Average Error: 21.4 → 0.0
Time: 1.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.34008686729452166 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.7383737548077102 \cdot 10^{147}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.34008686729452166 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.7383737548077102 \cdot 10^{147}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r498635 = x;
        double r498636 = r498635 * r498635;
        double r498637 = y;
        double r498638 = r498636 + r498637;
        double r498639 = sqrt(r498638);
        return r498639;
}
↓
double f(double x, double y) {
        double r498640 = x;
        double r498641 = -1.3400868672945217e+154;
        bool r498642 = r498640 <= r498641;
        double r498643 = 0.5;
        double r498644 = y;
        double r498645 = r498644 / r498640;
        double r498646 = r498643 * r498645;
        double r498647 = r498640 + r498646;
        double r498648 = -r498647;
        double r498649 = 1.73837375480771e+147;
        bool r498650 = r498640 <= r498649;
        double r498651 = r498640 * r498640;
        double r498652 = r498651 + r498644;
        double r498653 = sqrt(r498652);
        double r498654 = r498650 ? r498653 : r498647;
        double r498655 = r498642 ? r498648 : r498654;
        return r498655;
}

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
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3400868672945217e+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 -1.3400868672945217e+154 < x < 1.73837375480771e+147

    1. Initial program 0.0

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

    if 1.73837375480771e+147 < x

    1. Initial program 61.2

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

    2. Taylor expanded around inf 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.34008686729452166 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.7383737548077102 \cdot 10^{147}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)))