Average Error: 29.4 → 0.1
Time: 5.5s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -14614.459261386228 \lor \neg \left(x \leq 13111.01440601656\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{3} \cdot {\left(x + -1\right)}^{3} - {\left(x + 1\right)}^{6}}{{\left(-1 + x \cdot x\right)}^{3}}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x + -1}}{x + -1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -14614.459261386228 \lor \neg \left(x \leq 13111.01440601656\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{x}^{3} \cdot {\left(x + -1\right)}^{3} - {\left(x + 1\right)}^{6}}{{\left(-1 + x \cdot x\right)}^{3}}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{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 (or (<= x -14614.459261386228) (not (<= x 13111.01440601656)))
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (/
    (/
     (- (* (pow x 3.0) (pow (+ x -1.0) 3.0)) (pow (+ x 1.0) 6.0))
     (pow (+ -1.0 (* x x)) 3.0))
    (+
     (* (/ x (+ x 1.0)) (/ x (+ x 1.0)))
     (/ (+ x (* (+ x 1.0) (/ (+ 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 <= -14614.459261386228) || !(x <= 13111.01440601656)) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = (((pow(x, 3.0) * pow((x + -1.0), 3.0)) - pow((x + 1.0), 6.0)) / pow((-1.0 + (x * x)), 3.0)) / (((x / (x + 1.0)) * (x / (x + 1.0))) + ((x + ((x + 1.0) * ((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 x < -14614.4592613862278 or 13111.0144060165603 < x

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -14614.4592613862278 < x < 13111.0144060165603

    1. Initial program 0.1

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

    3. Applied flip3--_binary64_14590.1

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

    4. Simplified0.1

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

    6. Applied cube-div_binary64_14840.1

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

    7. Applied cube-div_binary64_14840.1

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

    8. Applied frac-sub_binary64_14640.1

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

    9. Simplified0.1

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

    10. Simplified0.1

      \[\leadsto \frac{\frac{{x}^{3} \cdot {\left(x - 1\right)}^{3} - {\left(x + 1\right)}^{6}}{\color{blue}{{\left(-1 + x \cdot x\right)}^{3}}}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14614.459261386228 \lor \neg \left(x \leq 13111.01440601656\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{3} \cdot {\left(x + -1\right)}^{3} - {\left(x + 1\right)}^{6}}{{\left(-1 + x \cdot x\right)}^{3}}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x + -1}}{x + -1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))