Average Error: 29.0 → 0.1
Time: 2.6s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -11903.191960040664 \lor \neg \left(x \leq 15314.37957646558\right):\\ \;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right) \cdot \sqrt[3]{\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -11903.191960040664 \lor \neg \left(x \leq 15314.37957646558\right):\\
\;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -11903.191960040664) (not (<= x 15314.37957646558)))
   (- (/ (- (+ (/ 1.0 x) 3.0)) x) (/ 3.0 (pow x 3.0)))
   (*
    (*
     (cbrt (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
     (cbrt (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))))
    (cbrt (log (exp (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))))))))
double code(double x) {
	return ((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	double tmp;
	if (((x <= -11903.191960040664) || !(x <= 15314.37957646558))) {
		tmp = ((double) ((((double) -(((double) ((1.0 / x) + 3.0)))) / x) - (3.0 / ((double) pow(x, 3.0)))));
	} else {
		tmp = ((double) (((double) (((double) cbrt(((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (x - 1.0))))))) * ((double) cbrt(((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (x - 1.0))))))))) * ((double) cbrt(((double) log(((double) exp(((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (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 < -11903.1919600406636 or 15314.37957646558 < x

    1. Initial program 59.1

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    5. Applied associate-*l/_binary640.0

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

    6. Simplified0.0

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

    if -11903.1919600406636 < x < 15314.37957646558

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt_binary640.1

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

    5. Applied add-log-exp_binary640.1

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

    6. Applied add-log-exp_binary640.1

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

    7. Applied diff-log_binary640.1

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

    8. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11903.191960040664 \lor \neg \left(x \leq 15314.37957646558\right):\\ \;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right) \cdot \sqrt[3]{\log \left(e^{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))