Average Error: 21.6 → 0.1
Time: 11.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.34741200166273024865549457096980522016 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 3.903488335015737711762269431799121933757 \cdot 10^{107}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.34741200166273024865549457096980522016 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 3.903488335015737711762269431799121933757 \cdot 10^{107}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r603358 = x;
        double r603359 = r603358 * r603358;
        double r603360 = y;
        double r603361 = r603359 + r603360;
        double r603362 = sqrt(r603361);
        return r603362;
}
↓
double f(double x, double y) {
        double r603363 = x;
        double r603364 = -1.3474120016627302e+154;
        bool r603365 = r603363 <= r603364;
        double r603366 = y;
        double r603367 = r603366 / r603363;
        double r603368 = -0.5;
        double r603369 = r603367 * r603368;
        double r603370 = r603369 - r603363;
        double r603371 = 3.903488335015738e+107;
        bool r603372 = r603363 <= r603371;
        double r603373 = r603363 * r603363;
        double r603374 = r603373 + r603366;
        double r603375 = sqrt(r603374);
        double r603376 = 0.5;
        double r603377 = r603376 * r603367;
        double r603378 = r603363 + r603377;
        double r603379 = r603372 ? r603375 : r603378;
        double r603380 = r603365 ? r603370 : r603379;
        return r603380;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.6
Target0.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3474120016627302e+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 -1.3474120016627302e+154 < x < 3.903488335015738e+107

    1. Initial program 0.0

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

    if 3.903488335015738e+107 < x

    1. Initial program 50.4

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

    2. Taylor expanded around inf 0.5

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.34741200166273024865549457096980522016 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 3.903488335015737711762269431799121933757 \cdot 10^{107}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))