Average Error: 14.3 → 0.1
Time: 9.6s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1061679461205214.1 \lor \neg \left(x \leq 13871.13225992715\right):\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(-2 - x\right)}{x \cdot x + -1}\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1061679461205214.1 \lor \neg \left(x \leq 13871.13225992715\right):\\
\;\;\;\;\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \left(-2 - x\right)}{x \cdot x + -1}\\

\end{array}
(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -1061679461205214.1) (not (<= x 13871.13225992715)))
   (- (- (/ (/ -2.0 x) x) (/ 2.0 (pow x 4.0))) (/ 2.0 (pow x 6.0)))
   (/ (+ x (- -2.0 x)) (+ (* x x) -1.0))))
double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -1061679461205214.1) || !(x <= 13871.13225992715)) {
		tmp = (((-2.0 / x) / x) - (2.0 / pow(x, 4.0))) - (2.0 / pow(x, 6.0));
	} else {
		tmp = (x + (-2.0 - x)) / ((x * x) + -1.0);
	}
	return tmp;
}

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 < -1061679461205214.12 or 13871.1322599271498 < x

    1. Initial program 28.9

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

    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. Simplified0.7

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

    5. Applied clear-num_binary64_38280.7

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

    6. Simplified0.8

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

    8. Applied associate-/r/_binary64_37750.7

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

    9. Applied associate-/r*_binary64_37730.1

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

    10. Simplified0.1

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

    if -1061679461205214.12 < x < 13871.1322599271498

    1. Initial program 0.6

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

    3. Applied frac-sub_binary64_38380.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1061679461205214.1 \lor \neg \left(x \leq 13871.13225992715\right):\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(-2 - x\right)}{x \cdot x + -1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))