Average Error: 14.6 → 0.2
Time: 4.6s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} - \frac{1}{x - 1}\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\
\;\;\;\;\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (+ 1.0 x)) (/ 1.0 (- x 1.0))) 0.0)
   (- (- (/ (/ -2.0 x) x) (/ 2.0 (pow x 4.0))) (/ 2.0 (pow x 6.0)))
   (- (/ 1.0 (+ 1.0 x)) (/ 1.0 (- 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 (((1.0 / (1.0 + x)) - (1.0 / (x - 1.0))) <= 0.0) {
		tmp = (((-2.0 / x) / x) - (2.0 / pow(x, 4.0))) - (2.0 / pow(x, 6.0));
	} else {
		tmp = (1.0 / (1.0 + x)) - (1.0 / (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 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 (-.f64 x 1))) < 0.0

    1. Initial program 30.1

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

    2. Taylor expanded around inf 1.0

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

    3. Simplified1.0

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

    5. Applied associate-/r*_binary640.3

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

    if 0.0 < (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 (-.f64 x 1)))

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity_binary640.0

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

    4. Applied associate-/r*_binary640.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} - \frac{1}{x - 1}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021110 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))