Average Error: 21.3 → 0.2
Time: 6.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.355324462692639940669848037627778133899 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.648277919935003443657636145779230905143 \cdot 10^{92}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{1}{2}}{x} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.355324462692639940669848037627778133899 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 2.648277919935003443657636145779230905143 \cdot 10^{92}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r391948 = x;
        double r391949 = r391948 * r391948;
        double r391950 = y;
        double r391951 = r391949 + r391950;
        double r391952 = sqrt(r391951);
        return r391952;
}
↓
double f(double x, double y) {
        double r391953 = x;
        double r391954 = -1.35532446269264e+154;
        bool r391955 = r391953 <= r391954;
        double r391956 = -0.5;
        double r391957 = y;
        double r391958 = r391953 / r391957;
        double r391959 = r391956 / r391958;
        double r391960 = r391959 - r391953;
        double r391961 = 2.6482779199350034e+92;
        bool r391962 = r391953 <= r391961;
        double r391963 = r391953 * r391953;
        double r391964 = r391963 + r391957;
        double r391965 = sqrt(r391964);
        double r391966 = 0.5;
        double r391967 = r391957 * r391966;
        double r391968 = r391967 / r391953;
        double r391969 = r391968 + r391953;
        double r391970 = r391962 ? r391965 : r391969;
        double r391971 = r391955 ? r391960 : r391970;
        return r391971;
}

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
Target0.6
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 < -1.35532446269264e+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 -1.35532446269264e+154 < x < 2.6482779199350034e+92

    1. Initial program 0.0

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

    if 2.6482779199350034e+92 < x

    1. Initial program 46.3

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

    2. Taylor expanded around inf 1.0

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

    3. Simplified1.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.355324462692639940669848037627778133899 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.648277919935003443657636145779230905143 \cdot 10^{92}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{1}{2}}{x} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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)))