Average Error: 31.5 → 17.9
Time: 12.0s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.914133431995725133760392243420299124786 \cdot 10^{-220}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.921239608950813885870067735992694561728 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le 1.914133431995725133760392243420299124786 \cdot 10^{-220}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 1.921239608950813885870067735992694561728 \cdot 10^{104}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

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

\end{array}
double f(double x, double y) {
        double r529531 = x;
        double r529532 = r529531 * r529531;
        double r529533 = y;
        double r529534 = r529533 * r529533;
        double r529535 = r529532 + r529534;
        double r529536 = sqrt(r529535);
        return r529536;
}
↓
double f(double x, double y) {
        double r529537 = x;
        double r529538 = -5.292452664608309e+126;
        bool r529539 = r529537 <= r529538;
        double r529540 = -r529537;
        double r529541 = -3.6872979313797757e-267;
        bool r529542 = r529537 <= r529541;
        double r529543 = r529537 * r529537;
        double r529544 = y;
        double r529545 = r529544 * r529544;
        double r529546 = r529543 + r529545;
        double r529547 = sqrt(r529546);
        double r529548 = 1.914133431995725e-220;
        bool r529549 = r529537 <= r529548;
        double r529550 = 1.921239608950814e+104;
        bool r529551 = r529537 <= r529550;
        double r529552 = r529551 ? r529547 : r529537;
        double r529553 = r529549 ? r529544 : r529552;
        double r529554 = r529542 ? r529547 : r529553;
        double r529555 = r529539 ? r529540 : r529554;
        return r529555;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.5
Target17.6
Herbie17.9
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659982632437974301616192301785 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.116557621183362039388201959321597704512 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -5.292452664608309e+126

    1. Initial program 57.1

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

    2. Taylor expanded around -inf 8.8

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

    3. Simplified8.8

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

    if -5.292452664608309e+126 < x < -3.6872979313797757e-267 or 1.914133431995725e-220 < x < 1.921239608950814e+104

    1. Initial program 19.7

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

    if -3.6872979313797757e-267 < x < 1.914133431995725e-220

    1. Initial program 30.3

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

    2. Taylor expanded around 0 32.9

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

    if 1.921239608950814e+104 < x

    1. Initial program 51.4

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.914133431995725133760392243420299124786 \cdot 10^{-220}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.921239608950813885870067735992694561728 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.123695082659983e145) (- x) (if (< x 1.11655762118336204e93) (sqrt (+ (* x x) (* y y))) x))

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