Average Error: 21.7 → 0.0
Time: 1.6s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.239102867687965121108359501827503075543 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\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 r461626 = x;
        double r461627 = r461626 * r461626;
        double r461628 = y;
        double r461629 = r461627 + r461628;
        double r461630 = sqrt(r461629);
        return r461630;
}
↓
double f(double x, double y) {
        double r461631 = x;
        double r461632 = -1.3305299901761994e+154;
        bool r461633 = r461631 <= r461632;
        double r461634 = 0.5;
        double r461635 = y;
        double r461636 = r461635 / r461631;
        double r461637 = r461634 * r461636;
        double r461638 = r461631 + r461637;
        double r461639 = -r461638;
        double r461640 = 1.2391028676879651e+132;
        bool r461641 = r461631 <= r461640;
        double r461642 = r461631 * r461631;
        double r461643 = r461642 + r461635;
        double r461644 = sqrt(r461643);
        double r461645 = r461641 ? r461644 : r461638;
        double r461646 = r461633 ? r461639 : r461645;
        return r461646;
}

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

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

    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

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.5823995511225407e57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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