Average Error: 21.5 → 0.1
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.37154299719310846 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.97039941263825195 \cdot 10^{110}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.37154299719310846 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 2.97039941263825195 \cdot 10^{110}:\\
\;\;\;\;\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 r452586 = x;
        double r452587 = r452586 * r452586;
        double r452588 = y;
        double r452589 = r452587 + r452588;
        double r452590 = sqrt(r452589);
        return r452590;
}
↓
double f(double x, double y) {
        double r452591 = x;
        double r452592 = -1.3715429971931085e+154;
        bool r452593 = r452591 <= r452592;
        double r452594 = 0.5;
        double r452595 = y;
        double r452596 = r452595 / r452591;
        double r452597 = r452594 * r452596;
        double r452598 = r452591 + r452597;
        double r452599 = -r452598;
        double r452600 = 2.970399412638252e+110;
        bool r452601 = r452591 <= r452600;
        double r452602 = r452591 * r452591;
        double r452603 = r452602 + r452595;
        double r452604 = sqrt(r452603);
        double r452605 = fma(r452594, r452596, r452591);
        double r452606 = r452601 ? r452604 : r452605;
        double r452607 = r452593 ? r452599 : r452606;
        return r452607;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.5
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 < -1.3715429971931085e+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 -1.3715429971931085e+154 < x < 2.970399412638252e+110

    1. Initial program 0.0

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

    if 2.970399412638252e+110 < x

    1. Initial program 49.8

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.37154299719310846 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 2.97039941263825195 \cdot 10^{110}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))