Average Error: 21.7 → 0.2
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3302483288514575 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.2824828758844473 \cdot 10^{94}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3302483288514575 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.2824828758844473 \cdot 10^{94}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r569236 = x;
        double r569237 = r569236 * r569236;
        double r569238 = y;
        double r569239 = r569237 + r569238;
        double r569240 = sqrt(r569239);
        return r569240;
}
↓
double f(double x, double y) {
        double r569241 = x;
        double r569242 = -1.3302483288514575e+154;
        bool r569243 = r569241 <= r569242;
        double r569244 = 0.5;
        double r569245 = y;
        double r569246 = r569245 / r569241;
        double r569247 = r569244 * r569246;
        double r569248 = r569241 + r569247;
        double r569249 = -r569248;
        double r569250 = 1.2824828758844473e+94;
        bool r569251 = r569241 <= r569250;
        double r569252 = r569241 * r569241;
        double r569253 = r569252 + r569245;
        double r569254 = sqrt(r569253);
        double r569255 = r569251 ? r569254 : r569248;
        double r569256 = r569243 ? r569249 : r569255;
        return r569256;
}

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
Target0.6
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3302483288514575e+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 -1.3302483288514575e+154 < x < 1.2824828758844473e+94

    1. Initial program 0.0

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

    if 1.2824828758844473e+94 < x

    1. Initial program 47.9

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

    2. Taylor expanded around inf 0.9

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3302483288514575 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.2824828758844473 \cdot 10^{94}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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)))