Average Error: 14.2 → 0.1
Time: 2.6s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.01819289813917391 \lor \neg \left(x \le 193.101107062296165\right):\\ \;\;\;\;2 \cdot \left(\frac{-1}{{x}^{6}} - \left({x}^{-2} + \frac{1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x + 1}} \cdot \sqrt{\frac{1}{x + 1}} - \frac{1}{x - 1}\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.01819289813917391 \lor \neg \left(x \le 193.101107062296165\right):\\
\;\;\;\;2 \cdot \left(\frac{-1}{{x}^{6}} - \left({x}^{-2} + \frac{1}{{x}^{4}}\right)\right)\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (1.0 / ((double) (x + 1.0)))) - ((double) (1.0 / ((double) (x - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -1.018192898139174) || !(x <= 193.10110706229617))) {
		VAR = ((double) (2.0 * ((double) (((double) (-1.0 / ((double) pow(x, 6.0)))) - ((double) (((double) pow(x, -2.0)) + ((double) (1.0 / ((double) pow(x, 4.0))))))))));
	} else {
		VAR = ((double) (((double) (((double) sqrt(((double) (1.0 / ((double) (x + 1.0)))))) * ((double) sqrt(((double) (1.0 / ((double) (x + 1.0)))))))) - ((double) (1.0 / ((double) (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 < -1.01819289813917391 or 193.101107062296165 < x

    1. Initial program 28.5

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

    2. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)}\]

    3. Simplified0.8

      \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{{x}^{6}} - \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)\right)}\]
    4. Using strategy rm

    5. Applied pow-flip0.1

      \[\leadsto 2 \cdot \left(\frac{-1}{{x}^{6}} - \left(\color{blue}{{x}^{\left(-2\right)}} + \frac{1}{{x}^{4}}\right)\right)\]

    6. Simplified0.1

      \[\leadsto 2 \cdot \left(\frac{-1}{{x}^{6}} - \left({x}^{\color{blue}{-2}} + \frac{1}{{x}^{4}}\right)\right)\]

    if -1.01819289813917391 < x < 193.101107062296165

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.01819289813917391 \lor \neg \left(x \le 193.101107062296165\right):\\ \;\;\;\;2 \cdot \left(\frac{-1}{{x}^{6}} - \left({x}^{-2} + \frac{1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x + 1}} \cdot \sqrt{\frac{1}{x + 1}} - \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))