Average Error: 21.4 → 0.0
Time: 2.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.319971728264327140890396761282936412133 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.027374157723114338036480360220146126719 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.319971728264327140890396761282936412133 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 2.027374157723114338036480360220146126719 \cdot 10^{123}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r488604 = x;
        double r488605 = r488604 * r488604;
        double r488606 = y;
        double r488607 = r488605 + r488606;
        double r488608 = sqrt(r488607);
        return r488608;
}
↓
double f(double x, double y) {
        double r488609 = x;
        double r488610 = -1.3199717282643271e+154;
        bool r488611 = r488609 <= r488610;
        double r488612 = 0.5;
        double r488613 = y;
        double r488614 = r488613 / r488609;
        double r488615 = r488612 * r488614;
        double r488616 = r488609 + r488615;
        double r488617 = -r488616;
        double r488618 = 2.0273741577231143e+123;
        bool r488619 = r488609 <= r488618;
        double r488620 = r488609 * r488609;
        double r488621 = r488620 + r488613;
        double r488622 = sqrt(r488621);
        double r488623 = r488619 ? r488622 : r488616;
        double r488624 = r488611 ? r488617 : r488623;
        return r488624;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.4
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 < -1.3199717282643271e+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 -1.3199717282643271e+154 < x < 2.0273741577231143e+123

    1. Initial program 0.0

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

    if 2.0273741577231143e+123 < x

    1. Initial program 54.3

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

    2. Taylor expanded around inf 0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.319971728264327140890396761282936412133 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.027374157723114338036480360220146126719 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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)))