Average Error: 19.8 → 0.7
Time: 9.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3323185489366894 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.3070827329489974 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3323185489366894 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 1.3070827329489974 \cdot 10^{+38}:\\
\;\;\;\;\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 r27730360 = x;
        double r27730361 = r27730360 * r27730360;
        double r27730362 = y;
        double r27730363 = r27730361 + r27730362;
        double r27730364 = sqrt(r27730363);
        return r27730364;
}
↓
double f(double x, double y) {
        double r27730365 = x;
        double r27730366 = -1.3323185489366894e+154;
        bool r27730367 = r27730365 <= r27730366;
        double r27730368 = -0.5;
        double r27730369 = y;
        double r27730370 = r27730365 / r27730369;
        double r27730371 = r27730368 / r27730370;
        double r27730372 = r27730371 - r27730365;
        double r27730373 = 1.3070827329489974e+38;
        bool r27730374 = r27730365 <= r27730373;
        double r27730375 = fma(r27730365, r27730365, r27730369);
        double r27730376 = sqrt(r27730375);
        double r27730377 = 0.5;
        double r27730378 = r27730377 / r27730365;
        double r27730379 = fma(r27730378, r27730369, r27730365);
        double r27730380 = r27730374 ? r27730376 : r27730379;
        double r27730381 = r27730367 ? r27730372 : r27730380;
        return r27730381;
}

Error

Bits error versus x

Bits error versus y

Target

Original19.8
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(\frac{1}{2} \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3323185489366894e+154

    1. Initial program 59.6

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

    2. Simplified59.6

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

    3. Taylor expanded around -inf 0

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

    4. Simplified0

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

    if -1.3323185489366894e+154 < x < 1.3070827329489974e+38

    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 1.3070827329489974e+38 < x

    1. Initial program 36.4

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

    2. Simplified36.4

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

    3. Taylor expanded around inf 2.6

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

    4. Simplified2.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3323185489366894 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.3070827329489974 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 1/2 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 1/2 (/ y x)) x)))

  (sqrt (+ (* x x) y)))