Average Error: 21.1 → 0.0
Time: 3.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.321181503567267682308379168130116570047 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.726425002480424766694997354741713243289 \cdot 10^{131}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.321181503567267682308379168130116570047 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.726425002480424766694997354741713243289 \cdot 10^{131}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r411143 = x;
        double r411144 = r411143 * r411143;
        double r411145 = y;
        double r411146 = r411144 + r411145;
        double r411147 = sqrt(r411146);
        return r411147;
}
↓
double f(double x, double y) {
        double r411148 = x;
        double r411149 = -1.3211815035672677e+154;
        bool r411150 = r411148 <= r411149;
        double r411151 = y;
        double r411152 = r411151 / r411148;
        double r411153 = -0.5;
        double r411154 = r411152 * r411153;
        double r411155 = r411154 - r411148;
        double r411156 = 1.7264250024804248e+131;
        bool r411157 = r411148 <= r411156;
        double r411158 = r411148 * r411148;
        double r411159 = r411158 + r411151;
        double r411160 = sqrt(r411159);
        double r411161 = 0.5;
        double r411162 = r411161 * r411152;
        double r411163 = r411148 + r411162;
        double r411164 = r411157 ? r411160 : r411163;
        double r411165 = r411150 ? r411155 : r411164;
        return r411165;
}

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

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3211815035672677e+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

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

    if -1.3211815035672677e+154 < x < 1.7264250024804248e+131

    1. Initial program 0.0

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

    if 1.7264250024804248e+131 < x

    1. Initial program 56.4

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

    2. Taylor expanded around inf 0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.321181503567267682308379168130116570047 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.726425002480424766694997354741713243289 \cdot 10^{131}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))