Average Error: 21.8 → 0.0
Time: 2.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.358575356778544888504370454055028257768 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.447947334872172544750837332377802276979 \cdot 10^{133}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.358575356778544888504370454055028257768 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.447947334872172544750837332377802276979 \cdot 10^{133}:\\
\;\;\;\;\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 r492318 = x;
        double r492319 = r492318 * r492318;
        double r492320 = y;
        double r492321 = r492319 + r492320;
        double r492322 = sqrt(r492321);
        return r492322;
}
↓
double f(double x, double y) {
        double r492323 = x;
        double r492324 = -1.358575356778545e+154;
        bool r492325 = r492323 <= r492324;
        double r492326 = 0.5;
        double r492327 = y;
        double r492328 = r492327 / r492323;
        double r492329 = r492326 * r492328;
        double r492330 = r492323 + r492329;
        double r492331 = -r492330;
        double r492332 = 1.4479473348721725e+133;
        bool r492333 = r492323 <= r492332;
        double r492334 = r492323 * r492323;
        double r492335 = r492334 + r492327;
        double r492336 = sqrt(r492335);
        double r492337 = fma(r492326, r492328, r492323);
        double r492338 = r492333 ? r492336 : r492337;
        double r492339 = r492325 ? r492331 : r492338;
        return r492339;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.8
Target0.5
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.358575356778545e+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 -1.358575356778545e+154 < x < 1.4479473348721725e+133

    1. Initial program 0.0

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

    if 1.4479473348721725e+133 < x

    1. Initial program 56.6

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.358575356778544888504370454055028257768 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.447947334872172544750837332377802276979 \cdot 10^{133}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))