Average Error: 21.8 → 0.7
Time: 2.6s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.35942609678321041 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.85694116631068637 \cdot 10^{48}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\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}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\end{array}
double f(double x, double y) {
        double r2615 = x;
        double r2616 = r2615 * r2615;
        double r2617 = y;
        double r2618 = r2616 + r2617;
        double r2619 = sqrt(r2618);
        return r2619;
}
↓
double f(double x, double y) {
        double r2620 = x;
        double r2621 = -1.3594260967832104e+154;
        bool r2622 = r2620 <= r2621;
        double r2623 = 0.5;
        double r2624 = y;
        double r2625 = r2624 / r2620;
        double r2626 = r2623 * r2625;
        double r2627 = r2620 + r2626;
        double r2628 = -r2627;
        double r2629 = 5.856941166310686e+48;
        bool r2630 = r2620 <= r2629;
        double r2631 = r2620 * r2620;
        double r2632 = r2631 + r2624;
        double r2633 = sqrt(r2632);
        double r2634 = fma(r2623, r2625, r2620);
        double r2635 = r2630 ? r2633 : r2634;
        double r2636 = r2622 ? r2628 : r2635;
        return r2636;
}

Error

Bits error versus x

Bits error versus y

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. Simplified2.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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)))