Average Error: 29.6 → 0.1
Time: 13.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8182.424782641311:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 10123.071980988583:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x \cdot x + \left(1 - x\right)\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8182.424782641311:\\
\;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\

\mathbf{elif}\;x \le 10123.071980988583:\\
\;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x \cdot x + \left(1 - x\right)\right) \cdot \left(x - 1\right)}\\

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

\end{array}
double f(double x) {
        double r2344531 = x;
        double r2344532 = 1.0;
        double r2344533 = r2344531 + r2344532;
        double r2344534 = r2344531 / r2344533;
        double r2344535 = r2344531 - r2344532;
        double r2344536 = r2344533 / r2344535;
        double r2344537 = r2344534 - r2344536;
        return r2344537;
}
↓
double f(double x) {
        double r2344538 = x;
        double r2344539 = -8182.424782641311;
        bool r2344540 = r2344538 <= r2344539;
        double r2344541 = -1.0;
        double r2344542 = r2344538 * r2344538;
        double r2344543 = r2344541 / r2344542;
        double r2344544 = -3.0;
        double r2344545 = r2344544 / r2344538;
        double r2344546 = r2344543 + r2344545;
        double r2344547 = r2344545 / r2344542;
        double r2344548 = r2344546 + r2344547;
        double r2344549 = 10123.071980988583;
        bool r2344550 = r2344538 <= r2344549;
        double r2344551 = 1.0;
        double r2344552 = r2344538 + r2344551;
        double r2344553 = r2344538 / r2344552;
        double r2344554 = 3.0;
        double r2344555 = pow(r2344538, r2344554);
        double r2344556 = r2344551 + r2344555;
        double r2344557 = r2344551 - r2344538;
        double r2344558 = r2344542 + r2344557;
        double r2344559 = r2344538 - r2344551;
        double r2344560 = r2344558 * r2344559;
        double r2344561 = r2344556 / r2344560;
        double r2344562 = r2344553 - r2344561;
        double r2344563 = r2344550 ? r2344562 : r2344548;
        double r2344564 = r2344540 ? r2344548 : r2344563;
        return r2344564;
}

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 < -8182.424782641311 or 10123.071980988583 < 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

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

    if -8182.424782641311 < x < 10123.071980988583

    1. Initial program 0.1

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

    3. Applied flip3-+0.1

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

    4. Applied associate-/l/0.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8182.424782641311:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 10123.071980988583:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x \cdot x + \left(1 - x\right)\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} + \frac{-3}{x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))