Average Error: 28.9 → 0.4
Time: 23.3s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;-\left(\left(\frac{3}{x \cdot \left(x \cdot x\right)} + \frac{1}{x \cdot x}\right) + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 1.021370211112824000210252961551304906607:\\ \;\;\;\;\mathsf{fma}\left(3, x, \mathsf{fma}\left(x \cdot x, 1, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(\frac{3}{x \cdot \left(x \cdot x\right)} + \frac{1}{x \cdot x}\right) + \frac{3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\
\;\;\;\;-\left(\left(\frac{3}{x \cdot \left(x \cdot x\right)} + \frac{1}{x \cdot x}\right) + \frac{3}{x}\right)\\

\mathbf{elif}\;x \le 1.021370211112824000210252961551304906607:\\
\;\;\;\;\mathsf{fma}\left(3, x, \mathsf{fma}\left(x \cdot x, 1, 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;-\left(\left(\frac{3}{x \cdot \left(x \cdot x\right)} + \frac{1}{x \cdot x}\right) + \frac{3}{x}\right)\\

\end{array}
double f(double x) {
        double r5258393 = x;
        double r5258394 = 1.0;
        double r5258395 = r5258393 + r5258394;
        double r5258396 = r5258393 / r5258395;
        double r5258397 = r5258393 - r5258394;
        double r5258398 = r5258395 / r5258397;
        double r5258399 = r5258396 - r5258398;
        return r5258399;
}
↓
double f(double x) {
        double r5258400 = x;
        double r5258401 = -0.9983730585657322;
        bool r5258402 = r5258400 <= r5258401;
        double r5258403 = 3.0;
        double r5258404 = r5258400 * r5258400;
        double r5258405 = r5258400 * r5258404;
        double r5258406 = r5258403 / r5258405;
        double r5258407 = 1.0;
        double r5258408 = r5258407 / r5258404;
        double r5258409 = r5258406 + r5258408;
        double r5258410 = r5258403 / r5258400;
        double r5258411 = r5258409 + r5258410;
        double r5258412 = -r5258411;
        double r5258413 = 1.021370211112824;
        bool r5258414 = r5258400 <= r5258413;
        double r5258415 = fma(r5258404, r5258407, r5258407);
        double r5258416 = fma(r5258403, r5258400, r5258415);
        double r5258417 = r5258414 ? r5258416 : r5258412;
        double r5258418 = r5258402 ? r5258412 : r5258417;
        return r5258418;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.9983730585657322 or 1.021370211112824 < x

    1. Initial program 58.6

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube58.6

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)\right) \cdot \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)}}\]

    4. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(1 \cdot \frac{1}{{x}^{2}} + 3 \cdot \frac{1}{x}\right)\right)}\]

    5. Simplified0.4

      \[\leadsto \color{blue}{-\left(\frac{3}{x} + \left(\frac{1}{x \cdot x} + \frac{3}{\left(x \cdot x\right) \cdot x}\right)\right)}\]

    if -0.9983730585657322 < x < 1.021370211112824

    1. Initial program 0.0

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube0.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)\right) \cdot \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)}}\]

    4. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{3 \cdot x + \left(1 \cdot {x}^{2} + 1\right)}\]

    5. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;-\left(\left(\frac{3}{x \cdot \left(x \cdot x\right)} + \frac{1}{x \cdot x}\right) + \frac{3}{x}\right)\\ \mathbf{elif}\;x \le 1.021370211112824000210252961551304906607:\\ \;\;\;\;\mathsf{fma}\left(3, x, \mathsf{fma}\left(x \cdot x, 1, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(\frac{3}{x \cdot \left(x \cdot x\right)} + \frac{1}{x \cdot x}\right) + \frac{3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))