Average Error: 21.1 → 0.1
Time: 22.8s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r324000 = x;
        double r324001 = r324000 * r324000;
        double r324002 = y;
        double r324003 = r324001 + r324002;
        double r324004 = sqrt(r324003);
        return r324004;
}
↓
double f(double x, double y) {
        double r324005 = x;
        double r324006 = -1.3349869326014939e+154;
        bool r324007 = r324005 <= r324006;
        double r324008 = y;
        double r324009 = r324008 / r324005;
        double r324010 = -0.5;
        double r324011 = r324009 * r324010;
        double r324012 = r324011 - r324005;
        double r324013 = 1.4388934535207275e+123;
        bool r324014 = r324005 <= r324013;
        double r324015 = r324005 * r324005;
        double r324016 = r324015 + r324008;
        double r324017 = sqrt(r324016);
        double r324018 = 0.5;
        double r324019 = r324018 * r324009;
        double r324020 = r324005 + r324019;
        double r324021 = r324014 ? r324017 : r324020;
        double r324022 = r324007 ? r324012 : r324021;
        return r324022;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.1
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 < -1.3349869326014939e+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 -1.3349869326014939e+154 < x < 1.4388934535207275e+123

    1. Initial program 0.0

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

    if 1.4388934535207275e+123 < x

    1. Initial program 53.4

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

    2. Taylor expanded around inf 0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))