Average Error: 21.0 → 0.0
Time: 2.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.34679050822059381152104109136094934248 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.483080572797596756164012838819236522397 \cdot 10^{140}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.34679050822059381152104109136094934248 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 7.483080572797596756164012838819236522397 \cdot 10^{140}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r545315 = x;
        double r545316 = r545315 * r545315;
        double r545317 = y;
        double r545318 = r545316 + r545317;
        double r545319 = sqrt(r545318);
        return r545319;
}
↓
double f(double x, double y) {
        double r545320 = x;
        double r545321 = -1.3467905082205938e+154;
        bool r545322 = r545320 <= r545321;
        double r545323 = 0.5;
        double r545324 = y;
        double r545325 = r545324 / r545320;
        double r545326 = r545323 * r545325;
        double r545327 = r545320 + r545326;
        double r545328 = -r545327;
        double r545329 = 7.483080572797597e+140;
        bool r545330 = r545320 <= r545329;
        double r545331 = r545320 * r545320;
        double r545332 = r545331 + r545324;
        double r545333 = sqrt(r545332);
        double r545334 = r545330 ? r545333 : r545327;
        double r545335 = r545322 ? r545328 : r545334;
        return r545335;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.0
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 < -1.3467905082205938e+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 -1.3467905082205938e+154 < x < 7.483080572797597e+140

    1. Initial program 0.0

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

    if 7.483080572797597e+140 < x

    1. Initial program 59.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.34679050822059381152104109136094934248 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.483080572797596756164012838819236522397 \cdot 10^{140}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))