Average Error: 21.4 → 0.1
Time: 1.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\
\;\;\;\;\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 r489260 = x;
        double r489261 = r489260 * r489260;
        double r489262 = y;
        double r489263 = r489261 + r489262;
        double r489264 = sqrt(r489263);
        return r489264;
}
↓
double f(double x, double y) {
        double r489265 = x;
        double r489266 = -1.3474626627347847e+154;
        bool r489267 = r489265 <= r489266;
        double r489268 = 0.5;
        double r489269 = y;
        double r489270 = r489269 / r489265;
        double r489271 = r489268 * r489270;
        double r489272 = r489265 + r489271;
        double r489273 = -r489272;
        double r489274 = 5.291435342096596e+124;
        bool r489275 = r489265 <= r489274;
        double r489276 = r489265 * r489265;
        double r489277 = r489276 + r489269;
        double r489278 = sqrt(r489277);
        double r489279 = fma(r489268, r489270, r489265);
        double r489280 = r489275 ? r489278 : r489279;
        double r489281 = r489267 ? r489273 : r489280;
        return r489281;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.4
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 < -1.3474626627347847e+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 -1.3474626627347847e+154 < x < 5.291435342096596e+124

    1. Initial program 0.0

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

    if 5.291435342096596e+124 < x

    1. Initial program 54.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.291435342096596 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))