Average Error: 14.6 → 0.2
Time: 7.2s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1} \]
↓
\[\begin{array}{l} t_0 := \frac{1}{x - 1}\\ \mathbf{if}\;\frac{1}{1 + x} - t_0 \leq 0:\\ \;\;\;\;\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(x\right)} - t_0\\ \end{array} \]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
t_0 := \frac{1}{x - 1}\\
\mathbf{if}\;\frac{1}{1 + x} - t_0 \leq 0:\\
\;\;\;\;\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\\

\mathbf{else}:\\
\;\;\;\;e^{-\mathsf{log1p}\left(x\right)} - t_0\\


\end{array}

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- x 1.0))))
   (if (<= (- (/ 1.0 (+ 1.0 x)) t_0) 0.0)
     (- (/ (/ -2.0 x) x) (/ 2.0 (pow x 4.0)))
     (- (exp (- (log1p x))) t_0))))
double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}
↓
double code(double x) {
	double t_0 = 1.0 / (x - 1.0);
	double tmp;
	if (((1.0 / (1.0 + x)) - t_0) <= 0.0) {
		tmp = ((-2.0 / x) / x) - (2.0 / pow(x, 4.0));
	} else {
		tmp = exp(-log1p(x)) - t_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 29.0

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

    2. Taylor expanded in x around inf 1.1

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

    3. Simplified1.1

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x} - \frac{2}{{x}^{4}}} \]

    4. Applied associate-/r*_binary640.4

      \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} - \frac{2}{{x}^{4}} \]

    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. Applied add-exp-log_binary640.0

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

    3. Applied 1-exp_binary640.0

      \[\leadsto \frac{\color{blue}{e^{0}}}{e^{\log \left(x + 1\right)}} - \frac{1}{x - 1} \]

    4. Applied div-exp_binary640.0

      \[\leadsto \color{blue}{e^{0 - \log \left(x + 1\right)}} - \frac{1}{x - 1} \]

    5. Simplified0.0

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(x\right)}} - \frac{1}{x - 1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(x\right)} - \frac{1}{x - 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))