Average Error: 21.7 → 0.2
Time: 10.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3300132858515627319920499059244220404 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 8.331092342254926651866005497662461340443 \cdot 10^{95}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3300132858515627319920499059244220404 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 8.331092342254926651866005497662461340443 \cdot 10^{95}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r387337 = x;
        double r387338 = r387337 * r387337;
        double r387339 = y;
        double r387340 = r387338 + r387339;
        double r387341 = sqrt(r387340);
        return r387341;
}
↓
double f(double x, double y) {
        double r387342 = x;
        double r387343 = -1.3300132858515627e+154;
        bool r387344 = r387342 <= r387343;
        double r387345 = y;
        double r387346 = r387345 / r387342;
        double r387347 = -0.5;
        double r387348 = r387346 * r387347;
        double r387349 = r387348 - r387342;
        double r387350 = 8.331092342254927e+95;
        bool r387351 = r387342 <= r387350;
        double r387352 = r387342 * r387342;
        double r387353 = r387352 + r387345;
        double r387354 = sqrt(r387353);
        double r387355 = 0.5;
        double r387356 = r387355 * r387346;
        double r387357 = r387342 + r387356;
        double r387358 = r387351 ? r387354 : r387357;
        double r387359 = r387344 ? r387349 : r387358;
        return r387359;
}

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
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3300132858515627e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{-1}{2} - x}\]

    if -1.3300132858515627e+154 < x < 8.331092342254927e+95

    1. Initial program 0.0

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

    if 8.331092342254927e+95 < x

    1. Initial program 46.8

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3300132858515627319920499059244220404 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 8.331092342254926651866005497662461340443 \cdot 10^{95}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))