Average Error: 21.1 → 0.2
Time: 12.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3584562935631266943400188789683880336 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3584562935631266943400188789683880336 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r22081686 = x;
        double r22081687 = r22081686 * r22081686;
        double r22081688 = y;
        double r22081689 = r22081687 + r22081688;
        double r22081690 = sqrt(r22081689);
        return r22081690;
}
↓
double f(double x, double y) {
        double r22081691 = x;
        double r22081692 = -1.3584562935631267e+154;
        bool r22081693 = r22081691 <= r22081692;
        double r22081694 = -0.5;
        double r22081695 = y;
        double r22081696 = r22081691 / r22081695;
        double r22081697 = r22081694 / r22081696;
        double r22081698 = r22081697 - r22081691;
        double r22081699 = 2.2470376765740678e+83;
        bool r22081700 = r22081691 <= r22081699;
        double r22081701 = r22081691 * r22081691;
        double r22081702 = r22081701 + r22081695;
        double r22081703 = sqrt(r22081702);
        double r22081704 = r22081695 / r22081691;
        double r22081705 = 0.5;
        double r22081706 = r22081704 * r22081705;
        double r22081707 = r22081706 + r22081691;
        double r22081708 = r22081700 ? r22081703 : r22081707;
        double r22081709 = r22081693 ? r22081698 : r22081708;
        return r22081709;
}

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
Target0.4
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 < -1.3584562935631267e+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 -1.3584562935631267e+154 < x < 2.2470376765740678e+83

    1. Initial program 0.0

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

    if 2.2470376765740678e+83 < x

    1. Initial program 43.9

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3584562935631266943400188789683880336 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))