Average Error: 19.8 → 0.1
Time: 9.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.187498495601377 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.666665935899027 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.187498495601377 \cdot 10^{+152}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 1.666665935899027 \cdot 10^{+110}:\\
\;\;\;\;\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 r22232983 = x;
        double r22232984 = r22232983 * r22232983;
        double r22232985 = y;
        double r22232986 = r22232984 + r22232985;
        double r22232987 = sqrt(r22232986);
        return r22232987;
}
↓
double f(double x, double y) {
        double r22232988 = x;
        double r22232989 = -3.187498495601377e+152;
        bool r22232990 = r22232988 <= r22232989;
        double r22232991 = -0.5;
        double r22232992 = y;
        double r22232993 = r22232988 / r22232992;
        double r22232994 = r22232991 / r22232993;
        double r22232995 = r22232994 - r22232988;
        double r22232996 = 1.666665935899027e+110;
        bool r22232997 = r22232988 <= r22232996;
        double r22232998 = fma(r22232988, r22232988, r22232992);
        double r22232999 = sqrt(r22232998);
        double r22233000 = 0.5;
        double r22233001 = r22233000 / r22232988;
        double r22233002 = fma(r22233001, r22232992, r22232988);
        double r22233003 = r22232997 ? r22232999 : r22233002;
        double r22233004 = r22232990 ? r22232995 : r22233003;
        return r22233004;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -3.187498495601377e+152

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded around -inf 0.0

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

    4. Simplified0.0

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

    if -3.187498495601377e+152 < x < 1.666665935899027e+110

    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 1.666665935899027e+110 < x

    1. Initial program 45.6

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

    2. Simplified45.6

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

    3. Taylor expanded around inf 0.5

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

    4. Simplified0.5

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.187498495601377 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.666665935899027 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))