Average Error: 21.5 → 0.0
Time: 3.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.352450667018653376213794623470338645249 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.090307459938937382017016771222292661016 \cdot 10^{121}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.352450667018653376213794623470338645249 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.090307459938937382017016771222292661016 \cdot 10^{121}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\

\end{array}
double f(double x, double y) {
        double r423911 = x;
        double r423912 = r423911 * r423911;
        double r423913 = y;
        double r423914 = r423912 + r423913;
        double r423915 = sqrt(r423914);
        return r423915;
}
↓
double f(double x, double y) {
        double r423916 = x;
        double r423917 = -1.3524506670186534e+154;
        bool r423918 = r423916 <= r423917;
        double r423919 = 0.5;
        double r423920 = y;
        double r423921 = r423920 / r423916;
        double r423922 = r423919 * r423921;
        double r423923 = r423916 + r423922;
        double r423924 = -r423923;
        double r423925 = 1.0903074599389374e+121;
        bool r423926 = r423916 <= r423925;
        double r423927 = r423916 * r423916;
        double r423928 = r423927 + r423920;
        double r423929 = sqrt(r423928);
        double r423930 = r423926 ? r423929 : r423923;
        double r423931 = r423918 ? r423924 : r423930;
        return r423931;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.5
Target0.5
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3524506670186534e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

    if -1.3524506670186534e+154 < x < 1.0903074599389374e+121

    1. Initial program 0.0

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

    if 1.0903074599389374e+121 < x

    1. Initial program 54.0

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

    2. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.352450667018653376213794623470338645249 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.090307459938937382017016771222292661016 \cdot 10^{121}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.5823995511225407e57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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