Average Error: 6.2 → 1.4
Time: 13.3s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -9.58097381083328705154269126658672692522 \cdot 10^{81}:\\ \;\;\;\;\frac{t - z}{\frac{a}{y}} + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 4.84668724673702282414318349461758654708 \cdot 10^{282}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -9.58097381083328705154269126658672692522 \cdot 10^{81}:\\
\;\;\;\;\frac{t - z}{\frac{a}{y}} + x\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 4.84668724673702282414318349461758654708 \cdot 10^{282}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r194716 = x;
        double r194717 = y;
        double r194718 = z;
        double r194719 = t;
        double r194720 = r194718 - r194719;
        double r194721 = r194717 * r194720;
        double r194722 = a;
        double r194723 = r194721 / r194722;
        double r194724 = r194716 - r194723;
        return r194724;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r194725 = y;
        double r194726 = z;
        double r194727 = t;
        double r194728 = r194726 - r194727;
        double r194729 = r194725 * r194728;
        double r194730 = a;
        double r194731 = r194729 / r194730;
        double r194732 = -9.580973810833287e+81;
        bool r194733 = r194731 <= r194732;
        double r194734 = r194727 - r194726;
        double r194735 = r194730 / r194725;
        double r194736 = r194734 / r194735;
        double r194737 = x;
        double r194738 = r194736 + r194737;
        double r194739 = 4.846687246737023e+282;
        bool r194740 = r194731 <= r194739;
        double r194741 = r194737 - r194731;
        double r194742 = r194734 / r194730;
        double r194743 = fma(r194742, r194725, r194737);
        double r194744 = r194740 ? r194741 : r194743;
        double r194745 = r194733 ? r194738 : r194744;
        return r194745;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) a) < -9.580973810833287e+81

    1. Initial program 14.7

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

    2. Simplified3.6

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

    4. Applied fma-udef3.6

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

    5. Simplified3.2

      \[\leadsto \color{blue}{\frac{t - z}{\frac{a}{y}}} + x\]

    if -9.580973810833287e+81 < (/ (* y (- z t)) a) < 4.846687246737023e+282

    1. Initial program 0.5

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

    if 4.846687246737023e+282 < (/ (* y (- z t)) a)

    1. Initial program 50.7

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

    2. Simplified1.0

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

    3. Taylor expanded around 0 50.7

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

    4. Simplified7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \le -9.58097381083328705154269126658672692522 \cdot 10^{81}:\\ \;\;\;\;\frac{t - z}{\frac{a}{y}} + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 4.84668724673702282414318349461758654708 \cdot 10^{282}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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 -1.07612662163899753e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))