Average Error: 31.4 → 17.2
Time: 3.6s
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 r1125804 = x;
        double r1125805 = r1125804 * r1125804;
        double r1125806 = y;
        double r1125807 = r1125806 * r1125806;
        double r1125808 = r1125805 + r1125807;
        double r1125809 = sqrt(r1125808);
        return r1125809;
}

double f(double x, double y) {
        double r1125810 = x;
        double r1125811 = -3.3032099476572043e+84;
        bool r1125812 = r1125810 <= r1125811;
        double r1125813 = -r1125810;
        double r1125814 = 1.943934747643532e+128;
        bool r1125815 = r1125810 <= r1125814;
        double r1125816 = r1125810 * r1125810;
        double r1125817 = y;
        double r1125818 = r1125817 * r1125817;
        double r1125819 = r1125816 + r1125818;
        double r1125820 = sqrt(r1125819);
        double r1125821 = r1125815 ? r1125820 : r1125810;
        double r1125822 = r1125812 ? r1125813 : r1125821;
        return r1125822;
}

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))))