Average Error: 11.4 → 1.9
Time: 11.0s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 6.1396716302029306 \cdot 10^{-218} \lor \neg \left(x \le 3.6778080321247172 \cdot 10^{22}\right):\\ \;\;\;\;x - \frac{y}{z - 0.5 \cdot \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\right)\\ \end{array}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;x \le 6.1396716302029306 \cdot 10^{-218} \lor \neg \left(x \le 3.6778080321247172 \cdot 10^{22}\right):\\
\;\;\;\;x - \frac{y}{z - 0.5 \cdot \frac{t}{\frac{z}{y}}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r509651 = x;
        double r509652 = y;
        double r509653 = 2.0;
        double r509654 = r509652 * r509653;
        double r509655 = z;
        double r509656 = r509654 * r509655;
        double r509657 = r509655 * r509653;
        double r509658 = r509657 * r509655;
        double r509659 = t;
        double r509660 = r509652 * r509659;
        double r509661 = r509658 - r509660;
        double r509662 = r509656 / r509661;
        double r509663 = r509651 - r509662;
        return r509663;
}
↓
double f(double x, double y, double z, double t) {
        double r509664 = x;
        double r509665 = 6.139671630202931e-218;
        bool r509666 = r509664 <= r509665;
        double r509667 = 3.677808032124717e+22;
        bool r509668 = r509664 <= r509667;
        double r509669 = !r509668;
        bool r509670 = r509666 || r509669;
        double r509671 = y;
        double r509672 = z;
        double r509673 = 0.5;
        double r509674 = t;
        double r509675 = r509672 / r509671;
        double r509676 = r509674 / r509675;
        double r509677 = r509673 * r509676;
        double r509678 = r509672 - r509677;
        double r509679 = r509671 / r509678;
        double r509680 = r509664 - r509679;
        double r509681 = sqrt(r509664);
        double r509682 = 2.0;
        double r509683 = r509671 / r509682;
        double r509684 = -r509674;
        double r509685 = r509684 / r509672;
        double r509686 = fma(r509683, r509685, r509672);
        double r509687 = r509671 / r509686;
        double r509688 = -r509687;
        double r509689 = fma(r509681, r509681, r509688);
        double r509690 = r509670 ? r509680 : r509689;
        return r509690;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.4
Target0.1
Herbie1.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 6.139671630202931e-218 or 3.677808032124717e+22 < x

    1. Initial program 11.4

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

    2. Simplified1.0

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity1.0

      \[\leadsto \color{blue}{1 \cdot \left(x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\right)}\]

    5. Taylor expanded around 0 2.4

      \[\leadsto 1 \cdot \left(x - \frac{y}{\color{blue}{z - 0.5 \cdot \frac{t \cdot y}{z}}}\right)\]
    6. Using strategy rm

    7. Applied associate-/l*2.1

      \[\leadsto 1 \cdot \left(x - \frac{y}{z - 0.5 \cdot \color{blue}{\frac{t}{\frac{z}{y}}}}\right)\]

    if 6.139671630202931e-218 < x < 3.677808032124717e+22

    1. Initial program 11.3

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

    2. Simplified1.1

      \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt1.4

      \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x}} - \frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\]

    5. Applied fma-neg1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 6.1396716302029306 \cdot 10^{-218} \lor \neg \left(x \le 3.6778080321247172 \cdot 10^{22}\right):\\ \;\;\;\;x - \frac{y}{z - 0.5 \cdot \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\frac{y}{\mathsf{fma}\left(\frac{y}{2}, \frac{-t}{z}, z\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2) z))))

  (- x (/ (* (* y 2) z) (- (* (* z 2) z) (* y t)))))