025e+178 < (/ (* y (- z t)) a) < 1.9891577630354806e+111

    1. Initial program 0.4

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -5.562195680964302490149328614550000366002 \cdot 10^{178} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 1.989157763035480554275360497772684535848 \cdot 10^{111}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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