Average Error: 14.5 → 0.0
Time: 1.7m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -82806819079.2980499 \lor \neg \left(x \le 424.61226587108411\right):\\ \;\;\;\;-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot {x}^{\left(-2\right)} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \frac{x - 1}{\sqrt[3]{1}} - \frac{x + 1}{\sqrt{1}} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{\frac{x + 1}{\sqrt{1}} \cdot \frac{x - 1}{\sqrt[3]{1}}}\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -82806819079.2980499 \lor \neg \left(x \le 424.61226587108411\right):\\
\;\;\;\;-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot {x}^{\left(-2\right)} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{1} \cdot \frac{x - 1}{\sqrt[3]{1}} - \frac{x + 1}{\sqrt{1}} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{\frac{x + 1}{\sqrt{1}} \cdot \frac{x - 1}{\sqrt[3]{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 <= -82806819079.29805) || !(x <= 424.6122658710841))) {
		VAR = ((double) -(((double) (((double) (2.0 * ((double) (1.0 / ((double) pow(x, 6.0)))))) + ((double) (((double) (2.0 * ((double) pow(x, ((double) -(2.0)))))) + ((double) (2.0 * ((double) (1.0 / ((double) pow(x, 4.0))))))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) sqrt(1.0)) * ((double) (((double) (x - 1.0)) / ((double) cbrt(1.0)))))) - ((double) (((double) (((double) (x + 1.0)) / ((double) sqrt(1.0)))) * ((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))))))) / ((double) (((double) (((double) (x + 1.0)) / ((double) sqrt(1.0)))) * ((double) (((double) (x - 1.0)) / ((double) cbrt(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 < -82806819079.29805 or 424.6122658710841 < x

    1. Initial program 29.4

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

    2. Taylor expanded around inf 0.7

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

    4. Applied pow-flip0.0

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

    if -82806819079.29805 < x < 424.6122658710841

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.3

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

    4. Applied associate-/l*0.3

      \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{x - 1}{\sqrt[3]{1}}}}\]

    5. Applied add-sqr-sqrt0.3

      \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{x + 1} - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{x - 1}{\sqrt[3]{1}}}\]

    6. Applied associate-/l*0.3

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\frac{x + 1}{\sqrt{1}}}} - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{x - 1}{\sqrt[3]{1}}}\]

    7. Applied frac-sub0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -82806819079.2980499 \lor \neg \left(x \le 424.61226587108411\right):\\ \;\;\;\;-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot {x}^{\left(-2\right)} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \frac{x - 1}{\sqrt[3]{1}} - \frac{x + 1}{\sqrt{1}} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{\frac{x + 1}{\sqrt{1}} \cdot \frac{x - 1}{\sqrt[3]{1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))