Average Error: 21.4 → 0.3
Time: 853.0ms
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.28151121097985566 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.3825854527583296 \cdot 10^{81}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.28151121097985566 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 3.3825854527583296 \cdot 10^{81}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r507198 = x;
        double r507199 = r507198 * r507198;
        double r507200 = y;
        double r507201 = r507199 + r507200;
        double r507202 = sqrt(r507201);
        return r507202;
}
↓
double f(double x, double y) {
        double r507203 = x;
        double r507204 = -1.2815112109798557e+154;
        bool r507205 = r507203 <= r507204;
        double r507206 = 0.5;
        double r507207 = y;
        double r507208 = r507207 / r507203;
        double r507209 = r507206 * r507208;
        double r507210 = r507203 + r507209;
        double r507211 = -r507210;
        double r507212 = 3.3825854527583296e+81;
        bool r507213 = r507203 <= r507212;
        double r507214 = r507203 * r507203;
        double r507215 = r507214 + r507207;
        double r507216 = sqrt(r507215);
        double r507217 = r507213 ? r507216 : r507210;
        double r507218 = r507205 ? r507211 : r507217;
        return r507218;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2815112109798557e+154

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 0.0

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

    if -1.2815112109798557e+154 < x < 3.3825854527583296e+81

    1. Initial program 0.0

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

    if 3.3825854527583296e+81 < x

    1. Initial program 44.2

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

    2. Taylor expanded around inf 1.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.28151121097985566 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.3825854527583296 \cdot 10^{81}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))