Average Error: 21.7 → 0.0
Time: 17.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.239102867687965121108359501827503075543 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.239102867687965121108359501827503075543 \cdot 10^{132}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r347051 = x;
        double r347052 = r347051 * r347051;
        double r347053 = y;
        double r347054 = r347052 + r347053;
        double r347055 = sqrt(r347054);
        return r347055;
}
↓
double f(double x, double y) {
        double r347056 = x;
        double r347057 = -1.3305299901761994e+154;
        bool r347058 = r347056 <= r347057;
        double r347059 = y;
        double r347060 = r347059 / r347056;
        double r347061 = -0.5;
        double r347062 = r347060 * r347061;
        double r347063 = r347062 - r347056;
        double r347064 = 1.2391028676879651e+132;
        bool r347065 = r347056 <= r347064;
        double r347066 = r347056 * r347056;
        double r347067 = r347066 + r347059;
        double r347068 = sqrt(r347067);
        double r347069 = 0.5;
        double r347070 = r347069 * r347060;
        double r347071 = r347056 + r347070;
        double r347072 = r347065 ? r347068 : r347071;
        double r347073 = r347058 ? r347063 : r347072;
        return r347073;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.7
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 < -1.3305299901761994e+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 -1.3305299901761994e+154 < x < 1.2391028676879651e+132

    1. Initial program 0.0

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

    if 1.2391028676879651e+132 < x

    1. Initial program 56.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.239102867687965121108359501827503075543 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))