Average Error: 21.4 → 0.4
Time: 1.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.35596794389876828 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 6.210525943263359 \cdot 10^{71}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.35596794389876828 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 6.210525943263359 \cdot 10^{71}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r497484 = x;
        double r497485 = r497484 * r497484;
        double r497486 = y;
        double r497487 = r497485 + r497486;
        double r497488 = sqrt(r497487);
        return r497488;
}
↓
double f(double x, double y) {
        double r497489 = x;
        double r497490 = -1.3559679438987683e+154;
        bool r497491 = r497489 <= r497490;
        double r497492 = 0.5;
        double r497493 = y;
        double r497494 = r497493 / r497489;
        double r497495 = r497492 * r497494;
        double r497496 = r497489 + r497495;
        double r497497 = -r497496;
        double r497498 = 6.210525943263359e+71;
        bool r497499 = r497489 <= r497498;
        double r497500 = r497489 * r497489;
        double r497501 = r497500 + r497493;
        double r497502 = sqrt(r497501);
        double r497503 = r497499 ? r497502 : r497496;
        double r497504 = r497491 ? r497497 : r497503;
        return r497504;
}

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
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3559679438987683e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

    if -1.3559679438987683e+154 < x < 6.210525943263359e+71

    1. Initial program 0.0

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

    if 6.210525943263359e+71 < x

    1. Initial program 42.3

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.35596794389876828 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 6.210525943263359 \cdot 10^{71}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))