Average Error: 6.3 → 0.9
Time: 13.9s
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 r390672 = x;
        double r390673 = y;
        double r390674 = z;
        double r390675 = t;
        double r390676 = r390674 - r390675;
        double r390677 = r390673 * r390676;
        double r390678 = a;
        double r390679 = r390677 / r390678;
        double r390680 = r390672 - r390679;
        return r390680;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r390681 = y;
        double r390682 = -792628.2817991371;
        bool r390683 = r390681 <= r390682;
        double r390684 = t;
        double r390685 = z;
        double r390686 = r390684 - r390685;
        double r390687 = 1.0;
        double r390688 = a;
        double r390689 = r390687 / r390688;
        double r390690 = r390686 * r390689;
        double r390691 = x;
        double r390692 = fma(r390690, r390681, r390691);
        double r390693 = 9.342267407344556e-106;
        bool r390694 = r390681 <= r390693;
        double r390695 = r390685 - r390684;
        double r390696 = r390681 * r390695;
        double r390697 = r390696 / r390688;
        double r390698 = r390691 - r390697;
        double r390699 = r390688 / r390695;
        double r390700 = r390681 / r390699;
        double r390701 = r390691 - r390700;
        double r390702 = r390694 ? r390698 : r390701;
        double r390703 = r390683 ? r390692 : r390702;
        return r390703;
}

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)))