Average Error: 29.2 → 0.2
Time: 20.5s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 4.386622570740073 \cdot 10^{-07}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 4.386622570740073 \cdot 10^{-07}:\\
\;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right)\\

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

\end{array}
double f(double x) {
        double r2461402 = x;
        double r2461403 = 1.0;
        double r2461404 = r2461402 + r2461403;
        double r2461405 = r2461402 / r2461404;
        double r2461406 = r2461402 - r2461403;
        double r2461407 = r2461404 / r2461406;
        double r2461408 = r2461405 - r2461407;
        return r2461408;
}
↓
double f(double x) {
        double r2461409 = x;
        double r2461410 = 1.0;
        double r2461411 = r2461410 + r2461409;
        double r2461412 = r2461409 / r2461411;
        double r2461413 = r2461409 - r2461410;
        double r2461414 = r2461411 / r2461413;
        double r2461415 = r2461412 - r2461414;
        double r2461416 = 4.386622570740073e-07;
        bool r2461417 = r2461415 <= r2461416;
        double r2461418 = -3.0;
        double r2461419 = r2461409 * r2461409;
        double r2461420 = r2461419 * r2461409;
        double r2461421 = r2461418 / r2461420;
        double r2461422 = r2461418 / r2461409;
        double r2461423 = -1.0;
        double r2461424 = r2461423 / r2461419;
        double r2461425 = r2461422 + r2461424;
        double r2461426 = r2461421 + r2461425;
        double r2461427 = r2461415 * r2461415;
        double r2461428 = r2461427 * r2461415;
        double r2461429 = cbrt(r2461428);
        double r2461430 = r2461417 ? r2461426 : r2461429;
        return r2461430;
}

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 (+ x 1)) (/ (+ x 1) (- x 1))) < 4.386622570740073e-07

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.2

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

    if 4.386622570740073e-07 < (- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 4.386622570740073 \cdot 10^{-07}:\\ \;\;\;\;\frac{-3}{\left(x \cdot x\right) \cdot x} + \left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)\right) \cdot \left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))