Average Error: 21.2 → 0.1
Time: 11.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3614720717698548 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 7.40557002165322956 \cdot 10^{112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3614720717698548 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\

\mathbf{elif}\;x \le 7.40557002165322956 \cdot 10^{112}:\\
\;\;\;\;\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 r564493 = x;
        double r564494 = r564493 * r564493;
        double r564495 = y;
        double r564496 = r564494 + r564495;
        double r564497 = sqrt(r564496);
        return r564497;
}
↓
double f(double x, double y) {
        double r564498 = x;
        double r564499 = -1.3614720717698548e+154;
        bool r564500 = r564498 <= r564499;
        double r564501 = y;
        double r564502 = r564501 / r564498;
        double r564503 = 0.5;
        double r564504 = fma(r564502, r564503, r564498);
        double r564505 = -r564504;
        double r564506 = 7.4055700216532296e+112;
        bool r564507 = r564498 <= r564506;
        double r564508 = fma(r564498, r564498, r564501);
        double r564509 = sqrt(r564508);
        double r564510 = r564507 ? r564509 : r564504;
        double r564511 = r564500 ? r564505 : r564510;
        return r564511;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.2
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3614720717698548e+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 -1.3614720717698548e+154 < x < 7.4055700216532296e+112

    1. Initial program 0.0

      \[\sqrt{x \cdot x + y}\]
    2. Using strategy rm

    3. Applied fma-def0.0

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

    if 7.4055700216532296e+112 < x

    1. Initial program 50.1

      \[\sqrt{x \cdot x + y}\]
    2. Using strategy rm

    3. Applied fma-def50.1

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

    4. Taylor expanded around inf 0.4

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

    5. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3614720717698548 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 7.40557002165322956 \cdot 10^{112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))