Average Error: 6.2 → 1.3
Time: 7.9s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -8.469201958944395 \cdot 10^{-176}:\\ \;\;\;\;x + \frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} = -\infty:\\
\;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\

\mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -8.469201958944395 \cdot 10^{-176}:\\
\;\;\;\;x + \frac{\left(z - x\right) \cdot y}{t}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r5782726 = x;
        double r5782727 = y;
        double r5782728 = z;
        double r5782729 = r5782728 - r5782726;
        double r5782730 = r5782727 * r5782729;
        double r5782731 = t;
        double r5782732 = r5782730 / r5782731;
        double r5782733 = r5782726 + r5782732;
        return r5782733;
}
↓
double f(double x, double y, double z, double t) {
        double r5782734 = x;
        double r5782735 = z;
        double r5782736 = r5782735 - r5782734;
        double r5782737 = y;
        double r5782738 = r5782736 * r5782737;
        double r5782739 = t;
        double r5782740 = r5782738 / r5782739;
        double r5782741 = r5782734 + r5782740;
        double r5782742 = -inf.0;
        bool r5782743 = r5782741 <= r5782742;
        double r5782744 = r5782739 / r5782736;
        double r5782745 = r5782737 / r5782744;
        double r5782746 = r5782734 + r5782745;
        double r5782747 = -8.469201958944395e-176;
        bool r5782748 = r5782741 <= r5782747;
        double r5782749 = r5782737 / r5782739;
        double r5782750 = fma(r5782749, r5782736, r5782734);
        double r5782751 = r5782748 ? r5782741 : r5782750;
        double r5782752 = r5782743 ? r5782746 : r5782751;
        return r5782752;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.2
Target1.9
Herbie1.3
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (/ (* y (- z x)) t)) < -inf.0

    1. Initial program 60.2

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

    3. Applied associate-/l*0.2

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

    if -inf.0 < (+ x (/ (* y (- z x)) t)) < -8.469201958944395e-176

    1. Initial program 0.4

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

    if -8.469201958944395e-176 < (+ x (/ (* y (- z x)) t))

    1. Initial program 5.9

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

    2. Simplified2.0

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -8.469201958944395 \cdot 10^{-176}:\\ \;\;\;\;x + \frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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