Average Error: 21.1 → 0.2
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r603305 = x;
        double r603306 = r603305 * r603305;
        double r603307 = y;
        double r603308 = r603306 + r603307;
        double r603309 = sqrt(r603308);
        return r603309;
}
↓
double f(double x, double y) {
        double r603310 = x;
        double r603311 = -1.3694640831062883e+154;
        bool r603312 = r603310 <= r603311;
        double r603313 = 0.5;
        double r603314 = y;
        double r603315 = r603314 / r603310;
        double r603316 = r603313 * r603315;
        double r603317 = r603310 + r603316;
        double r603318 = -r603317;
        double r603319 = 8.439081781731016e+104;
        bool r603320 = r603310 <= r603319;
        double r603321 = r603310 * r603310;
        double r603322 = r603321 + r603314;
        double r603323 = sqrt(r603322);
        double r603324 = r603320 ? r603323 : r603317;
        double r603325 = r603312 ? r603318 : r603324;
        return r603325;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.1
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3694640831062883e+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 -1.3694640831062883e+154 < x < 8.439081781731016e+104

    1. Initial program 0.0

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

    if 8.439081781731016e+104 < x

    1. Initial program 49.3

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

    2. Taylor expanded around inf 0.9

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))