Average Error: 21.4 → 0.4
Time: 1.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.34738501180888645 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.33688706983331878 \cdot 10^{74}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.34738501180888645 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.33688706983331878 \cdot 10^{74}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r463154 = x;
        double r463155 = r463154 * r463154;
        double r463156 = y;
        double r463157 = r463155 + r463156;
        double r463158 = sqrt(r463157);
        return r463158;
}
↓
double f(double x, double y) {
        double r463159 = x;
        double r463160 = -1.3473850118088864e+154;
        bool r463161 = r463159 <= r463160;
        double r463162 = 0.5;
        double r463163 = y;
        double r463164 = r463163 / r463159;
        double r463165 = r463162 * r463164;
        double r463166 = r463159 + r463165;
        double r463167 = -r463166;
        double r463168 = 1.3368870698333188e+74;
        bool r463169 = r463159 <= r463168;
        double r463170 = r463159 * r463159;
        double r463171 = r463170 + r463163;
        double r463172 = sqrt(r463171);
        double r463173 = r463169 ? r463172 : r463166;
        double r463174 = r463161 ? r463167 : r463173;
        return r463174;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.4
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3473850118088864e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

    if -1.3473850118088864e+154 < x < 1.3368870698333188e+74

    1. Initial program 0.0

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

    if 1.3368870698333188e+74 < x

    1. Initial program 43.5

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.34738501180888645 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.33688706983331878 \cdot 10^{74}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))