Average Error: 21.4 → 0.0
Time: 9.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.117615036957898616945829215156814760292 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.574212501990470180534105460064927660599 \cdot 10^{131}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{1}{2}}{x} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.117615036957898616945829215156814760292 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 1.574212501990470180534105460064927660599 \cdot 10^{131}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r465474 = x;
        double r465475 = r465474 * r465474;
        double r465476 = y;
        double r465477 = r465475 + r465476;
        double r465478 = sqrt(r465477);
        return r465478;
}
↓
double f(double x, double y) {
        double r465479 = x;
        double r465480 = -1.1176150369578986e+154;
        bool r465481 = r465479 <= r465480;
        double r465482 = -0.5;
        double r465483 = y;
        double r465484 = r465479 / r465483;
        double r465485 = r465482 / r465484;
        double r465486 = r465485 - r465479;
        double r465487 = 1.5742125019904702e+131;
        bool r465488 = r465479 <= r465487;
        double r465489 = r465479 * r465479;
        double r465490 = r465489 + r465483;
        double r465491 = sqrt(r465490);
        double r465492 = 0.5;
        double r465493 = r465483 * r465492;
        double r465494 = r465493 / r465479;
        double r465495 = r465494 + r465479;
        double r465496 = r465488 ? r465491 : r465495;
        double r465497 = r465481 ? r465486 : r465496;
        return r465497;
}

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
Target0.4
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 < -1.1176150369578986e+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

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

    if -1.1176150369578986e+154 < x < 1.5742125019904702e+131

    1. Initial program 0.0

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

    if 1.5742125019904702e+131 < x

    1. Initial program 55.8

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.117615036957898616945829215156814760292 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.574212501990470180534105460064927660599 \cdot 10^{131}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{1}{2}}{x} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :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)))