Average Error: 28.8 → 0.1
Time: 8.1s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -29075.666064525398:\\ \;\;\;\;\frac{\frac{-5}{x \cdot x} - \left(\frac{6}{x} + \frac{16}{{x}^{3}}\right)}{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}\\ \mathbf{elif}\;x \leq 1630.155029845001:\\ \;\;\;\;\frac{x}{x + 1} - \sqrt[3]{{\left(\frac{x + 1}{x + -1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -29075.666064525398:\\
\;\;\;\;\frac{\frac{-5}{x \cdot x} - \left(\frac{6}{x} + \frac{16}{{x}^{3}}\right)}{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}\\

\mathbf{elif}\;x \leq 1630.155029845001:\\
\;\;\;\;\frac{x}{x + 1} - \sqrt[3]{{\left(\frac{x + 1}{x + -1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -29075.666064525398)
   (/
    (- (/ -5.0 (* x x)) (+ (/ 6.0 x) (/ 16.0 (pow x 3.0))))
    (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))))
   (if (<= x 1630.155029845001)
     (- (/ x (+ x 1.0)) (cbrt (pow (/ (+ x 1.0) (+ x -1.0)) 3.0)))
     (-
      (- (/ -3.0 x) (/ 1.0 (* x x)))
      (+ (/ 3.0 (pow x 3.0)) (/ 1.0 (pow x 4.0)))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -29075.666064525398) {
		tmp = ((-5.0 / (x * x)) - ((6.0 / x) + (16.0 / pow(x, 3.0)))) / ((x / (x + 1.0)) + ((x + 1.0) / (x + -1.0)));
	} else if (x <= 1630.155029845001) {
		tmp = (x / (x + 1.0)) - cbrt(pow(((x + 1.0) / (x + -1.0)), 3.0));
	} else {
		tmp = ((-3.0 / x) - (1.0 / (x * x))) - ((3.0 / pow(x, 3.0)) + (1.0 / pow(x, 4.0)));
	}
	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 3 regimes

  2. if x < -29075.6660645253978

    1. Initial program 59.3

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

    3. Applied flip--_binary64_278159.3

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

    4. Simplified59.3

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

    5. Simplified59.3

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

    6. Taylor expanded around inf 0.3

      \[\leadsto \frac{\color{blue}{-\left(5 \cdot \frac{1}{{x}^{2}} + \left(6 \cdot \frac{1}{x} + 16 \cdot \frac{1}{{x}^{3}}\right)\right)}}{\frac{x}{1 + x} + \frac{1 + x}{x + -1}}\]

    7. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{-5}{x \cdot x} - \left(\frac{6}{x} + \frac{16}{{x}^{3}}\right)}}{\frac{x}{1 + x} + \frac{1 + x}{x + -1}}\]

    if -29075.6660645253978 < x < 1630.1550298450011

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube_binary64_28420.1

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

    4. Simplified0.1

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

    if 1630.1550298450011 < x

    1. Initial program 59.1

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -29075.666064525398:\\ \;\;\;\;\frac{\frac{-5}{x \cdot x} - \left(\frac{6}{x} + \frac{16}{{x}^{3}}\right)}{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}\\ \mathbf{elif}\;x \leq 1630.155029845001:\\ \;\;\;\;\frac{x}{x + 1} - \sqrt[3]{{\left(\frac{x + 1}{x + -1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021013 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))