Average Error: 21.3 → 0.4
Time: 8.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

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

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

\end{array}
double f(double x, double y) {
        double r567626 = x;
        double r567627 = r567626 * r567626;
        double r567628 = y;
        double r567629 = r567627 + r567628;
        double r567630 = sqrt(r567629);
        return r567630;
}
↓
double f(double x, double y) {
        double r567631 = x;
        double r567632 = -1.3778733035656446e+154;
        bool r567633 = r567631 <= r567632;
        double r567634 = 0.5;
        double r567635 = y;
        double r567636 = r567635 / r567631;
        double r567637 = fma(r567634, r567636, r567631);
        double r567638 = -r567637;
        double r567639 = 1.2922566123944575e+80;
        bool r567640 = r567631 <= r567639;
        double r567641 = fma(r567631, r567631, r567635);
        double r567642 = sqrt(r567641);
        double r567643 = r567640 ? r567642 : r567637;
        double r567644 = r567633 ? r567638 : r567643;
        return r567644;
}

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. Using strategy rm

    4. Applied add-sqr-sqrt64.0

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

    5. Applied sqrt-prod64.0

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

    6. Taylor expanded around -inf 0

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

    7. Simplified0

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

    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 add-sqr-sqrt44.1

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

    5. Applied sqrt-prod44.3

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

    6. Taylor expanded around inf 1.5

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

    7. Simplified1.5

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

  4. Final simplification0.4

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

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)))