Average Error: 29.6 → 0.1
Time: 33.1s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8182.424782641311:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 10123.071980988583:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1 + {x}^{3}}{\left(\left(x \cdot x + 1\right) - x\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) - \frac{\frac{3}{x}}{x \cdot x}\\ \end{array}\]
\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}{1 + x} - \frac{1 + {x}^{3}}{\left(\left(x \cdot x + 1\right) - x\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 r5012721 = x;
        double r5012722 = 1.0;
        double r5012723 = r5012721 + r5012722;
        double r5012724 = r5012721 / r5012723;
        double r5012725 = r5012721 - r5012722;
        double r5012726 = r5012723 / r5012725;
        double r5012727 = r5012724 - r5012726;
        return r5012727;
}
↓
double f(double x) {
        double r5012728 = x;
        double r5012729 = -8182.424782641311;
        bool r5012730 = r5012728 <= r5012729;
        double r5012731 = -1.0;
        double r5012732 = r5012728 * r5012728;
        double r5012733 = r5012731 / r5012732;
        double r5012734 = 3.0;
        double r5012735 = r5012734 / r5012728;
        double r5012736 = r5012733 - r5012735;
        double r5012737 = r5012735 / r5012732;
        double r5012738 = r5012736 - r5012737;
        double r5012739 = 10123.071980988583;
        bool r5012740 = r5012728 <= r5012739;
        double r5012741 = 1.0;
        double r5012742 = r5012741 + r5012728;
        double r5012743 = r5012728 / r5012742;
        double r5012744 = pow(r5012728, r5012734);
        double r5012745 = r5012741 + r5012744;
        double r5012746 = r5012732 + r5012741;
        double r5012747 = r5012746 - r5012728;
        double r5012748 = r5012728 - r5012741;
        double r5012749 = r5012747 * r5012748;
        double r5012750 = r5012745 / r5012749;
        double r5012751 = r5012743 - r5012750;
        double r5012752 = r5012740 ? r5012751 : r5012738;
        double r5012753 = r5012730 ? r5012738 : r5012752;
        return r5012753;
}

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

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

    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

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

  4. Final simplification0.1

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

Reproduce

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