Average Error: 21.7 → 0.0
Time: 4.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \le 1.239102867687965121108359501827503075543 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\mathbf{elif}\;x \le 1.239102867687965121108359501827503075543 \cdot 10^{132}:\\
\;\;\;\;\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 r477617 = x;
        double r477618 = r477617 * r477617;
        double r477619 = y;
        double r477620 = r477618 + r477619;
        double r477621 = sqrt(r477620);
        return r477621;
}
↓
double f(double x, double y) {
        double r477622 = x;
        double r477623 = -1.3305299901761994e+154;
        bool r477624 = r477622 <= r477623;
        double r477625 = 0.5;
        double r477626 = y;
        double r477627 = r477626 / r477622;
        double r477628 = fma(r477625, r477627, r477622);
        double r477629 = -r477628;
        double r477630 = 1.2391028676879651e+132;
        bool r477631 = r477622 <= r477630;
        double r477632 = r477622 * r477622;
        double r477633 = r477632 + r477626;
        double r477634 = sqrt(r477633);
        double r477635 = r477631 ? r477634 : r477628;
        double r477636 = r477624 ? r477629 : r477635;
        return r477636;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.7
Target0.5
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3305299901761994e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

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

    3. Simplified0

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)}\]

    if -1.3305299901761994e+154 < x < 1.2391028676879651e+132

    1. Initial program 0.0

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

    if 1.2391028676879651e+132 < x

    1. Initial program 56.0

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.330529990176199361196485578032770005542 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \le 1.239102867687965121108359501827503075543 \cdot 10^{132}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))