Average Error: 29.3 → 0.1
Time: 6.0m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9041.134741942875:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 8821.768090851323:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\left(x \cdot x\right) \cdot x + 1}{\left(x - 1\right) \cdot \left(\left(x \cdot x + 1\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9041.134741942875:\\
\;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\

\mathbf{elif}\;x \le 8821.768090851323:\\
\;\;\;\;\frac{x}{x + 1} - \frac{\left(x \cdot x\right) \cdot x + 1}{\left(x - 1\right) \cdot \left(\left(x \cdot x + 1\right) - x\right)}\\

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

\end{array}
double f(double x) {
        double r62226412 = x;
        double r62226413 = 1.0;
        double r62226414 = r62226412 + r62226413;
        double r62226415 = r62226412 / r62226414;
        double r62226416 = r62226412 - r62226413;
        double r62226417 = r62226414 / r62226416;
        double r62226418 = r62226415 - r62226417;
        return r62226418;
}
↓
double f(double x) {
        double r62226419 = x;
        double r62226420 = -9041.134741942875;
        bool r62226421 = r62226419 <= r62226420;
        double r62226422 = -3.0;
        double r62226423 = r62226422 / r62226419;
        double r62226424 = r62226419 * r62226419;
        double r62226425 = r62226423 / r62226424;
        double r62226426 = 1.0;
        double r62226427 = r62226426 / r62226424;
        double r62226428 = r62226427 - r62226423;
        double r62226429 = r62226425 - r62226428;
        double r62226430 = 8821.768090851323;
        bool r62226431 = r62226419 <= r62226430;
        double r62226432 = r62226419 + r62226426;
        double r62226433 = r62226419 / r62226432;
        double r62226434 = r62226424 * r62226419;
        double r62226435 = r62226434 + r62226426;
        double r62226436 = r62226419 - r62226426;
        double r62226437 = r62226424 + r62226426;
        double r62226438 = r62226437 - r62226419;
        double r62226439 = r62226436 * r62226438;
        double r62226440 = r62226435 / r62226439;
        double r62226441 = r62226433 - r62226440;
        double r62226442 = r62226431 ? r62226441 : r62226429;
        double r62226443 = r62226421 ? r62226429 : r62226442;
        return r62226443;
}

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 < -9041.134741942875 or 8821.768090851323 < x

    1. Initial program 59.3

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

    3. Applied div-inv59.5

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\left(x + 1\right) \cdot \frac{1}{x - 1}}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube59.5

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

    6. Taylor expanded around inf 0.3

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

    7. Simplified0.0

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

    if -9041.134741942875 < x < 8821.768090851323

    1. Initial program 0.1

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

    3. Applied div-inv0.1

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\left(x + 1\right) \cdot \frac{1}{x - 1}}\]
    4. Using strategy rm

    5. Applied flip3-+0.1

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

    6. Applied frac-times0.1

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

    7. Simplified0.1

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

    8. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9041.134741942875:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 8821.768090851323:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\left(x \cdot x\right) \cdot x + 1}{\left(x - 1\right) \cdot \left(\left(x \cdot x + 1\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{x}}{x \cdot x} - \left(\frac{1}{x \cdot x} - \frac{-3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))