Average Error: 21.5 → 0.1
Time: 1.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33994248806749521 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.30747845872698201 \cdot 10^{114}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33994248806749521 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.30747845872698201 \cdot 10^{114}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r502001 = x;
        double r502002 = r502001 * r502001;
        double r502003 = y;
        double r502004 = r502002 + r502003;
        double r502005 = sqrt(r502004);
        return r502005;
}
↓
double f(double x, double y) {
        double r502006 = x;
        double r502007 = -1.3399424880674952e+154;
        bool r502008 = r502006 <= r502007;
        double r502009 = 0.5;
        double r502010 = y;
        double r502011 = r502010 / r502006;
        double r502012 = r502009 * r502011;
        double r502013 = r502006 + r502012;
        double r502014 = -r502013;
        double r502015 = 1.307478458726982e+114;
        bool r502016 = r502006 <= r502015;
        double r502017 = r502006 * r502006;
        double r502018 = r502017 + r502010;
        double r502019 = sqrt(r502018);
        double r502020 = r502016 ? r502019 : r502013;
        double r502021 = r502008 ? r502014 : r502020;
        return r502021;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.5
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3399424880674952e+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 -1.3399424880674952e+154 < x < 1.307478458726982e+114

    1. Initial program 0.0

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

    if 1.307478458726982e+114 < x

    1. Initial program 50.5

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

    2. Taylor expanded around inf 0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33994248806749521 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.30747845872698201 \cdot 10^{114}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))