Average Error: 21.3 → 0.0
Time: 9.6s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3374804328684521 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.7963572810955448 \cdot 10^{152}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3374804328684521 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.7963572810955448 \cdot 10^{152}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r519842 = x;
        double r519843 = r519842 * r519842;
        double r519844 = y;
        double r519845 = r519843 + r519844;
        double r519846 = sqrt(r519845);
        return r519846;
}
↓
double f(double x, double y) {
        double r519847 = x;
        double r519848 = -1.3374804328684521e+154;
        bool r519849 = r519847 <= r519848;
        double r519850 = y;
        double r519851 = r519850 / r519847;
        double r519852 = -0.5;
        double r519853 = r519851 * r519852;
        double r519854 = r519853 - r519847;
        double r519855 = 1.7963572810955448e+152;
        bool r519856 = r519847 <= r519855;
        double r519857 = r519847 * r519847;
        double r519858 = r519857 + r519850;
        double r519859 = sqrt(r519858);
        double r519860 = 0.5;
        double r519861 = r519860 * r519851;
        double r519862 = r519847 + r519861;
        double r519863 = r519856 ? r519859 : r519862;
        double r519864 = r519849 ? r519854 : r519863;
        return r519864;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.3
Target0.4
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3374804328684521e+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 -1.3374804328684521e+154 < x < 1.7963572810955448e+152

    1. Initial program 0.0

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

    if 1.7963572810955448e+152 < x

    1. Initial program 62.8

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

    2. Taylor expanded around inf 0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3374804328684521 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.7963572810955448 \cdot 10^{152}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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)))