Average Error: 21.7 → 0.1
Time: 967.0ms
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3296513343657604 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.6165462206006295 \cdot 10^{113}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3296513343657604 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 7.6165462206006295 \cdot 10^{113}:\\
\;\;\;\;\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 r496852 = x;
        double r496853 = r496852 * r496852;
        double r496854 = y;
        double r496855 = r496853 + r496854;
        double r496856 = sqrt(r496855);
        return r496856;
}
↓
double f(double x, double y) {
        double r496857 = x;
        double r496858 = -1.3296513343657604e+154;
        bool r496859 = r496857 <= r496858;
        double r496860 = 0.5;
        double r496861 = y;
        double r496862 = r496861 / r496857;
        double r496863 = r496860 * r496862;
        double r496864 = r496857 + r496863;
        double r496865 = -r496864;
        double r496866 = 7.61654622060063e+113;
        bool r496867 = r496857 <= r496866;
        double r496868 = r496857 * r496857;
        double r496869 = r496868 + r496861;
        double r496870 = sqrt(r496869);
        double r496871 = fma(r496860, r496862, r496857);
        double r496872 = r496867 ? r496870 : r496871;
        double r496873 = r496859 ? r496865 : r496872;
        return r496873;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.7
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 < -1.3296513343657604e+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 -1.3296513343657604e+154 < x < 7.61654622060063e+113

    1. Initial program 0.0

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

    if 7.61654622060063e+113 < x

    1. Initial program 52.5

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

    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3296513343657604 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.6165462206006295 \cdot 10^{113}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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)))