Average Error: 29.4 → 0.0
Time: 34.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10145.89298673575:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 22336.500710864904:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot x - \left(1 + x\right) \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10145.89298673575:\\
\;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\

\mathbf{elif}\;x \le 22336.500710864904:\\
\;\;\;\;\frac{\left(x - 1\right) \cdot x - \left(1 + x\right) \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(1 + x\right)}\\

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

\end{array}
double f(double x) {
        double r17232007 = x;
        double r17232008 = 1.0;
        double r17232009 = r17232007 + r17232008;
        double r17232010 = r17232007 / r17232009;
        double r17232011 = r17232007 - r17232008;
        double r17232012 = r17232009 / r17232011;
        double r17232013 = r17232010 - r17232012;
        return r17232013;
}
↓
double f(double x) {
        double r17232014 = x;
        double r17232015 = -10145.89298673575;
        bool r17232016 = r17232014 <= r17232015;
        double r17232017 = -1.0;
        double r17232018 = r17232014 * r17232014;
        double r17232019 = r17232017 / r17232018;
        double r17232020 = -3.0;
        double r17232021 = r17232018 * r17232014;
        double r17232022 = r17232020 / r17232021;
        double r17232023 = r17232020 / r17232014;
        double r17232024 = r17232022 + r17232023;
        double r17232025 = r17232019 + r17232024;
        double r17232026 = 22336.500710864904;
        bool r17232027 = r17232014 <= r17232026;
        double r17232028 = 1.0;
        double r17232029 = r17232014 - r17232028;
        double r17232030 = r17232029 * r17232014;
        double r17232031 = r17232028 + r17232014;
        double r17232032 = r17232031 * r17232031;
        double r17232033 = r17232030 - r17232032;
        double r17232034 = r17232029 * r17232031;
        double r17232035 = r17232033 / r17232034;
        double r17232036 = r17232027 ? r17232035 : r17232025;
        double r17232037 = r17232016 ? r17232025 : r17232036;
        return r17232037;
}

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 < -10145.89298673575 or 22336.500710864904 < x

    1. Initial program 59.1

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

    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. Simplified0.3

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

    4. Taylor expanded around -inf 0.3

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

    5. Simplified0.0

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

    if -10145.89298673575 < x < 22336.500710864904

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10145.89298673575:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \mathbf{elif}\;x \le 22336.500710864904:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot x - \left(1 + x\right) \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-3}{\left(x \cdot x\right) \cdot x} + \frac{-3}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))