Average Error: 21.4 → 0.6
Time: 1.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.2288156672747498 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.2801916827935602 \cdot 10^{51}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.2288156672747498 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.2801916827935602 \cdot 10^{51}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r458392 = x;
        double r458393 = r458392 * r458392;
        double r458394 = y;
        double r458395 = r458393 + r458394;
        double r458396 = sqrt(r458395);
        return r458396;
}
↓
double f(double x, double y) {
        double r458397 = x;
        double r458398 = -1.2288156672747498e+154;
        bool r458399 = r458397 <= r458398;
        double r458400 = 0.5;
        double r458401 = y;
        double r458402 = r458401 / r458397;
        double r458403 = r458400 * r458402;
        double r458404 = r458397 + r458403;
        double r458405 = -r458404;
        double r458406 = 1.2801916827935602e+51;
        bool r458407 = r458397 <= r458406;
        double r458408 = r458397 * r458397;
        double r458409 = r458408 + r458401;
        double r458410 = sqrt(r458409);
        double r458411 = r458407 ? r458410 : r458404;
        double r458412 = r458399 ? r458405 : r458411;
        return r458412;
}

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
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2288156672747498e+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 -1.2288156672747498e+154 < x < 1.2801916827935602e+51

    1. Initial program 0.0

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

    if 1.2801916827935602e+51 < x

    1. Initial program 39.4

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

    2. Taylor expanded around inf 2.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2288156672747498 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.2801916827935602 \cdot 10^{51}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))