Average Error: 21.4 → 0.1
Time: 1.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r463718 = x;
        double r463719 = r463718 * r463718;
        double r463720 = y;
        double r463721 = r463719 + r463720;
        double r463722 = sqrt(r463721);
        return r463722;
}
↓
double f(double x, double y) {
        double r463723 = x;
        double r463724 = -1.3474626627347847e+154;
        bool r463725 = r463723 <= r463724;
        double r463726 = 0.5;
        double r463727 = y;
        double r463728 = r463727 / r463723;
        double r463729 = r463726 * r463728;
        double r463730 = r463723 + r463729;
        double r463731 = -r463730;
        double r463732 = 5.291435342096596e+124;
        bool r463733 = r463723 <= r463732;
        double r463734 = r463723 * r463723;
        double r463735 = r463734 + r463727;
        double r463736 = sqrt(r463735);
        double r463737 = r463733 ? r463736 : r463730;
        double r463738 = r463725 ? r463731 : r463737;
        return r463738;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.4
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 < -1.3474626627347847e+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 -1.3474626627347847e+154 < x < 5.291435342096596e+124

    1. Initial program 0.0

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

    if 5.291435342096596e+124 < x

    1. Initial program 54.2

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

    2. Taylor expanded around inf 0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))