Average Error: 21.8 → 0.1
Time: 3.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33806932031074013067314768498084378419 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 6.747088751163897916312969619772228458059 \cdot 10^{113}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33806932031074013067314768498084378419 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 6.747088751163897916312969619772228458059 \cdot 10^{113}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r355246 = x;
        double r355247 = r355246 * r355246;
        double r355248 = y;
        double r355249 = r355247 + r355248;
        double r355250 = sqrt(r355249);
        return r355250;
}
↓
double f(double x, double y) {
        double r355251 = x;
        double r355252 = -1.3380693203107401e+154;
        bool r355253 = r355251 <= r355252;
        double r355254 = 0.5;
        double r355255 = y;
        double r355256 = r355255 / r355251;
        double r355257 = r355254 * r355256;
        double r355258 = r355251 + r355257;
        double r355259 = -r355258;
        double r355260 = 6.747088751163898e+113;
        bool r355261 = r355251 <= r355260;
        double r355262 = r355251 * r355251;
        double r355263 = r355262 + r355255;
        double r355264 = sqrt(r355263);
        double r355265 = r355261 ? r355264 : r355258;
        double r355266 = r355253 ? r355259 : r355265;
        return r355266;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.8
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3380693203107401e+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 -1.3380693203107401e+154 < x < 6.747088751163898e+113

    1. Initial program 0.0

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

    if 6.747088751163898e+113 < x

    1. Initial program 50.1

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

    2. Taylor expanded around inf 0.5

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33806932031074013067314768498084378419 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 6.747088751163897916312969619772228458059 \cdot 10^{113}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 1978988140 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.5823995511225407e57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

  (sqrt (+ (* x x) y)))