Average Error: 21.4 → 0.1
Time: 1.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\
\;\;\;\;\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 r500555 = x;
        double r500556 = r500555 * r500555;
        double r500557 = y;
        double r500558 = r500556 + r500557;
        double r500559 = sqrt(r500558);
        return r500559;
}
↓
double f(double x, double y) {
        double r500560 = x;
        double r500561 = -1.3474626627347847e+154;
        bool r500562 = r500560 <= r500561;
        double r500563 = 0.5;
        double r500564 = y;
        double r500565 = r500564 / r500560;
        double r500566 = r500563 * r500565;
        double r500567 = r500560 + r500566;
        double r500568 = -r500567;
        double r500569 = 5.291435342096596e+124;
        bool r500570 = r500560 <= r500569;
        double r500571 = r500560 * r500560;
        double r500572 = r500571 + r500564;
        double r500573 = sqrt(r500572);
        double r500574 = fma(r500563, r500565, r500560);
        double r500575 = r500570 ? r500573 : r500574;
        double r500576 = r500562 ? r500568 : r500575;
        return r500576;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.4
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3474626627347847e+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 -1.3474626627347847e+154 < x < 5.291435342096596e+124

    1. Initial program 0.0

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

    if 5.291435342096596e+124 < x

    1. Initial program 54.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))