Average Error: 22.0 → 0.3
Time: 6.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.179358339742639161356757845424881409648 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.967804778338946091999022758751187720718 \cdot 10^{86}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.179358339742639161356757845424881409648 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 3.967804778338946091999022758751187720718 \cdot 10^{86}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r430960 = x;
        double r430961 = r430960 * r430960;
        double r430962 = y;
        double r430963 = r430961 + r430962;
        double r430964 = sqrt(r430963);
        return r430964;
}
↓
double f(double x, double y) {
        double r430965 = x;
        double r430966 = -1.1793583397426392e+154;
        bool r430967 = r430965 <= r430966;
        double r430968 = 0.5;
        double r430969 = y;
        double r430970 = r430969 / r430965;
        double r430971 = r430968 * r430970;
        double r430972 = r430965 + r430971;
        double r430973 = -r430972;
        double r430974 = 3.967804778338946e+86;
        bool r430975 = r430965 <= r430974;
        double r430976 = r430965 * r430965;
        double r430977 = r430976 + r430969;
        double r430978 = sqrt(r430977);
        double r430979 = r430975 ? r430978 : r430972;
        double r430980 = r430967 ? r430973 : r430979;
        return r430980;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.1793583397426392e+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 -1.1793583397426392e+154 < x < 3.967804778338946e+86

    1. Initial program 0.0

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

    if 3.967804778338946e+86 < x

    1. Initial program 46.1

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

    2. Taylor expanded around inf 1.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.179358339742639161356757845424881409648 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.967804778338946091999022758751187720718 \cdot 10^{86}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.5823995511225407e57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

  (sqrt (+ (* x x) y)))