Average Error: 6.3 → 0.9
Time: 10.1s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -792628.28179913713:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a}, y, x\right)\\ \mathbf{elif}\;y \le 9.3422674073445563 \cdot 10^{-106}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -792628.28179913713:\\
\;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a}, y, x\right)\\

\mathbf{elif}\;y \le 9.3422674073445563 \cdot 10^{-106}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r293277 = x;
        double r293278 = y;
        double r293279 = z;
        double r293280 = t;
        double r293281 = r293279 - r293280;
        double r293282 = r293278 * r293281;
        double r293283 = a;
        double r293284 = r293282 / r293283;
        double r293285 = r293277 - r293284;
        return r293285;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r293286 = y;
        double r293287 = -792628.2817991371;
        bool r293288 = r293286 <= r293287;
        double r293289 = t;
        double r293290 = z;
        double r293291 = r293289 - r293290;
        double r293292 = 1.0;
        double r293293 = a;
        double r293294 = r293292 / r293293;
        double r293295 = r293291 * r293294;
        double r293296 = x;
        double r293297 = fma(r293295, r293286, r293296);
        double r293298 = 9.342267407344556e-106;
        bool r293299 = r293286 <= r293298;
        double r293300 = r293290 - r293289;
        double r293301 = r293286 * r293300;
        double r293302 = r293301 / r293293;
        double r293303 = r293296 - r293302;
        double r293304 = r293293 / r293300;
        double r293305 = r293286 / r293304;
        double r293306 = r293296 - r293305;
        double r293307 = r293299 ? r293303 : r293306;
        double r293308 = r293288 ? r293297 : r293307;
        return r293308;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -792628.2817991371

    1. Initial program 15.9

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

    2. Simplified0.7

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

    4. Applied div-inv0.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t - z\right) \cdot \frac{1}{a}}, y, x\right)\]

    if -792628.2817991371 < y < 9.342267407344556e-106

    1. Initial program 0.6

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

    if 9.342267407344556e-106 < y

    1. Initial program 10.7

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied associate-/l*1.6

      \[\leadsto x - \color{blue}{\frac{y}{\frac{a}{z - t}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -792628.28179913713:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a}, y, x\right)\\ \mathbf{elif}\;y \le 9.3422674073445563 \cdot 10^{-106}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))