Average Error: 29.9 → 0.1
Time: 17.4s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10256.035048410798:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} + \frac{\frac{-1}{x}}{x} \cdot \frac{3}{x}\right) - \frac{3}{x}\\ \mathbf{elif}\;x \le 11268.875490349343:\\ \;\;\;\;\frac{1}{1 + x} \cdot x - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} + \frac{\frac{-1}{x}}{x} \cdot \frac{3}{x}\right) - \frac{3}{x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10256.035048410798:\\
\;\;\;\;\left(\frac{\frac{-1}{x}}{x} + \frac{\frac{-1}{x}}{x} \cdot \frac{3}{x}\right) - \frac{3}{x}\\

\mathbf{elif}\;x \le 11268.875490349343:\\
\;\;\;\;\frac{1}{1 + x} \cdot x - \frac{1 + x}{x - 1}\\

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

\end{array}
double f(double x) {
        double r2594025 = x;
        double r2594026 = 1.0;
        double r2594027 = r2594025 + r2594026;
        double r2594028 = r2594025 / r2594027;
        double r2594029 = r2594025 - r2594026;
        double r2594030 = r2594027 / r2594029;
        double r2594031 = r2594028 - r2594030;
        return r2594031;
}
↓
double f(double x) {
        double r2594032 = x;
        double r2594033 = -10256.035048410798;
        bool r2594034 = r2594032 <= r2594033;
        double r2594035 = -1.0;
        double r2594036 = r2594035 / r2594032;
        double r2594037 = r2594036 / r2594032;
        double r2594038 = 3.0;
        double r2594039 = r2594038 / r2594032;
        double r2594040 = r2594037 * r2594039;
        double r2594041 = r2594037 + r2594040;
        double r2594042 = r2594041 - r2594039;
        double r2594043 = 11268.875490349343;
        bool r2594044 = r2594032 <= r2594043;
        double r2594045 = 1.0;
        double r2594046 = r2594045 + r2594032;
        double r2594047 = r2594045 / r2594046;
        double r2594048 = r2594047 * r2594032;
        double r2594049 = r2594032 - r2594045;
        double r2594050 = r2594046 / r2594049;
        double r2594051 = r2594048 - r2594050;
        double r2594052 = r2594044 ? r2594051 : r2594042;
        double r2594053 = r2594034 ? r2594042 : r2594052;
        return r2594053;
}

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 < -10256.035048410798 or 11268.875490349343 < 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

      \[\leadsto \color{blue}{\frac{\frac{-1}{x \cdot x}}{x} \cdot 3 + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right)}\]
    4. Using strategy rm

    5. Applied associate-+r-0.0

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

    6. Simplified0.0

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

    if -10256.035048410798 < x < 11268.875490349343

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10256.035048410798:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} + \frac{\frac{-1}{x}}{x} \cdot \frac{3}{x}\right) - \frac{3}{x}\\ \mathbf{elif}\;x \le 11268.875490349343:\\ \;\;\;\;\frac{1}{1 + x} \cdot x - \frac{1 + x}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} + \frac{\frac{-1}{x}}{x} \cdot \frac{3}{x}\right) - \frac{3}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))