Average Error: 6.1 → 0.5
Time: 3.6s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.086588240667298 \cdot 10^{268} \lor \neg \left(y \cdot \left(z - t\right) \le 4.3668041271782376 \cdot 10^{219}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -1.086588240667298 \cdot 10^{268} \lor \neg \left(y \cdot \left(z - t\right) \le 4.3668041271782376 \cdot 10^{219}\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 r434376 = x;
        double r434377 = y;
        double r434378 = z;
        double r434379 = t;
        double r434380 = r434378 - r434379;
        double r434381 = r434377 * r434380;
        double r434382 = a;
        double r434383 = r434381 / r434382;
        double r434384 = r434376 - r434383;
        return r434384;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r434385 = y;
        double r434386 = z;
        double r434387 = t;
        double r434388 = r434386 - r434387;
        double r434389 = r434385 * r434388;
        double r434390 = -1.0865882406672983e+268;
        bool r434391 = r434389 <= r434390;
        double r434392 = 4.3668041271782376e+219;
        bool r434393 = r434389 <= r434392;
        double r434394 = !r434393;
        bool r434395 = r434391 || r434394;
        double r434396 = a;
        double r434397 = r434385 / r434396;
        double r434398 = r434387 - r434386;
        double r434399 = x;
        double r434400 = fma(r434397, r434398, r434399);
        double r434401 = r434389 / r434396;
        double r434402 = r434399 - r434401;
        double r434403 = r434395 ? r434400 : r434402;
        return r434403;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -1.0865882406672983e+268 or 4.3668041271782376e+219 < (* y (- z t))

    1. Initial program 38.7

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

    2. Simplified0.3

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

    if -1.0865882406672983e+268 < (* y (- z t)) < 4.3668041271782376e+219

    1. Initial program 0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.086588240667298 \cdot 10^{268} \lor \neg \left(y \cdot \left(z - t\right) \le 4.3668041271782376 \cdot 10^{219}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))