Average Error: 21.4 → 0.4
Time: 17.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.352540539223698945459720582374994240123 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 4.721377972444830028288106556093954402552 \cdot 10^{71}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.352540539223698945459720582374994240123 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 4.721377972444830028288106556093954402552 \cdot 10^{71}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r357753 = x;
        double r357754 = r357753 * r357753;
        double r357755 = y;
        double r357756 = r357754 + r357755;
        double r357757 = sqrt(r357756);
        return r357757;
}
↓
double f(double x, double y) {
        double r357758 = x;
        double r357759 = -1.352540539223699e+154;
        bool r357760 = r357758 <= r357759;
        double r357761 = y;
        double r357762 = r357761 / r357758;
        double r357763 = -0.5;
        double r357764 = r357762 * r357763;
        double r357765 = r357764 - r357758;
        double r357766 = 4.72137797244483e+71;
        bool r357767 = r357758 <= r357766;
        double r357768 = r357758 * r357758;
        double r357769 = r357768 + r357761;
        double r357770 = sqrt(r357769);
        double r357771 = 0.5;
        double r357772 = r357771 * r357762;
        double r357773 = r357758 + r357772;
        double r357774 = r357767 ? r357770 : r357773;
        double r357775 = r357760 ? r357765 : r357774;
        return r357775;
}

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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.352540539223699e+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 -1.352540539223699e+154 < x < 4.72137797244483e+71

    1. Initial program 0.0

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

    if 4.72137797244483e+71 < x

    1. Initial program 42.1

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

    2. Taylor expanded around inf 1.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.352540539223698945459720582374994240123 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 4.721377972444830028288106556093954402552 \cdot 10^{71}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))