Average Error: 21.8 → 0.0
Time: 1.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.358575356778544888504370454055028257768 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.447947334872172544750837332377802276979 \cdot 10^{133}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.358575356778544888504370454055028257768 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.447947334872172544750837332377802276979 \cdot 10^{133}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r736750 = x;
        double r736751 = r736750 * r736750;
        double r736752 = y;
        double r736753 = r736751 + r736752;
        double r736754 = sqrt(r736753);
        return r736754;
}
↓
double f(double x, double y) {
        double r736755 = x;
        double r736756 = -1.358575356778545e+154;
        bool r736757 = r736755 <= r736756;
        double r736758 = 0.5;
        double r736759 = y;
        double r736760 = r736759 / r736755;
        double r736761 = r736758 * r736760;
        double r736762 = r736755 + r736761;
        double r736763 = -r736762;
        double r736764 = 1.4479473348721725e+133;
        bool r736765 = r736755 <= r736764;
        double r736766 = r736755 * r736755;
        double r736767 = r736766 + r736759;
        double r736768 = sqrt(r736767);
        double r736769 = r736765 ? r736768 : r736762;
        double r736770 = r736757 ? r736763 : r736769;
        return r736770;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.8
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 < -1.358575356778545e+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 -1.358575356778545e+154 < x < 1.4479473348721725e+133

    1. Initial program 0.0

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

    if 1.4479473348721725e+133 < x

    1. Initial program 56.6

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

    2. Taylor expanded around inf 0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.358575356778544888504370454055028257768 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.447947334872172544750837332377802276979 \cdot 10^{133}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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