Average Error: 21.9 → 0.2
Time: 1.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.26339127879627209 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 4.85789658872666293 \cdot 10^{84}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.26339127879627209 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 4.85789658872666293 \cdot 10^{84}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r497498 = x;
        double r497499 = r497498 * r497498;
        double r497500 = y;
        double r497501 = r497499 + r497500;
        double r497502 = sqrt(r497501);
        return r497502;
}
↓
double f(double x, double y) {
        double r497503 = x;
        double r497504 = -1.2633912787962721e+154;
        bool r497505 = r497503 <= r497504;
        double r497506 = 0.5;
        double r497507 = y;
        double r497508 = r497507 / r497503;
        double r497509 = r497506 * r497508;
        double r497510 = r497503 + r497509;
        double r497511 = -r497510;
        double r497512 = 4.857896588726663e+84;
        bool r497513 = r497503 <= r497512;
        double r497514 = r497503 * r497503;
        double r497515 = r497514 + r497507;
        double r497516 = sqrt(r497515);
        double r497517 = r497513 ? r497516 : r497510;
        double r497518 = r497505 ? r497511 : r497517;
        return r497518;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.9
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2633912787962721e+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 -1.2633912787962721e+154 < x < 4.857896588726663e+84

    1. Initial program 0.0

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

    if 4.857896588726663e+84 < x

    1. Initial program 45.0

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

    2. Taylor expanded around inf 1.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.26339127879627209 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 4.85789658872666293 \cdot 10^{84}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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)))