Average Error: 6.2 → 0.4
Time: 3.3s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.6079589991147918 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 6.345330527820856 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -3.6079589991147918 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 6.345330527820856 \cdot 10^{275}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, 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 r268209 = x;
        double r268210 = y;
        double r268211 = z;
        double r268212 = t;
        double r268213 = r268211 - r268212;
        double r268214 = r268210 * r268213;
        double r268215 = a;
        double r268216 = r268214 / r268215;
        double r268217 = r268209 - r268216;
        return r268217;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r268218 = y;
        double r268219 = z;
        double r268220 = t;
        double r268221 = r268219 - r268220;
        double r268222 = r268218 * r268221;
        double r268223 = -3.607958999114792e+296;
        bool r268224 = r268222 <= r268223;
        double r268225 = 6.345330527820856e+275;
        bool r268226 = r268222 <= r268225;
        double r268227 = !r268226;
        bool r268228 = r268224 || r268227;
        double r268229 = a;
        double r268230 = r268218 / r268229;
        double r268231 = r268220 - r268219;
        double r268232 = x;
        double r268233 = fma(r268230, r268231, r268232);
        double r268234 = r268222 / r268229;
        double r268235 = r268232 - r268234;
        double r268236 = r268228 ? r268233 : r268235;
        return r268236;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -3.607958999114792e+296 or 6.345330527820856e+275 < (* y (- z t))

    1. Initial program 53.2

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

    2. Simplified0.3

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

    if -3.607958999114792e+296 < (* y (- z t)) < 6.345330527820856e+275

    1. Initial program 0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -3.6079589991147918 \cdot 10^{296} \lor \neg \left(y \cdot \left(z - t\right) \le 6.345330527820856 \cdot 10^{275}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))