Average Error: 21.6 → 0.0
Time: 7.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.354574227411527670224267982161705278163 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 8.970759006124063597546869044071596379693 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.354574227411527670224267982161705278163 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 8.970759006124063597546869044071596379693 \cdot 10^{123}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r418372 = x;
        double r418373 = r418372 * r418372;
        double r418374 = y;
        double r418375 = r418373 + r418374;
        double r418376 = sqrt(r418375);
        return r418376;
}
↓
double f(double x, double y) {
        double r418377 = x;
        double r418378 = -1.3545742274115277e+154;
        bool r418379 = r418377 <= r418378;
        double r418380 = y;
        double r418381 = r418380 / r418377;
        double r418382 = -0.5;
        double r418383 = r418381 * r418382;
        double r418384 = r418383 - r418377;
        double r418385 = 8.970759006124064e+123;
        bool r418386 = r418377 <= r418385;
        double r418387 = r418377 * r418377;
        double r418388 = r418387 + r418380;
        double r418389 = sqrt(r418388);
        double r418390 = 0.5;
        double r418391 = r418390 * r418381;
        double r418392 = r418377 + r418391;
        double r418393 = r418386 ? r418389 : r418392;
        double r418394 = r418379 ? r418384 : r418393;
        return r418394;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.6
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 < -1.3545742274115277e+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 -1.3545742274115277e+154 < x < 8.970759006124064e+123

    1. Initial program 0.0

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

    if 8.970759006124064e+123 < x

    1. Initial program 54.2

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.354574227411527670224267982161705278163 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 8.970759006124063597546869044071596379693 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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)))