Average Error: 21.0 → 0.0
Time: 11.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.375458412520537572590774977918936206023 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 2.058549686456957362651677908428126133886 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.375458412520537572590774977918936206023 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 2.058549686456957362651677908428126133886 \cdot 10^{132}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r346461 = x;
        double r346462 = r346461 * r346461;
        double r346463 = y;
        double r346464 = r346462 + r346463;
        double r346465 = sqrt(r346464);
        return r346465;
}
↓
double f(double x, double y) {
        double r346466 = x;
        double r346467 = -1.3754584125205376e+154;
        bool r346468 = r346466 <= r346467;
        double r346469 = y;
        double r346470 = r346469 / r346466;
        double r346471 = -0.5;
        double r346472 = r346470 * r346471;
        double r346473 = r346472 - r346466;
        double r346474 = 2.0585496864569574e+132;
        bool r346475 = r346466 <= r346474;
        double r346476 = r346466 * r346466;
        double r346477 = r346476 + r346469;
        double r346478 = sqrt(r346477);
        double r346479 = 0.5;
        double r346480 = r346479 * r346470;
        double r346481 = r346466 + r346480;
        double r346482 = r346475 ? r346478 : r346481;
        double r346483 = r346468 ? r346473 : r346482;
        return r346483;
}

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
Target0.6
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 < -1.3754584125205376e+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 -1.3754584125205376e+154 < x < 2.0585496864569574e+132

    1. Initial program 0.0

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

    if 2.0585496864569574e+132 < x

    1. Initial program 56.3

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

    2. Taylor expanded around inf 0.2

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.375458412520537572590774977918936206023 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 2.058549686456957362651677908428126133886 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))