Average Error: 21.2 → 0.1
Time: 11.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3614720717698548 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \le 7.40557002165322956 \cdot 10^{112}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3614720717698548 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\mathbf{elif}\;x \le 7.40557002165322956 \cdot 10^{112}:\\
\;\;\;\;\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 r644523 = x;
        double r644524 = r644523 * r644523;
        double r644525 = y;
        double r644526 = r644524 + r644525;
        double r644527 = sqrt(r644526);
        return r644527;
}
↓
double f(double x, double y) {
        double r644528 = x;
        double r644529 = -1.3614720717698548e+154;
        bool r644530 = r644528 <= r644529;
        double r644531 = 0.5;
        double r644532 = y;
        double r644533 = r644532 / r644528;
        double r644534 = fma(r644531, r644533, r644528);
        double r644535 = -r644534;
        double r644536 = 7.4055700216532296e+112;
        bool r644537 = r644528 <= r644536;
        double r644538 = r644528 * r644528;
        double r644539 = r644538 + r644532;
        double r644540 = sqrt(r644539);
        double r644541 = r644537 ? r644540 : r644534;
        double r644542 = r644530 ? r644535 : r644541;
        return r644542;
}

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

    1. Initial program 0.0

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

    if 7.4055700216532296e+112 < x

    1. Initial program 50.1

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.1

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

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)))