Average Error: 29.5 → 0.2
Time: 2.6s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.00048455376280864826:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + {x}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.00048455376280864826:\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + {x}^{-4}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)\right)}\\

\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))) 0.00048455376280864826)
   (- (- (/ -3.0 x) (/ 1.0 (* x x))) (+ (/ 3.0 (pow x 3.0)) (pow x -4.0)))
   (cbrt
    (*
     (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))
     (*
      (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))
      (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- 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))) <= 0.00048455376280864826) {
		tmp = ((-3.0 / x) - (1.0 / (x * x))) - ((3.0 / pow(x, 3.0)) + pow(x, -4.0));
	} else {
		tmp = cbrt(((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) * (((x / (x + 1.0)) - ((x + 1.0) / (x - 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 (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 4.8455376280865e-4

    1. Initial program 58.8

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.3

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

    5. Applied pow-flip_binary640.3

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

    6. Simplified0.3

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

    if 4.8455376280865e-4 < (-.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-cbrt-cube_binary640.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.00048455376280864826:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + {x}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right) \cdot \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)\right)}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021113 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))