Average Error: 6.5 → 0.4
Time: 2.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -8.981442880940557861413613942616636285951 \cdot 10^{292} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 8.973171892835292774846855805866219421622 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -8.981442880940557861413613942616636285951 \cdot 10^{292} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 8.973171892835292774846855805866219421622 \cdot 10^{305}\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 r310271 = x;
        double r310272 = y;
        double r310273 = z;
        double r310274 = t;
        double r310275 = r310273 - r310274;
        double r310276 = r310272 * r310275;
        double r310277 = a;
        double r310278 = r310276 / r310277;
        double r310279 = r310271 - r310278;
        return r310279;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r310280 = y;
        double r310281 = z;
        double r310282 = t;
        double r310283 = r310281 - r310282;
        double r310284 = r310280 * r310283;
        double r310285 = a;
        double r310286 = r310284 / r310285;
        double r310287 = -8.981442880940558e+292;
        bool r310288 = r310286 <= r310287;
        double r310289 = 8.973171892835293e+305;
        bool r310290 = r310286 <= r310289;
        double r310291 = !r310290;
        bool r310292 = r310288 || r310291;
        double r310293 = r310280 / r310285;
        double r310294 = r310282 - r310281;
        double r310295 = x;
        double r310296 = fma(r310293, r310294, r310295);
        double r310297 = r310295 - r310286;
        double r310298 = r310292 ? r310296 : r310297;
        return r310298;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.5
Target0.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* y (- z t)) a) < -8.981442880940558e+292 or 8.973171892835293e+305 < (/ (* y (- z t)) a)

    1. Initial program 57.8

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

    2. Simplified1.0

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

    if -8.981442880940558e+292 < (/ (* y (- z t)) a) < 8.973171892835293e+305

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -8.981442880940557861413613942616636285951 \cdot 10^{292} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \le 8.973171892835292774846855805866219421622 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))