Average Error: 21.0 → 0.2
Time: 12.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r310667 = x;
        double r310668 = r310667 * r310667;
        double r310669 = y;
        double r310670 = r310668 + r310669;
        double r310671 = sqrt(r310670);
        return r310671;
}
↓
double f(double x, double y) {
        double r310672 = x;
        double r310673 = -1.3221955759293222e+154;
        bool r310674 = r310672 <= r310673;
        double r310675 = y;
        double r310676 = r310675 / r310672;
        double r310677 = -0.5;
        double r310678 = r310676 * r310677;
        double r310679 = r310678 - r310672;
        double r310680 = 1.892549585482312e+97;
        bool r310681 = r310672 <= r310680;
        double r310682 = r310672 * r310672;
        double r310683 = r310682 + r310675;
        double r310684 = sqrt(r310683);
        double r310685 = 0.5;
        double r310686 = r310685 * r310676;
        double r310687 = r310672 + r310686;
        double r310688 = r310681 ? r310684 : r310687;
        double r310689 = r310674 ? r310679 : r310688;
        return r310689;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.0
Target0.5
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 < -1.3221955759293222e+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 -1.3221955759293222e+154 < x < 1.892549585482312e+97

    1. Initial program 0.0

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

    if 1.892549585482312e+97 < x

    1. Initial program 47.8

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))