Average Error: 6.2 → 0.5
Time: 3.7s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.8448302221257819 \cdot 10^{209}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.3471532654452293 \cdot 10^{177}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -1.8448302221257819 \cdot 10^{209}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.3471532654452293 \cdot 10^{177}:\\
\;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r262611 = x;
        double r262612 = y;
        double r262613 = z;
        double r262614 = t;
        double r262615 = r262613 - r262614;
        double r262616 = r262612 * r262615;
        double r262617 = a;
        double r262618 = r262616 / r262617;
        double r262619 = r262611 - r262618;
        return r262619;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r262620 = y;
        double r262621 = z;
        double r262622 = t;
        double r262623 = r262621 - r262622;
        double r262624 = r262620 * r262623;
        double r262625 = -1.844830222125782e+209;
        bool r262626 = r262624 <= r262625;
        double r262627 = a;
        double r262628 = r262620 / r262627;
        double r262629 = r262622 - r262621;
        double r262630 = x;
        double r262631 = fma(r262628, r262629, r262630);
        double r262632 = 2.3471532654452293e+177;
        bool r262633 = r262624 <= r262632;
        double r262634 = 1.0;
        double r262635 = r262627 / r262624;
        double r262636 = r262634 / r262635;
        double r262637 = r262630 - r262636;
        double r262638 = r262627 / r262623;
        double r262639 = r262620 / r262638;
        double r262640 = r262630 - r262639;
        double r262641 = r262633 ? r262637 : r262640;
        double r262642 = r262626 ? r262631 : r262641;
        return r262642;
}

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
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 3 regimes

  2. if (* y (- z t)) < -1.844830222125782e+209

    1. Initial program 30.2

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

    2. Simplified0.4

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

    if -1.844830222125782e+209 < (* y (- z t)) < 2.3471532654452293e+177

    1. Initial program 0.4

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

    3. Applied clear-num0.4

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

    if 2.3471532654452293e+177 < (* y (- z t))

    1. Initial program 25.1

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

    3. Applied associate-/l*1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.8448302221257819 \cdot 10^{209}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.3471532654452293 \cdot 10^{177}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))