Average Error: 21.5 → 0.1
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.37154299719310846 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.97039941263825195 \cdot 10^{110}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.37154299719310846 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 2.97039941263825195 \cdot 10^{110}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r326129 = x;
        double r326130 = r326129 * r326129;
        double r326131 = y;
        double r326132 = r326130 + r326131;
        double r326133 = sqrt(r326132);
        return r326133;
}
↓
double f(double x, double y) {
        double r326134 = x;
        double r326135 = -1.3715429971931085e+154;
        bool r326136 = r326134 <= r326135;
        double r326137 = 0.5;
        double r326138 = y;
        double r326139 = r326138 / r326134;
        double r326140 = r326137 * r326139;
        double r326141 = r326134 + r326140;
        double r326142 = -r326141;
        double r326143 = 2.970399412638252e+110;
        bool r326144 = r326134 <= r326143;
        double r326145 = r326134 * r326134;
        double r326146 = r326145 + r326138;
        double r326147 = sqrt(r326146);
        double r326148 = r326144 ? r326147 : r326141;
        double r326149 = r326136 ? r326142 : r326148;
        return r326149;
}

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 < -1.3715429971931085e+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 -1.3715429971931085e+154 < x < 2.970399412638252e+110

    1. Initial program 0.0

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

    if 2.970399412638252e+110 < x

    1. Initial program 49.8

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

    2. Taylor expanded around inf 0.6

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.37154299719310846 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.97039941263825195 \cdot 10^{110}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(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)))