Average Error: 21.3 → 0.4
Time: 16.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\

\mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y) {
        double r497148 = x;
        double r497149 = r497148 * r497148;
        double r497150 = y;
        double r497151 = r497149 + r497150;
        double r497152 = sqrt(r497151);
        return r497152;
}
↓
double f(double x, double y) {
        double r497153 = x;
        double r497154 = -1.3778733035656446e+154;
        bool r497155 = r497153 <= r497154;
        double r497156 = y;
        double r497157 = r497156 / r497153;
        double r497158 = 0.5;
        double r497159 = fma(r497157, r497158, r497153);
        double r497160 = -r497159;
        double r497161 = 1.2922566123944575e+80;
        bool r497162 = r497153 <= r497161;
        double r497163 = fma(r497153, r497153, r497156);
        double r497164 = sqrt(r497163);
        double r497165 = r497162 ? r497164 : r497153;
        double r497166 = r497155 ? r497160 : r497165;
        return r497166;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.3
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3778733035656446e+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

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)}\]

    if -1.3778733035656446e+154 < x < 1.2922566123944575e+80

    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 1.2922566123944575e+80 < x

    1. Initial program 44.1

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

    2. Simplified44.1

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity44.1

      \[\leadsto \sqrt{\color{blue}{1 \cdot \mathsf{fma}\left(x, x, y\right)}}\]

    5. Applied sqrt-prod44.1

      \[\leadsto \color{blue}{\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(x, x, y\right)}}\]

    6. Simplified44.1

      \[\leadsto \color{blue}{1} \cdot \sqrt{\mathsf{fma}\left(x, x, y\right)}\]

    7. Simplified31.5

      \[\leadsto 1 \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{y}, x\right)}\]

    8. Taylor expanded around 0 1.6

      \[\leadsto 1 \cdot \color{blue}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))