Average Error: 21.4 → 0.3
Time: 930.0ms
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.28151121097985566 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.3825854527583296 \cdot 10^{81}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.28151121097985566 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 3.3825854527583296 \cdot 10^{81}:\\
\;\;\;\;\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 r514091 = x;
        double r514092 = r514091 * r514091;
        double r514093 = y;
        double r514094 = r514092 + r514093;
        double r514095 = sqrt(r514094);
        return r514095;
}
↓
double f(double x, double y) {
        double r514096 = x;
        double r514097 = -1.2815112109798557e+154;
        bool r514098 = r514096 <= r514097;
        double r514099 = 0.5;
        double r514100 = y;
        double r514101 = r514100 / r514096;
        double r514102 = r514099 * r514101;
        double r514103 = r514096 + r514102;
        double r514104 = -r514103;
        double r514105 = 3.3825854527583296e+81;
        bool r514106 = r514096 <= r514105;
        double r514107 = r514096 * r514096;
        double r514108 = r514107 + r514100;
        double r514109 = sqrt(r514108);
        double r514110 = fma(r514099, r514101, r514096);
        double r514111 = r514106 ? r514109 : r514110;
        double r514112 = r514098 ? r514104 : r514111;
        return r514112;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.4
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.2815112109798557e+154

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 0.0

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

    if -1.2815112109798557e+154 < x < 3.3825854527583296e+81

    1. Initial program 0.0

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

    if 3.3825854527583296e+81 < x

    1. Initial program 44.2

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

    2. Taylor expanded around inf 1.1

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

    3. Simplified1.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.28151121097985566 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.3825854527583296 \cdot 10^{81}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))