Average Error: 6.3 → 0.5
Time: 2.6s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.5554972572940787 \cdot 10^{247} \lor \neg \left(y \cdot \left(z - t\right) \le 2.18276991070108086 \cdot 10^{197}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -2.5554972572940787 \cdot 10^{247} \lor \neg \left(y \cdot \left(z - t\right) \le 2.18276991070108086 \cdot 10^{197}\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 r264260 = x;
        double r264261 = y;
        double r264262 = z;
        double r264263 = t;
        double r264264 = r264262 - r264263;
        double r264265 = r264261 * r264264;
        double r264266 = a;
        double r264267 = r264265 / r264266;
        double r264268 = r264260 + r264267;
        return r264268;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r264269 = y;
        double r264270 = z;
        double r264271 = t;
        double r264272 = r264270 - r264271;
        double r264273 = r264269 * r264272;
        double r264274 = -2.5554972572940787e+247;
        bool r264275 = r264273 <= r264274;
        double r264276 = 2.182769910701081e+197;
        bool r264277 = r264273 <= r264276;
        double r264278 = !r264277;
        bool r264279 = r264275 || r264278;
        double r264280 = a;
        double r264281 = r264269 / r264280;
        double r264282 = x;
        double r264283 = fma(r264281, r264272, r264282);
        double r264284 = r264273 / r264280;
        double r264285 = r264282 + r264284;
        double r264286 = r264279 ? r264283 : r264285;
        return r264286;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -2.5554972572940787e+247 or 2.182769910701081e+197 < (* y (- z t))

    1. Initial program 31.8

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

    2. Simplified0.8

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

    if -2.5554972572940787e+247 < (* y (- z t)) < 2.182769910701081e+197

    1. Initial program 0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.5554972572940787 \cdot 10^{247} \lor \neg \left(y \cdot \left(z - t\right) \le 2.18276991070108086 \cdot 10^{197}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))