Average Error: 20.9 → 0.3
Time: 12.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33039994920999637206017606321533586726 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 6.063771965228404863100273443341838455211 \cdot 10^{84}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33039994920999637206017606321533586726 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 6.063771965228404863100273443341838455211 \cdot 10^{84}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r426187 = x;
        double r426188 = r426187 * r426187;
        double r426189 = y;
        double r426190 = r426188 + r426189;
        double r426191 = sqrt(r426190);
        return r426191;
}
↓
double f(double x, double y) {
        double r426192 = x;
        double r426193 = -1.3303999492099964e+154;
        bool r426194 = r426192 <= r426193;
        double r426195 = y;
        double r426196 = r426195 / r426192;
        double r426197 = -0.5;
        double r426198 = r426196 * r426197;
        double r426199 = r426198 - r426192;
        double r426200 = 6.063771965228405e+84;
        bool r426201 = r426192 <= r426200;
        double r426202 = r426192 * r426192;
        double r426203 = r426202 + r426195;
        double r426204 = sqrt(r426203);
        double r426205 = 0.5;
        double r426206 = r426205 * r426196;
        double r426207 = r426192 + r426206;
        double r426208 = r426201 ? r426204 : r426207;
        double r426209 = r426194 ? r426199 : r426208;
        return r426209;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.9
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3303999492099964e+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 -1.3303999492099964e+154 < x < 6.063771965228405e+84

    1. Initial program 0.0

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

    if 6.063771965228405e+84 < x

    1. Initial program 43.9

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33039994920999637206017606321533586726 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 6.063771965228404863100273443341838455211 \cdot 10^{84}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))