Average Error: 28.7 → 0.1
Time: 2.1s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.00029194385033270986:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + {x}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.00029194385033270986:\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + {x}^{-4}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{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))) 0.00029194385033270986)
   (- (- (/ -3.0 x) (/ 1.0 (* x x))) (+ (/ 3.0 (pow x 3.0)) (pow x -4.0)))
   (-
    (/ (/ x (* (cbrt (+ x 1.0)) (cbrt (+ x 1.0)))) (cbrt (+ 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))) <= 0.00029194385033270986) {
		tmp = ((-3.0 / x) - (1.0 / (x * x))) - ((3.0 / pow(x, 3.0)) + pow(x, -4.0));
	} else {
		tmp = ((x / (cbrt(x + 1.0) * cbrt(x + 1.0))) / cbrt(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))) < 2.9194385033271e-4

    1. Initial program 58.9

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

    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

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

    5. Applied pow-flip_binary640.2

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

    6. Simplified0.2

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

    if 2.9194385033271e-4 < (-.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 add-cube-cbrt_binary640.1

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

    4. Applied associate-/r*_binary640.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.00029194385033270986:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + {x}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021148 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))