Average Error: 6.3 → 1.2
Time: 2.7s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x - \frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(x - \frac{y \cdot \left(z - t\right)}{a} \le 7.1248143761836939 \cdot 10^{-232}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;x - \frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(x - \frac{y \cdot \left(z - t\right)}{a} \le 7.1248143761836939 \cdot 10^{-232}\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 r265631 = x;
        double r265632 = y;
        double r265633 = z;
        double r265634 = t;
        double r265635 = r265633 - r265634;
        double r265636 = r265632 * r265635;
        double r265637 = a;
        double r265638 = r265636 / r265637;
        double r265639 = r265631 - r265638;
        return r265639;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r265640 = x;
        double r265641 = y;
        double r265642 = z;
        double r265643 = t;
        double r265644 = r265642 - r265643;
        double r265645 = r265641 * r265644;
        double r265646 = a;
        double r265647 = r265645 / r265646;
        double r265648 = r265640 - r265647;
        double r265649 = -inf.0;
        bool r265650 = r265648 <= r265649;
        double r265651 = 7.124814376183694e-232;
        bool r265652 = r265648 <= r265651;
        double r265653 = !r265652;
        bool r265654 = r265650 || r265653;
        double r265655 = r265641 / r265646;
        double r265656 = r265643 - r265642;
        double r265657 = fma(r265655, r265656, r265640);
        double r265658 = r265654 ? r265657 : r265648;
        return r265658;
}

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
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- x (/ (* y (- z t)) a)) < -inf.0 or 7.124814376183694e-232 < (- x (/ (* y (- z t)) a))

    1. Initial program 11.3

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

    2. Simplified1.8

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

    if -inf.0 < (- x (/ (* y (- z t)) a)) < 7.124814376183694e-232

    1. Initial program 0.5

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{y \cdot \left(z - t\right)}{a} = -\infty \lor \neg \left(x - \frac{y \cdot \left(z - t\right)}{a} \le 7.1248143761836939 \cdot 10^{-232}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))