Average Error: 21.5 → 0.0
Time: 9.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.192129562741525672838937725985142908574 \cdot 10^{154}:\\ \;\;\;\;\frac{y \cdot \frac{-1}{2}}{x} - x\\ \mathbf{elif}\;x \le 1.720238385409512324136337202681508109959 \cdot 10^{146}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.192129562741525672838937725985142908574 \cdot 10^{154}:\\
\;\;\;\;\frac{y \cdot \frac{-1}{2}}{x} - x\\

\mathbf{elif}\;x \le 1.720238385409512324136337202681508109959 \cdot 10^{146}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\

\end{array}
double f(double x, double y) {
        double r25634355 = x;
        double r25634356 = r25634355 * r25634355;
        double r25634357 = y;
        double r25634358 = r25634356 + r25634357;
        double r25634359 = sqrt(r25634358);
        return r25634359;
}
↓
double f(double x, double y) {
        double r25634360 = x;
        double r25634361 = -1.1921295627415257e+154;
        bool r25634362 = r25634360 <= r25634361;
        double r25634363 = y;
        double r25634364 = -0.5;
        double r25634365 = r25634363 * r25634364;
        double r25634366 = r25634365 / r25634360;
        double r25634367 = r25634366 - r25634360;
        double r25634368 = 1.7202383854095123e+146;
        bool r25634369 = r25634360 <= r25634368;
        double r25634370 = fma(r25634360, r25634360, r25634363);
        double r25634371 = sqrt(r25634370);
        double r25634372 = 0.5;
        double r25634373 = r25634372 / r25634360;
        double r25634374 = fma(r25634373, r25634363, r25634360);
        double r25634375 = r25634369 ? r25634371 : r25634374;
        double r25634376 = r25634362 ? r25634367 : r25634375;
        return r25634376;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.5
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 < -1.1921295627415257e+154

    1. Initial program 64.0

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

    2. Simplified64.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}}\]

    3. Taylor expanded around -inf 0

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

    4. Simplified0

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

    if -1.1921295627415257e+154 < x < 1.7202383854095123e+146

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}}\]

    if 1.7202383854095123e+146 < x

    1. Initial program 61.3

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

    2. Simplified61.3

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}}\]

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.192129562741525672838937725985142908574 \cdot 10^{154}:\\ \;\;\;\;\frac{y \cdot \frac{-1}{2}}{x} - x\\ \mathbf{elif}\;x \le 1.720238385409512324136337202681508109959 \cdot 10^{146}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :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)))