Average Error: 29.7 → 0.1
Time: 44.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13076.124426981547:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 13803.703572778493:\\ \;\;\;\;\mathsf{fma}\left(1, \left(\frac{x}{1 + x}\right), \left(\frac{-1}{x - 1} \cdot \left(1 + x\right)\right)\right) + \mathsf{fma}\left(\left(\frac{-1}{x - 1}\right), \left(1 + x\right), \left(\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13076.124426981547:\\
\;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right)\\

\mathbf{elif}\;x \le 13803.703572778493:\\
\;\;\;\;\mathsf{fma}\left(1, \left(\frac{x}{1 + x}\right), \left(\frac{-1}{x - 1} \cdot \left(1 + x\right)\right)\right) + \mathsf{fma}\left(\left(\frac{-1}{x - 1}\right), \left(1 + x\right), \left(\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)\\

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

\end{array}
double f(double x) {
        double r4546370 = x;
        double r4546371 = 1.0;
        double r4546372 = r4546370 + r4546371;
        double r4546373 = r4546370 / r4546372;
        double r4546374 = r4546370 - r4546371;
        double r4546375 = r4546372 / r4546374;
        double r4546376 = r4546373 - r4546375;
        return r4546376;
}
↓
double f(double x) {
        double r4546377 = x;
        double r4546378 = -13076.124426981547;
        bool r4546379 = r4546377 <= r4546378;
        double r4546380 = -3.0;
        double r4546381 = r4546377 * r4546377;
        double r4546382 = r4546381 * r4546377;
        double r4546383 = r4546380 / r4546382;
        double r4546384 = -1.0;
        double r4546385 = r4546384 / r4546381;
        double r4546386 = r4546380 / r4546377;
        double r4546387 = r4546385 + r4546386;
        double r4546388 = r4546383 + r4546387;
        double r4546389 = 13803.703572778493;
        bool r4546390 = r4546377 <= r4546389;
        double r4546391 = 1.0;
        double r4546392 = r4546391 + r4546377;
        double r4546393 = r4546377 / r4546392;
        double r4546394 = r4546377 - r4546391;
        double r4546395 = r4546384 / r4546394;
        double r4546396 = r4546395 * r4546392;
        double r4546397 = fma(r4546391, r4546393, r4546396);
        double r4546398 = r4546391 / r4546394;
        double r4546399 = r4546398 * r4546392;
        double r4546400 = fma(r4546395, r4546392, r4546399);
        double r4546401 = r4546397 + r4546400;
        double r4546402 = r4546390 ? r4546401 : r4546388;
        double r4546403 = r4546379 ? r4546388 : r4546402;
        return r4546403;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -13076.124426981547 or 13803.703572778493 < x

    1. Initial program 59.2

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.0

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

    if -13076.124426981547 < x < 13803.703572778493

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\log \left(e^{\frac{x + 1}{x - 1}}\right)}\]

    4. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{x}{x + 1}}\right)} - \log \left(e^{\frac{x + 1}{x - 1}}\right)\]

    5. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{x}{x + 1}}}{e^{\frac{x + 1}{x - 1}}}\right)}\]

    6. Simplified0.1

      \[\leadsto \log \color{blue}{\left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right)}\]
    7. Using strategy rm

    8. Applied div-inv0.1

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

    9. Applied *-un-lft-identity0.1

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

    10. Applied *-un-lft-identity0.1

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

    11. Applied times-frac0.1

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

    12. Applied prod-diff0.1

      \[\leadsto \log \left(e^{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{1}\right), \left(\frac{x}{1 + x}\right), \left(-\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right) + \mathsf{fma}\left(\left(-\frac{1}{x - 1}\right), \left(1 + x\right), \left(\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)}}\right)\]

    13. Applied exp-sum0.1

      \[\leadsto \log \color{blue}{\left(e^{\mathsf{fma}\left(\left(\frac{1}{1}\right), \left(\frac{x}{1 + x}\right), \left(-\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)} \cdot e^{\mathsf{fma}\left(\left(-\frac{1}{x - 1}\right), \left(1 + x\right), \left(\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)}\right)}\]

    14. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(\left(\frac{1}{1}\right), \left(\frac{x}{1 + x}\right), \left(-\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)}\right) + \log \left(e^{\mathsf{fma}\left(\left(-\frac{1}{x - 1}\right), \left(1 + x\right), \left(\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)}\right)}\]

    15. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \left(\frac{x}{1 + x}\right), \left(\frac{-1}{x - 1} \cdot \left(1 + x\right)\right)\right)} + \log \left(e^{\mathsf{fma}\left(\left(-\frac{1}{x - 1}\right), \left(1 + x\right), \left(\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)}\right)\]

    16. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13076.124426981547:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 13803.703572778493:\\ \;\;\;\;\mathsf{fma}\left(1, \left(\frac{x}{1 + x}\right), \left(\frac{-1}{x - 1} \cdot \left(1 + x\right)\right)\right) + \mathsf{fma}\left(\left(\frac{-1}{x - 1}\right), \left(1 + x\right), \left(\frac{1}{x - 1} \cdot \left(1 + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))