Average Error: 21.2 → 0.5
Time: 1.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.380447448834878 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.56561154983368585 \cdot 10^{62}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.380447448834878 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.56561154983368585 \cdot 10^{62}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r596830 = x;
        double r596831 = r596830 * r596830;
        double r596832 = y;
        double r596833 = r596831 + r596832;
        double r596834 = sqrt(r596833);
        return r596834;
}
↓
double f(double x, double y) {
        double r596835 = x;
        double r596836 = -1.380447448834878e+154;
        bool r596837 = r596835 <= r596836;
        double r596838 = 0.5;
        double r596839 = y;
        double r596840 = r596839 / r596835;
        double r596841 = r596838 * r596840;
        double r596842 = r596835 + r596841;
        double r596843 = -r596842;
        double r596844 = 1.5656115498336859e+62;
        bool r596845 = r596835 <= r596844;
        double r596846 = r596835 * r596835;
        double r596847 = r596846 + r596839;
        double r596848 = sqrt(r596847);
        double r596849 = r596845 ? r596848 : r596842;
        double r596850 = r596837 ? r596843 : r596849;
        return r596850;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.2
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.380447448834878e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    if -1.380447448834878e+154 < x < 1.5656115498336859e+62

    1. Initial program 0.0

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

    if 1.5656115498336859e+62 < x

    1. Initial program 41.5

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.380447448834878 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.56561154983368585 \cdot 10^{62}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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)))