Average Error: 21.8 → 0.7
Time: 1.6s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.35942609678321041 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.85694116631068637 \cdot 10^{48}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.35942609678321041 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 5.85694116631068637 \cdot 10^{48}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r532705 = x;
        double r532706 = r532705 * r532705;
        double r532707 = y;
        double r532708 = r532706 + r532707;
        double r532709 = sqrt(r532708);
        return r532709;
}
↓
double f(double x, double y) {
        double r532710 = x;
        double r532711 = -1.3594260967832104e+154;
        bool r532712 = r532710 <= r532711;
        double r532713 = 0.5;
        double r532714 = y;
        double r532715 = r532714 / r532710;
        double r532716 = r532713 * r532715;
        double r532717 = r532710 + r532716;
        double r532718 = -r532717;
        double r532719 = 5.856941166310686e+48;
        bool r532720 = r532710 <= r532719;
        double r532721 = r532710 * r532710;
        double r532722 = r532721 + r532714;
        double r532723 = sqrt(r532722);
        double r532724 = r532720 ? r532723 : r532717;
        double r532725 = r532712 ? r532718 : r532724;
        return r532725;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.8
Target0.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3594260967832104e+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 -1.3594260967832104e+154 < x < 5.856941166310686e+48

    1. Initial program 0.0

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

    if 5.856941166310686e+48 < x

    1. Initial program 39.2

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.35942609678321041 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.85694116631068637 \cdot 10^{48}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))