Average Error: 32.2 → 18.4
Time: 1.9s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 2.2436091775473112 \cdot 10^{-248}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 6.3015272029718245 \cdot 10^{96}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.2696195727379345 \cdot 10^{139}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le -3.5543765182763856 \cdot 10^{-161}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le 2.2436091775473112 \cdot 10^{-248}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 6.3015272029718245 \cdot 10^{96}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y) {
        double r895855 = x;
        double r895856 = r895855 * r895855;
        double r895857 = y;
        double r895858 = r895857 * r895857;
        double r895859 = r895856 + r895858;
        double r895860 = sqrt(r895859);
        return r895860;
}
↓
double f(double x, double y) {
        double r895861 = x;
        double r895862 = -4.2696195727379345e+139;
        bool r895863 = r895861 <= r895862;
        double r895864 = -r895861;
        double r895865 = -3.5543765182763856e-161;
        bool r895866 = r895861 <= r895865;
        double r895867 = r895861 * r895861;
        double r895868 = y;
        double r895869 = r895868 * r895868;
        double r895870 = r895867 + r895869;
        double r895871 = sqrt(r895870);
        double r895872 = 2.243609177547311e-248;
        bool r895873 = r895861 <= r895872;
        double r895874 = 6.3015272029718245e+96;
        bool r895875 = r895861 <= r895874;
        double r895876 = r895875 ? r895871 : r895861;
        double r895877 = r895873 ? r895868 : r895876;
        double r895878 = r895866 ? r895871 : r895877;
        double r895879 = r895863 ? r895864 : r895878;
        return r895879;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.2
Target18.2
Herbie18.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -4.2696195727379345e+139

    1. Initial program 59.5

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around -inf 8.4

      \[\leadsto \color{blue}{-1 \cdot x}\]

    3. Simplified8.4

      \[\leadsto \color{blue}{-x}\]

    if -4.2696195727379345e+139 < x < -3.5543765182763856e-161 or 2.243609177547311e-248 < x < 6.3015272029718245e+96

    1. Initial program 18.8

      \[\sqrt{x \cdot x + y \cdot y}\]

    if -3.5543765182763856e-161 < x < 2.243609177547311e-248

    1. Initial program 32.3

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around 0 33.8

      \[\leadsto \color{blue}{y}\]

    if 6.3015272029718245e+96 < x

    1. Initial program 51.2

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around inf 10.7

      \[\leadsto \color{blue}{x}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 2.2436091775473112 \cdot 10^{-248}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 6.3015272029718245 \cdot 10^{96}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.123695082659983e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

  (sqrt (+ (* x x) (* y y))))