Average Error: 21.8 → 0.7
Time: 1.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.35942609678321041 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.85694116631068637 \cdot 10^{48}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.35942609678321041 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 5.85694116631068637 \cdot 10^{48}:\\
\;\;\;\;\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 r692476 = x;
        double r692477 = r692476 * r692476;
        double r692478 = y;
        double r692479 = r692477 + r692478;
        double r692480 = sqrt(r692479);
        return r692480;
}
↓
double f(double x, double y) {
        double r692481 = x;
        double r692482 = -1.3594260967832104e+154;
        bool r692483 = r692481 <= r692482;
        double r692484 = 0.5;
        double r692485 = y;
        double r692486 = r692485 / r692481;
        double r692487 = r692484 * r692486;
        double r692488 = r692481 + r692487;
        double r692489 = -r692488;
        double r692490 = 5.856941166310686e+48;
        bool r692491 = r692481 <= r692490;
        double r692492 = r692481 * r692481;
        double r692493 = r692492 + r692485;
        double r692494 = sqrt(r692493);
        double r692495 = fma(r692484, r692486, r692481);
        double r692496 = r692491 ? r692494 : r692495;
        double r692497 = r692483 ? r692489 : r692496;
        return r692497;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.8
Target0.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3594260967832104e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

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

    if -1.3594260967832104e+154 < x < 5.856941166310686e+48

    1. Initial program 0.0

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

    if 5.856941166310686e+48 < x

    1. Initial program 39.2

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

    2. Taylor expanded around inf 2.4

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

    3. Simplified2.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.35942609678321041 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.85694116631068637 \cdot 10^{48}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))