Average Error: 14.0 → 0.1
Time: 5.8s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4125149662004715 \lor \neg \left(x \leq 114.8142218867796\right):\\ \;\;\;\;\left(\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\right) - \frac{2}{{x}^{8}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x - 1} \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4125149662004715 \lor \neg \left(x \leq 114.8142218867796\right):\\
\;\;\;\;\left(\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\right) - \frac{2}{{x}^{8}}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -4125149662004715.0) (not (<= x 114.8142218867796)))
   (-
    (- (- (/ (/ -2.0 x) x) (/ 2.0 (pow x 4.0))) (/ 2.0 (pow x 6.0)))
    (/ 2.0 (pow x 8.0)))
   (* (/ 1.0 (- (* x x) 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 ((x <= -4125149662004715.0) || !(x <= 114.8142218867796)) {
		tmp = ((((-2.0 / x) / x) - (2.0 / pow(x, 4.0))) - (2.0 / pow(x, 6.0))) - (2.0 / pow(x, 8.0));
	} else {
		tmp = (1.0 / ((x * x) - 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 x < -4125149662004715 or 114.814221886779606 < x

    1. Initial program 28.3

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

    2. Taylor expanded around inf 0.9

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \frac{1}{{x}^{4}} + 2 \cdot \frac{1}{{x}^{8}}\right)\right)\right)}\]

    3. Simplified0.9

      \[\leadsto \color{blue}{\left(\left(\frac{-2}{x \cdot x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\right) - \frac{2}{{x}^{8}}}\]
    4. Using strategy rm

    5. Applied associate-/r*_binary64_24090.1

      \[\leadsto \left(\left(\color{blue}{\frac{\frac{-2}{x}}{x}} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\right) - \frac{2}{{x}^{8}}\]

    if -4125149662004715 < x < 114.814221886779606

    1. Initial program 0.7

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

    3. Applied flip--_binary64_24400.7

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

    4. Applied associate-/r/_binary64_24110.7

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

    5. Applied flip-+_binary64_24390.7

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

    6. Applied associate-/r/_binary64_24110.7

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

    7. Applied distribute-lft-out--_binary64_24170.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4125149662004715 \lor \neg \left(x \leq 114.8142218867796\right):\\ \;\;\;\;\left(\left(\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\right) - \frac{2}{{x}^{6}}\right) - \frac{2}{{x}^{8}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x - 1} \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))