Average Error: 6.1 → 0.6
Time: 3.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 3.04145977992159801 \cdot 10^{266}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 3.04145977992159801 \cdot 10^{266}\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 r269224 = x;
        double r269225 = y;
        double r269226 = z;
        double r269227 = t;
        double r269228 = r269226 - r269227;
        double r269229 = r269225 * r269228;
        double r269230 = a;
        double r269231 = r269229 / r269230;
        double r269232 = r269224 + r269231;
        return r269232;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r269233 = y;
        double r269234 = z;
        double r269235 = t;
        double r269236 = r269234 - r269235;
        double r269237 = r269233 * r269236;
        double r269238 = a;
        double r269239 = r269237 / r269238;
        double r269240 = -inf.0;
        bool r269241 = r269239 <= r269240;
        double r269242 = 3.041459779921598e+266;
        bool r269243 = r269239 <= r269242;
        double r269244 = !r269243;
        bool r269245 = r269241 || r269244;
        double r269246 = r269233 / r269238;
        double r269247 = x;
        double r269248 = fma(r269246, r269236, r269247);
        double r269249 = r269247 + r269239;
        double r269250 = r269245 ? r269248 : r269249;
        return r269250;
}

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
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* y (- z t)) a) < -inf.0 or 3.041459779921598e+266 < (/ (* y (- z t)) a)

    1. Initial program 52.1

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

    2. Simplified1.5

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

    if -inf.0 < (/ (* y (- z t)) a) < 3.041459779921598e+266

    1. Initial program 0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 3.04145977992159801 \cdot 10^{266}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))