Average Error: 14.2 → 0.1
Time: 9.3s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9927999296991189:\\ \;\;\;\;\frac{-2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{\frac{-2}{x}}{x}\right)\\ \mathbf{elif}\;x \le 199.8576714437756:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x}}}{\sqrt{1 + x}} - \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{\frac{-2}{x}}{x}\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.9927999296991189:\\
\;\;\;\;\frac{-2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{\frac{-2}{x}}{x}\right)\\

\mathbf{elif}\;x \le 199.8576714437756:\\
\;\;\;\;\frac{\frac{1}{\sqrt{1 + x}}}{\sqrt{1 + x}} - \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{\frac{-2}{x}}{x}\right)\\

\end{array}
double f(double x) {
        double r2140092 = 1.0;
        double r2140093 = x;
        double r2140094 = r2140093 + r2140092;
        double r2140095 = r2140092 / r2140094;
        double r2140096 = r2140093 - r2140092;
        double r2140097 = r2140092 / r2140096;
        double r2140098 = r2140095 - r2140097;
        return r2140098;
}
↓
double f(double x) {
        double r2140099 = x;
        double r2140100 = -0.9927999296991189;
        bool r2140101 = r2140099 <= r2140100;
        double r2140102 = -2.0;
        double r2140103 = r2140099 * r2140099;
        double r2140104 = r2140103 * r2140099;
        double r2140105 = r2140104 * r2140104;
        double r2140106 = r2140102 / r2140105;
        double r2140107 = r2140102 / r2140103;
        double r2140108 = r2140107 / r2140103;
        double r2140109 = r2140102 / r2140099;
        double r2140110 = r2140109 / r2140099;
        double r2140111 = r2140108 + r2140110;
        double r2140112 = r2140106 + r2140111;
        double r2140113 = 199.8576714437756;
        bool r2140114 = r2140099 <= r2140113;
        double r2140115 = 1.0;
        double r2140116 = r2140115 + r2140099;
        double r2140117 = sqrt(r2140116);
        double r2140118 = r2140115 / r2140117;
        double r2140119 = r2140118 / r2140117;
        double r2140120 = r2140099 - r2140115;
        double r2140121 = r2140115 / r2140120;
        double r2140122 = r2140119 - r2140121;
        double r2140123 = r2140114 ? r2140122 : r2140112;
        double r2140124 = r2140101 ? r2140112 : r2140123;
        return r2140124;
}

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 < -0.9927999296991189 or 199.8576714437756 < x

    1. Initial program 28.3

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

    2. Taylor expanded around inf 0.9

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{4}} + \left(2 \cdot \frac{1}{{x}^{6}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

    3. Simplified0.9

      \[\leadsto \color{blue}{-\left(\left(\frac{\frac{2}{x \cdot x}}{x \cdot x} + \frac{2}{x \cdot x}\right) + \frac{2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)}\]
    4. Using strategy rm

    5. Applied associate-/r*0.3

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

    if -0.9927999296991189 < x < 199.8576714437756

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{1}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} - \frac{1}{x - 1}\]

    4. Applied associate-/r*0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9927999296991189:\\ \;\;\;\;\frac{-2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{\frac{-2}{x}}{x}\right)\\ \mathbf{elif}\;x \le 199.8576714437756:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x}}}{\sqrt{1 + x}} - \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{-2}{x \cdot x}}{x \cdot x} + \frac{\frac{-2}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))