Average Error: 21.1 → 0.3
Time: 1.8s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.298475003874099941155891200323427793939 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.116552450926022022585758234935512177514 \cdot 10^{82}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.298475003874099941155891200323427793939 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 2.116552450926022022585758234935512177514 \cdot 10^{82}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r474950 = x;
        double r474951 = r474950 * r474950;
        double r474952 = y;
        double r474953 = r474951 + r474952;
        double r474954 = sqrt(r474953);
        return r474954;
}
↓
double f(double x, double y) {
        double r474955 = x;
        double r474956 = -1.2984750038741e+154;
        bool r474957 = r474955 <= r474956;
        double r474958 = 0.5;
        double r474959 = y;
        double r474960 = r474959 / r474955;
        double r474961 = r474958 * r474960;
        double r474962 = r474955 + r474961;
        double r474963 = -r474962;
        double r474964 = 2.116552450926022e+82;
        bool r474965 = r474955 <= r474964;
        double r474966 = r474955 * r474955;
        double r474967 = r474966 + r474959;
        double r474968 = sqrt(r474967);
        double r474969 = r474965 ? r474968 : r474962;
        double r474970 = r474957 ? r474963 : r474969;
        return r474970;
}

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

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2984750038741e+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 -1.2984750038741e+154 < x < 2.116552450926022e+82

    1. Initial program 0.0

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

    if 2.116552450926022e+82 < x

    1. Initial program 44.6

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

    2. Taylor expanded around inf 1.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.298475003874099941155891200323427793939 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.116552450926022022585758234935512177514 \cdot 10^{82}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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)))