Average Error: 20.7 → 0.0
Time: 6.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.321177529973866349487419790268642426336 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.785490461401806573968894092684210024398 \cdot 10^{149}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{1}{2}}{x} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.321177529973866349487419790268642426336 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 1.785490461401806573968894092684210024398 \cdot 10^{149}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r321055 = x;
        double r321056 = r321055 * r321055;
        double r321057 = y;
        double r321058 = r321056 + r321057;
        double r321059 = sqrt(r321058);
        return r321059;
}
↓
double f(double x, double y) {
        double r321060 = x;
        double r321061 = -1.3211775299738663e+154;
        bool r321062 = r321060 <= r321061;
        double r321063 = -0.5;
        double r321064 = y;
        double r321065 = r321060 / r321064;
        double r321066 = r321063 / r321065;
        double r321067 = r321066 - r321060;
        double r321068 = 1.7854904614018066e+149;
        bool r321069 = r321060 <= r321068;
        double r321070 = r321060 * r321060;
        double r321071 = r321070 + r321064;
        double r321072 = sqrt(r321071);
        double r321073 = 0.5;
        double r321074 = r321064 * r321073;
        double r321075 = r321074 / r321060;
        double r321076 = r321075 + r321060;
        double r321077 = r321069 ? r321072 : r321076;
        double r321078 = r321062 ? r321067 : r321077;
        return r321078;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.7
Target0.6
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3211775299738663e+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

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{\frac{x}{y}} - x}\]

    if -1.3211775299738663e+154 < x < 1.7854904614018066e+149

    1. Initial program 0.0

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

    if 1.7854904614018066e+149 < x

    1. Initial program 61.8

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.321177529973866349487419790268642426336 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.785490461401806573968894092684210024398 \cdot 10^{149}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{1}{2}}{x} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :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)))