Average Error: 20.9 → 0.3
Time: 3.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33039994920999637206017606321533586726 \cdot 10^{154}:\\ \;\;\;\;\frac{-\frac{y}{x}}{2} - x\\ \mathbf{elif}\;x \le 6.063771965228404863100273443341838455211 \cdot 10^{84}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{x}}{2}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33039994920999637206017606321533586726 \cdot 10^{154}:\\
\;\;\;\;\frac{-\frac{y}{x}}{2} - x\\

\mathbf{elif}\;x \le 6.063771965228404863100273443341838455211 \cdot 10^{84}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r325678 = x;
        double r325679 = r325678 * r325678;
        double r325680 = y;
        double r325681 = r325679 + r325680;
        double r325682 = sqrt(r325681);
        return r325682;
}
↓
double f(double x, double y) {
        double r325683 = x;
        double r325684 = -1.3303999492099964e+154;
        bool r325685 = r325683 <= r325684;
        double r325686 = y;
        double r325687 = r325686 / r325683;
        double r325688 = -r325687;
        double r325689 = 2.0;
        double r325690 = r325688 / r325689;
        double r325691 = r325690 - r325683;
        double r325692 = 6.063771965228405e+84;
        bool r325693 = r325683 <= r325692;
        double r325694 = r325683 * r325683;
        double r325695 = r325694 + r325686;
        double r325696 = sqrt(r325695);
        double r325697 = r325687 / r325689;
        double r325698 = r325683 + r325697;
        double r325699 = r325693 ? r325696 : r325698;
        double r325700 = r325685 ? r325691 : r325699;
        return r325700;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.9
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3303999492099964e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

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

    3. Simplified0

      \[\leadsto \color{blue}{\frac{-\frac{y}{x}}{2} - x}\]

    if -1.3303999492099964e+154 < x < 6.063771965228405e+84

    1. Initial program 0.0

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

    if 6.063771965228405e+84 < x

    1. Initial program 43.9

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

    2. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]

    3. Simplified1.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33039994920999637206017606321533586726 \cdot 10^{154}:\\ \;\;\;\;\frac{-\frac{y}{x}}{2} - x\\ \mathbf{elif}\;x \le 6.063771965228404863100273443341838455211 \cdot 10^{84}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))