Average Error: 29.3 → 0.3
Time: 3.6s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 8.384351546375512 \cdot 10^{-08}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 8.384351546375512 \cdot 10^{-08}:\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \frac{x + 1}{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))) 8.384351546375512e-08)
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (-
    (* (* (cbrt x) (cbrt x)) (/ (cbrt 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))) <= 8.384351546375512e-08) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = ((cbrt(x) * cbrt(x)) * (cbrt(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))) < 8.3843515464e-8

    1. Initial program 58.9

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.4

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

    if 8.3843515464e-8 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity_binary64_17830.1

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

    4. Applied add-cube-cbrt_binary64_18180.2

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

    5. Applied times-frac_binary64_17890.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{x + 1}} - \frac{x + 1}{x - 1}\]

    6. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 8.384351546375512 \cdot 10^{-08}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))