Average Error: 31.4 → 17.2
Time: 3.5s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
\[\begin{array}{l} \mathbf{if}\;x \le -3.303209947657204342880384766533788360025 \cdot 10^{84}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 1.943934747643531875760607600817076209382 \cdot 10^{128}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
\begin{array}{l}
\mathbf{if}\;x \le -3.303209947657204342880384766533788360025 \cdot 10^{84}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le 1.943934747643531875760607600817076209382 \cdot 10^{128}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

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

\end{array}
double f(double x, double y) {
        double r794397 = x;
        double r794398 = r794397 * r794397;
        double r794399 = y;
        double r794400 = r794399 * r794399;
        double r794401 = r794398 + r794400;
        double r794402 = sqrt(r794401);
        return r794402;
}

double f(double x, double y) {
        double r794403 = x;
        double r794404 = -3.3032099476572043e+84;
        bool r794405 = r794403 <= r794404;
        double r794406 = -r794403;
        double r794407 = 1.943934747643532e+128;
        bool r794408 = r794403 <= r794407;
        double r794409 = r794403 * r794403;
        double r794410 = y;
        double r794411 = r794410 * r794410;
        double r794412 = r794409 + r794411;
        double r794413 = sqrt(r794412);
        double r794414 = r794408 ? r794413 : r794403;
        double r794415 = r794405 ? r794406 : r794414;
        return r794415;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if x < -3.3032099476572043e+84

    1. Initial program 48.4

      \[\sqrt{x \cdot x + y \cdot y}\]
    2. Taylor expanded around -inf 10.3

      \[\leadsto \color{blue}{-1 \cdot x}\]
    3. Simplified10.3

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

    if -3.3032099476572043e+84 < x < 1.943934747643532e+128

    1. Initial program 21.1

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

    if 1.943934747643532e+128 < x

    1. Initial program 57.0

      \[\sqrt{x \cdot x + y \cdot y}\]
    2. Taylor expanded around inf 8.2

      \[\leadsto \color{blue}{x}\]
  3. Recombined 3 regimes into one program.
  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.303209947657204342880384766533788360025 \cdot 10^{84}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 1.943934747643531875760607600817076209382 \cdot 10^{128}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 1 
(FPCore (x y)
  :name "sqrt(x*x + y*y)"
  :precision binary64
  (sqrt (+ (* x x) (* y y))))