Average Error: 14.5 → 0.2
Time: 3.8s
Precision: binary64
\[\frac{1}{x + 1} - \frac{1}{x - 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\ \;\;\;\;\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \left(x + 2\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{1 + x} - \frac{1}{x - 1} \leq 0:\\
\;\;\;\;\frac{\frac{-2}{x}}{x} - \frac{2}{{x}^{4}}\\

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


\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)))
   (/ (- x (+ x 2.0)) (fma 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 (((1.0 / (1.0 + x)) - (1.0 / (x - 1.0))) <= 0.0) {
		tmp = ((-2.0 / x) / x) - (2.0 / pow(x, 4.0));
	} else {
		tmp = (x - (x + 2.0)) / fma(x, x, -1.0);
	}
	return tmp;
}

Error

Bits error versus x

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 28.7

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

    2. Taylor expanded in x around inf 1.0

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

    3. Simplified1.0

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

    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 frac-sub_binary640.0

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

    3. Simplified0.0

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

    4. Simplified0.0

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022125 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))