Average Error: 21.0 → 0.2
Time: 10.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r301490 = x;
        double r301491 = r301490 * r301490;
        double r301492 = y;
        double r301493 = r301491 + r301492;
        double r301494 = sqrt(r301493);
        return r301494;
}
↓
double f(double x, double y) {
        double r301495 = x;
        double r301496 = -1.3221955759293222e+154;
        bool r301497 = r301495 <= r301496;
        double r301498 = 0.5;
        double r301499 = y;
        double r301500 = r301499 / r301495;
        double r301501 = fma(r301498, r301500, r301495);
        double r301502 = -r301501;
        double r301503 = 1.892549585482312e+97;
        bool r301504 = r301495 <= r301503;
        double r301505 = r301495 * r301495;
        double r301506 = r301505 + r301499;
        double r301507 = sqrt(r301506);
        double r301508 = r301504 ? r301507 : r301501;
        double r301509 = r301497 ? r301502 : r301508;
        return r301509;
}

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. Taylor expanded around -inf 0

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

    3. Simplified0

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

    if -1.3221955759293222e+154 < x < 1.892549585482312e+97

    1. Initial program 0.0

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

    if 1.892549585482312e+97 < x

    1. Initial program 47.8

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

    2. Taylor expanded around inf 0.8

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

    3. Simplified0.8

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

  4. Final simplification0.2

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

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)))