Average Error: 29.2 → 0.1
Time: 4.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -19531.3574826783042 \lor \neg \left(x \le 11598.163947811103\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -19531.3574826783042 \lor \neg \left(x \le 11598.163947811103\right):\\
\;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r120053 = x;
        double r120054 = 1.0;
        double r120055 = r120053 + r120054;
        double r120056 = r120053 / r120055;
        double r120057 = r120053 - r120054;
        double r120058 = r120055 / r120057;
        double r120059 = r120056 - r120058;
        return r120059;
}
↓
double f(double x) {
        double r120060 = x;
        double r120061 = -19531.357482678304;
        bool r120062 = r120060 <= r120061;
        double r120063 = 11598.163947811103;
        bool r120064 = r120060 <= r120063;
        double r120065 = !r120064;
        bool r120066 = r120062 || r120065;
        double r120067 = 1.0;
        double r120068 = 2.0;
        double r120069 = pow(r120060, r120068);
        double r120070 = r120067 / r120069;
        double r120071 = 3.0;
        double r120072 = r120071 / r120060;
        double r120073 = r120070 + r120072;
        double r120074 = -r120073;
        double r120075 = 1.0;
        double r120076 = 3.0;
        double r120077 = pow(r120060, r120076);
        double r120078 = r120075 / r120077;
        double r120079 = r120071 * r120078;
        double r120080 = r120074 - r120079;
        double r120081 = r120060 - r120067;
        double r120082 = r120060 * r120081;
        double r120083 = r120060 + r120067;
        double r120084 = r120083 * r120083;
        double r120085 = r120082 - r120084;
        double r120086 = r120060 * r120060;
        double r120087 = r120067 * r120067;
        double r120088 = r120086 - r120087;
        double r120089 = r120085 / r120088;
        double r120090 = r120066 ? r120080 : r120089;
        return r120090;
}

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 < -19531.357482678304 or 11598.163947811103 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)}\]
    4. Using strategy rm

    5. Applied fma-udef0.3

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

    6. Applied associate--r+0.3

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

    7. Simplified0.0

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

    if -19531.357482678304 < x < 11598.163947811103

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -19531.3574826783042 \lor \neg \left(x \le 11598.163947811103\right):\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))