Average Error: 21.8 → 0.2
Time: 6.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.359438392215608298210869874698329828323 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.772185700281653014225614062032486953916 \cdot 10^{80}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.359438392215608298210869874698329828323 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 2.772185700281653014225614062032486953916 \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 r442046 = x;
        double r442047 = r442046 * r442046;
        double r442048 = y;
        double r442049 = r442047 + r442048;
        double r442050 = sqrt(r442049);
        return r442050;
}
↓
double f(double x, double y) {
        double r442051 = x;
        double r442052 = -1.3594383922156083e+154;
        bool r442053 = r442051 <= r442052;
        double r442054 = 0.5;
        double r442055 = y;
        double r442056 = r442055 / r442051;
        double r442057 = r442054 * r442056;
        double r442058 = r442051 + r442057;
        double r442059 = -r442058;
        double r442060 = 2.772185700281653e+80;
        bool r442061 = r442051 <= r442060;
        double r442062 = r442051 * r442051;
        double r442063 = r442062 + r442055;
        double r442064 = sqrt(r442063);
        double r442065 = r442061 ? r442064 : r442058;
        double r442066 = r442053 ? r442059 : r442065;
        return r442066;
}

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
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3594383922156083e+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 -1.3594383922156083e+154 < x < 2.772185700281653e+80

    1. Initial program 0.0

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

    if 2.772185700281653e+80 < x

    1. Initial program 44.1

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

    2. Taylor expanded around inf 0.8

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.359438392215608298210869874698329828323 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.772185700281653014225614062032486953916 \cdot 10^{80}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(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)))