Average Error: 21.5 → 0.2
Time: 12.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.335728053209796214179137167831695743037 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 3.200301816232115926663695869716028224145 \cdot 10^{89}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.335728053209796214179137167831695743037 \cdot 10^{154}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\

\mathbf{elif}\;x \le 3.200301816232115926663695869716028224145 \cdot 10^{89}:\\
\;\;\;\;\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 r23721723 = x;
        double r23721724 = r23721723 * r23721723;
        double r23721725 = y;
        double r23721726 = r23721724 + r23721725;
        double r23721727 = sqrt(r23721726);
        return r23721727;
}
↓
double f(double x, double y) {
        double r23721728 = x;
        double r23721729 = -1.3357280532097962e+154;
        bool r23721730 = r23721728 <= r23721729;
        double r23721731 = -0.5;
        double r23721732 = y;
        double r23721733 = r23721732 / r23721728;
        double r23721734 = r23721731 * r23721733;
        double r23721735 = r23721734 - r23721728;
        double r23721736 = 3.200301816232116e+89;
        bool r23721737 = r23721728 <= r23721736;
        double r23721738 = fma(r23721728, r23721728, r23721732);
        double r23721739 = sqrt(r23721738);
        double r23721740 = 0.5;
        double r23721741 = r23721740 / r23721728;
        double r23721742 = fma(r23721741, r23721732, r23721728);
        double r23721743 = r23721737 ? r23721739 : r23721742;
        double r23721744 = r23721730 ? r23721735 : r23721743;
        return r23721744;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.5
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3357280532097962e+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-cbrt-cube64.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left(x, x, y\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, y\right)}\right) \cdot \sqrt{\mathsf{fma}\left(x, x, y\right)}}}\]

    5. Simplified64.0

      \[\leadsto \sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, y\right) \cdot \sqrt{\mathsf{fma}\left(x, x, y\right)}}}\]

    6. Taylor expanded around -inf 0

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

    7. Simplified0

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

    if -1.3357280532097962e+154 < x < 3.200301816232116e+89

    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 3.200301816232116e+89 < x

    1. Initial program 45.8

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

    2. Simplified45.8

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

    3. Taylor expanded around inf 0.9

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

    4. Simplified0.9

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.335728053209796214179137167831695743037 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 3.200301816232115926663695869716028224145 \cdot 10^{89}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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)))