Average Error: 21.2 → 0.5
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.380447448834878 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.56561154983368585 \cdot 10^{62}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.380447448834878 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.56561154983368585 \cdot 10^{62}:\\
\;\;\;\;\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 r483503 = x;
        double r483504 = r483503 * r483503;
        double r483505 = y;
        double r483506 = r483504 + r483505;
        double r483507 = sqrt(r483506);
        return r483507;
}
↓
double f(double x, double y) {
        double r483508 = x;
        double r483509 = -1.380447448834878e+154;
        bool r483510 = r483508 <= r483509;
        double r483511 = 0.5;
        double r483512 = y;
        double r483513 = r483512 / r483508;
        double r483514 = r483511 * r483513;
        double r483515 = r483508 + r483514;
        double r483516 = -r483515;
        double r483517 = 1.5656115498336859e+62;
        bool r483518 = r483508 <= r483517;
        double r483519 = r483508 * r483508;
        double r483520 = r483519 + r483512;
        double r483521 = sqrt(r483520);
        double r483522 = fma(r483511, r483513, r483508);
        double r483523 = r483518 ? r483521 : r483522;
        double r483524 = r483510 ? r483516 : r483523;
        return r483524;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.2
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.380447448834878e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    if -1.380447448834878e+154 < x < 1.5656115498336859e+62

    1. Initial program 0.0

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

    if 1.5656115498336859e+62 < x

    1. Initial program 41.5

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

    2. Taylor expanded around inf 1.8

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

    3. Simplified1.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.380447448834878 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.56561154983368585 \cdot 10^{62}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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)))