Average Error: 28.7 → 0.2
Time: 7.9s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.0010948536261903286:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\sqrt{x + 1}}{\frac{x - 1}{\sqrt{x + 1}}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.0010948536261903286:\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 0.0010948536261903286)
   (-
    (- (/ -3.0 x) (/ 1.0 (* x x)))
    (+ (/ 3.0 (pow x 3.0)) (/ 1.0 (pow x 4.0))))
   (- (/ x (+ x 1.0)) (/ (sqrt (+ x 1.0)) (/ (- x 1.0) (sqrt (+ x 1.0)))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 0.0010948536261903286) {
		tmp = ((-3.0 / x) - (1.0 / (x * x))) - ((3.0 / pow(x, 3.0)) + (1.0 / pow(x, 4.0)));
	} else {
		tmp = (x / (x + 1.0)) - (sqrt(x + 1.0) / ((x - 1.0) / sqrt(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 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 0.0010948536261903

    1. Initial program 58.8

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

    2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{4}} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)}\]

    if 0.0010948536261903 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary64_28280.3

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

    4. Applied associate-/l*_binary64_27510.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.0010948536261903286:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\sqrt{x + 1}}{\frac{x - 1}{\sqrt{x + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))