Average Error: 29.2 → 0.3
Time: 7.4s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.82386745995 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.82386745995 \cdot 10^{-5}:\\
\;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}\\

\end{array}
double code(double x) {
	return ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)));
}
↓
double code(double x) {
	double VAR;
	if ((((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 1.8238674599491134e-05)) {
		VAR = ((-1.0 / pow(x, 2.0)) - fma(3.0, (1.0 / x), (3.0 * (1.0 / pow(x, 3.0)))));
	} else {
		VAR = ((x / (x + 1.0)) - (1.0 / ((x - 1.0) / (x + 1.0))));
	}
	return VAR;
}

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 1.0)) (/ (+ x 1.0) (- x 1.0))) < 1.8238674599491134e-05

    1. Initial program 59.0

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    if 1.8238674599491134e-05 < (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))

    1. Initial program 0.1

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

    3. Applied clear-num0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.82386745995 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))