Average Error: 21.3 → 0.4
Time: 7.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r537914 = x;
        double r537915 = r537914 * r537914;
        double r537916 = y;
        double r537917 = r537915 + r537916;
        double r537918 = sqrt(r537917);
        return r537918;
}
↓
double f(double x, double y) {
        double r537919 = x;
        double r537920 = -1.3778733035656446e+154;
        bool r537921 = r537919 <= r537920;
        double r537922 = y;
        double r537923 = r537922 / r537919;
        double r537924 = -0.5;
        double r537925 = r537923 * r537924;
        double r537926 = r537925 - r537919;
        double r537927 = 1.2922566123944575e+80;
        bool r537928 = r537919 <= r537927;
        double r537929 = r537919 * r537919;
        double r537930 = r537929 + r537922;
        double r537931 = sqrt(r537930);
        double r537932 = 0.5;
        double r537933 = r537932 * r537923;
        double r537934 = r537919 + r537933;
        double r537935 = r537928 ? r537931 : r537934;
        double r537936 = r537921 ? r537926 : r537935;
        return r537936;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.3
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3778733035656446e+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

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

    if -1.3778733035656446e+154 < x < 1.2922566123944575e+80

    1. Initial program 0.0

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

    if 1.2922566123944575e+80 < x

    1. Initial program 44.1

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

    2. Taylor expanded around inf 1.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))