Average Error: 21.3 → 0.1
Time: 1.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3498784301045228 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.19727351594512604 \cdot 10^{126}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3498784301045228 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 7.19727351594512604 \cdot 10^{126}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

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

\end{array}
double f(double x, double y) {
        double r527544 = x;
        double r527545 = r527544 * r527544;
        double r527546 = y;
        double r527547 = r527545 + r527546;
        double r527548 = sqrt(r527547);
        return r527548;
}
↓
double f(double x, double y) {
        double r527549 = x;
        double r527550 = -1.3498784301045228e+154;
        bool r527551 = r527549 <= r527550;
        double r527552 = 0.5;
        double r527553 = y;
        double r527554 = r527553 / r527549;
        double r527555 = r527552 * r527554;
        double r527556 = r527549 + r527555;
        double r527557 = -r527556;
        double r527558 = 7.197273515945126e+126;
        bool r527559 = r527549 <= r527558;
        double r527560 = fma(r527549, r527549, r527553);
        double r527561 = sqrt(r527560);
        double r527562 = fma(r527552, r527554, r527549);
        double r527563 = r527559 ? r527561 : r527562;
        double r527564 = r527551 ? r527557 : r527563;
        return r527564;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.3
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 < -1.3498784301045228e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    if -1.3498784301045228e+154 < x < 7.197273515945126e+126

    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 7.197273515945126e+126 < x

    1. Initial program 54.9

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3498784301045228 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.19727351594512604 \cdot 10^{126}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))