Average Error: 21.2 → 0.6
Time: 1.8s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.375014865247395747540068707733175756025 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.065884434570755342390558398880505157434 \cdot 10^{46}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.375014865247395747540068707733175756025 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 3.065884434570755342390558398880505157434 \cdot 10^{46}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r547396 = x;
        double r547397 = r547396 * r547396;
        double r547398 = y;
        double r547399 = r547397 + r547398;
        double r547400 = sqrt(r547399);
        return r547400;
}
↓
double f(double x, double y) {
        double r547401 = x;
        double r547402 = -1.3750148652473957e+154;
        bool r547403 = r547401 <= r547402;
        double r547404 = 0.5;
        double r547405 = y;
        double r547406 = r547405 / r547401;
        double r547407 = r547404 * r547406;
        double r547408 = r547401 + r547407;
        double r547409 = -r547408;
        double r547410 = 3.0658844345707553e+46;
        bool r547411 = r547401 <= r547410;
        double r547412 = r547401 * r547401;
        double r547413 = r547412 + r547405;
        double r547414 = sqrt(r547413);
        double r547415 = r547411 ? r547414 : r547408;
        double r547416 = r547403 ? r547409 : r547415;
        return r547416;
}

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
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3750148652473957e+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 -1.3750148652473957e+154 < x < 3.0658844345707553e+46

    1. Initial program 0.0

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

    if 3.0658844345707553e+46 < x

    1. Initial program 39.7

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

    2. Taylor expanded around inf 2.3

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.375014865247395747540068707733175756025 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.065884434570755342390558398880505157434 \cdot 10^{46}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))