Average Error: 21.0 → 0.2
Time: 10.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\

\mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

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

\end{array}
double f(double x, double y) {
        double r337539 = x;
        double r337540 = r337539 * r337539;
        double r337541 = y;
        double r337542 = r337540 + r337541;
        double r337543 = sqrt(r337542);
        return r337543;
}
↓
double f(double x, double y) {
        double r337544 = x;
        double r337545 = -1.3221955759293222e+154;
        bool r337546 = r337544 <= r337545;
        double r337547 = y;
        double r337548 = r337547 / r337544;
        double r337549 = 0.5;
        double r337550 = fma(r337548, r337549, r337544);
        double r337551 = -r337550;
        double r337552 = 1.892549585482312e+97;
        bool r337553 = r337544 <= r337552;
        double r337554 = fma(r337544, r337544, r337547);
        double r337555 = sqrt(r337554);
        double r337556 = r337553 ? r337555 : r337550;
        double r337557 = r337546 ? r337551 : r337556;
        return r337557;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.0
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 < -1.3221955759293222e+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 -1.3221955759293222e+154 < x < 1.892549585482312e+97

    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 1.892549585482312e+97 < x

    1. Initial program 47.8

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

    2. Simplified47.8

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

    3. Taylor expanded around inf 0.8

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

    4. Simplified0.8

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))