Average Error: 32.2 → 18.4
Time: 2.3s
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 r819504 = x;
        double r819505 = r819504 * r819504;
        double r819506 = y;
        double r819507 = r819506 * r819506;
        double r819508 = r819505 + r819507;
        double r819509 = sqrt(r819508);
        return r819509;
}
↓
double f(double x, double y) {
        double r819510 = x;
        double r819511 = -4.2696195727379345e+139;
        bool r819512 = r819510 <= r819511;
        double r819513 = -r819510;
        double r819514 = -3.5543765182763856e-161;
        bool r819515 = r819510 <= r819514;
        double r819516 = r819510 * r819510;
        double r819517 = y;
        double r819518 = r819517 * r819517;
        double r819519 = r819516 + r819518;
        double r819520 = sqrt(r819519);
        double r819521 = 2.243609177547311e-248;
        bool r819522 = r819510 <= r819521;
        double r819523 = 6.3015272029718245e+96;
        bool r819524 = r819510 <= r819523;
        double r819525 = r819524 ? r819520 : r819510;
        double r819526 = r819522 ? r819517 : r819525;
        double r819527 = r819515 ? r819520 : r819526;
        double r819528 = r819512 ? r819513 : r819527;
        return r819528;
}

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))))