Average Error: 6.2 → 2.5
Time: 2.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.3249082459376733 \cdot 10^{-158} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.3249082459376733 \cdot 10^{-158} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{else}:\\
\;\;\;\;x - y \cdot \frac{z - t}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r260459 = x;
        double r260460 = y;
        double r260461 = z;
        double r260462 = t;
        double r260463 = r260461 - r260462;
        double r260464 = r260460 * r260463;
        double r260465 = a;
        double r260466 = r260464 / r260465;
        double r260467 = r260459 - r260466;
        return r260467;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r260468 = z;
        double r260469 = -5.324908245937673e-158;
        bool r260470 = r260468 <= r260469;
        double r260471 = 6.939075397429748e-149;
        bool r260472 = r260468 <= r260471;
        double r260473 = !r260472;
        bool r260474 = r260470 || r260473;
        double r260475 = y;
        double r260476 = a;
        double r260477 = r260475 / r260476;
        double r260478 = t;
        double r260479 = r260478 - r260468;
        double r260480 = x;
        double r260481 = fma(r260477, r260479, r260480);
        double r260482 = r260468 - r260478;
        double r260483 = r260482 / r260476;
        double r260484 = r260475 * r260483;
        double r260485 = r260480 - r260484;
        double r260486 = r260474 ? r260481 : r260485;
        return r260486;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.324908245937673e-158 or 6.939075397429748e-149 < z

    1. Initial program 6.9

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

    2. Simplified2.1

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

    if -5.324908245937673e-158 < z < 6.939075397429748e-149

    1. Initial program 4.2

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity4.2

      \[\leadsto x - \frac{y \cdot \left(z - t\right)}{\color{blue}{1 \cdot a}}\]

    4. Applied times-frac3.7

      \[\leadsto x - \color{blue}{\frac{y}{1} \cdot \frac{z - t}{a}}\]

    5. Simplified3.7

      \[\leadsto x - \color{blue}{y} \cdot \frac{z - t}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.3249082459376733 \cdot 10^{-158} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))