Average Error: 31.4 → 17.2
Time: 9.6s
Precision: 64
\[\sqrt{{x}^{2} + {y}^{2}}\]
\[\begin{array}{l} \mathbf{if}\;x \le -3.303209947657204342880384766533788360025 \cdot 10^{84}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 1.943934747643531875760607600817076209382 \cdot 10^{128}:\\ \;\;\;\;\sqrt{{x}^{2} + {y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{{x}^{2} + {y}^{2}}
\begin{array}{l}
\mathbf{if}\;x \le -3.303209947657204342880384766533788360025 \cdot 10^{84}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le 1.943934747643531875760607600817076209382 \cdot 10^{128}:\\
\;\;\;\;\sqrt{{x}^{2} + {y}^{2}}\\

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

\end{array}
double f(double x, double y) {
        double r928526 = x;
        double r928527 = 2.0;
        double r928528 = pow(r928526, r928527);
        double r928529 = y;
        double r928530 = pow(r928529, r928527);
        double r928531 = r928528 + r928530;
        double r928532 = sqrt(r928531);
        return r928532;
}

double f(double x, double y) {
        double r928533 = x;
        double r928534 = -3.3032099476572043e+84;
        bool r928535 = r928533 <= r928534;
        double r928536 = -r928533;
        double r928537 = 1.943934747643532e+128;
        bool r928538 = r928533 <= r928537;
        double r928539 = 2.0;
        double r928540 = pow(r928533, r928539);
        double r928541 = y;
        double r928542 = pow(r928541, r928539);
        double r928543 = r928540 + r928542;
        double r928544 = sqrt(r928543);
        double r928545 = r928538 ? r928544 : r928533;
        double r928546 = r928535 ? r928536 : r928545;
        return r928546;
}

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

      \[\sqrt{{x}^{2} + {y}^{2}}\]
    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

      \[\sqrt{{x}^{2} + {y}^{2}}\]

    if 1.943934747643532e+128 < x

    1. Initial program 57.0

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

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.303209947657204342880384766533788360025 \cdot 10^{84}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le 1.943934747643531875760607600817076209382 \cdot 10^{128}:\\ \;\;\;\;\sqrt{{x}^{2} + {y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 1 
(FPCore (x y)
  :name "sqrt(x^2+y^2)"
  :precision binary64
  (sqrt (+ (pow x 2) (pow y 2))))