Average Error: 21.3 → 0.4
Time: 10.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r634248 = x;
        double r634249 = r634248 * r634248;
        double r634250 = y;
        double r634251 = r634249 + r634250;
        double r634252 = sqrt(r634251);
        return r634252;
}
↓
double f(double x, double y) {
        double r634253 = x;
        double r634254 = -1.3778733035656446e+154;
        bool r634255 = r634253 <= r634254;
        double r634256 = y;
        double r634257 = r634256 / r634253;
        double r634258 = -0.5;
        double r634259 = r634257 * r634258;
        double r634260 = r634259 - r634253;
        double r634261 = 1.2922566123944575e+80;
        bool r634262 = r634253 <= r634261;
        double r634263 = r634253 * r634253;
        double r634264 = r634263 + r634256;
        double r634265 = sqrt(r634264);
        double r634266 = 0.5;
        double r634267 = r634266 * r634257;
        double r634268 = r634253 + r634267;
        double r634269 = r634262 ? r634265 : r634268;
        double r634270 = r634255 ? r634260 : r634269;
        return r634270;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.3
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3778733035656446e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

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

    3. Simplified0

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{-1}{2} - x}\]

    if -1.3778733035656446e+154 < x < 1.2922566123944575e+80

    1. Initial program 0.0

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

    if 1.2922566123944575e+80 < x

    1. Initial program 44.1

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

    2. Taylor expanded around inf 1.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))