Average Error: 29.4 → 0.1
Time: 4.3s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -14610.34704770997814193833619356155395508 \lor \neg \left(x \le 11258.21718550441983097698539495468139648\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -14610.34704770997814193833619356155395508 \lor \neg \left(x \le 11258.21718550441983097698539495468139648\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r115056 = x;
        double r115057 = 1.0;
        double r115058 = r115056 + r115057;
        double r115059 = r115056 / r115058;
        double r115060 = r115056 - r115057;
        double r115061 = r115058 / r115060;
        double r115062 = r115059 - r115061;
        return r115062;
}
↓
double f(double x) {
        double r115063 = x;
        double r115064 = -14610.347047709978;
        bool r115065 = r115063 <= r115064;
        double r115066 = 11258.21718550442;
        bool r115067 = r115063 <= r115066;
        double r115068 = !r115067;
        bool r115069 = r115065 || r115068;
        double r115070 = 1.0;
        double r115071 = -r115070;
        double r115072 = 2.0;
        double r115073 = pow(r115063, r115072);
        double r115074 = r115071 / r115073;
        double r115075 = 3.0;
        double r115076 = r115075 / r115063;
        double r115077 = r115074 - r115076;
        double r115078 = 3.0;
        double r115079 = pow(r115063, r115078);
        double r115080 = r115075 / r115079;
        double r115081 = r115077 - r115080;
        double r115082 = r115063 - r115070;
        double r115083 = r115063 * r115082;
        double r115084 = r115063 + r115070;
        double r115085 = r115084 * r115084;
        double r115086 = r115083 - r115085;
        double r115087 = r115063 * r115063;
        double r115088 = r115070 * r115070;
        double r115089 = r115087 - r115088;
        double r115090 = r115086 / r115089;
        double r115091 = exp(r115090);
        double r115092 = log(r115091);
        double r115093 = r115069 ? r115081 : r115092;
        return r115093;
}

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 < -14610.347047709978 or 11258.21718550442 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}}\]

    if -14610.347047709978 < x < 11258.21718550442

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

      \[\leadsto \log \color{blue}{\left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)}\]
    7. Using strategy rm

    8. Applied frac-sub0.1

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

    9. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14610.34704770997814193833619356155395508 \lor \neg \left(x \le 11258.21718550441983097698539495468139648\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))