Average Error: 21.4 → 0.0
Time: 1.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.34008686729452166 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.7383737548077102 \cdot 10^{147}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.34008686729452166 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.7383737548077102 \cdot 10^{147}:\\
\;\;\;\;\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 r484619 = x;
        double r484620 = r484619 * r484619;
        double r484621 = y;
        double r484622 = r484620 + r484621;
        double r484623 = sqrt(r484622);
        return r484623;
}
↓
double f(double x, double y) {
        double r484624 = x;
        double r484625 = -1.3400868672945217e+154;
        bool r484626 = r484624 <= r484625;
        double r484627 = 0.5;
        double r484628 = y;
        double r484629 = r484628 / r484624;
        double r484630 = r484627 * r484629;
        double r484631 = r484624 + r484630;
        double r484632 = -r484631;
        double r484633 = 1.73837375480771e+147;
        bool r484634 = r484624 <= r484633;
        double r484635 = r484624 * r484624;
        double r484636 = r484635 + r484628;
        double r484637 = sqrt(r484636);
        double r484638 = fma(r484627, r484629, r484624);
        double r484639 = r484634 ? r484637 : r484638;
        double r484640 = r484626 ? r484632 : r484639;
        return r484640;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.4
Target0.5
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3400868672945217e+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 -1.3400868672945217e+154 < x < 1.73837375480771e+147

    1. Initial program 0.0

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

    if 1.73837375480771e+147 < x

    1. Initial program 61.2

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.34008686729452166 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.7383737548077102 \cdot 10^{147}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))