Average Error: 28.9 → 0.4
Time: 15.6s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 1.021370211112824000210252961551304906607:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 1 + 1\right) + 3 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9983730585657322187387308076722547411919:\\
\;\;\;\;\left(-\frac{3}{\left(x \cdot x\right) \cdot x}\right) - \left(\frac{3}{x} + \frac{\frac{1}{x}}{x}\right)\\

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

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

\end{array}
double f(double x) {
        double r5866256 = x;
        double r5866257 = 1.0;
        double r5866258 = r5866256 + r5866257;
        double r5866259 = r5866256 / r5866258;
        double r5866260 = r5866256 - r5866257;
        double r5866261 = r5866258 / r5866260;
        double r5866262 = r5866259 - r5866261;
        return r5866262;
}
↓
double f(double x) {
        double r5866263 = x;
        double r5866264 = -0.9983730585657322;
        bool r5866265 = r5866263 <= r5866264;
        double r5866266 = 3.0;
        double r5866267 = r5866263 * r5866263;
        double r5866268 = r5866267 * r5866263;
        double r5866269 = r5866266 / r5866268;
        double r5866270 = -r5866269;
        double r5866271 = r5866266 / r5866263;
        double r5866272 = 1.0;
        double r5866273 = r5866272 / r5866263;
        double r5866274 = r5866273 / r5866263;
        double r5866275 = r5866271 + r5866274;
        double r5866276 = r5866270 - r5866275;
        double r5866277 = 1.021370211112824;
        bool r5866278 = r5866263 <= r5866277;
        double r5866279 = r5866267 * r5866272;
        double r5866280 = r5866279 + r5866272;
        double r5866281 = r5866266 * r5866263;
        double r5866282 = r5866280 + r5866281;
        double r5866283 = r5866278 ? r5866282 : r5866276;
        double r5866284 = r5866265 ? r5866276 : r5866283;
        return r5866284;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around inf 0.7

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

    3. Simplified0.4

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

    if -0.9983730585657322 < x < 1.021370211112824

    1. Initial program 0.0

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.4

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

Reproduce

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