Average Error: 29.7 → 0.1
Time: 8.4s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9937.5860968431371 \lor \neg \left(x \le 8870.39112568465316\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{x}{x \cdot x - 1 \cdot 1} \cdot \left({x}^{3} - {1}^{3}\right)\right) \cdot \left(x - 1\right) - x \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\right) - \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right) \cdot 1}{\left(x - 1\right) \cdot \left(1 \cdot \left(1 + x\right) + {x}^{2}\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9937.5860968431371 \lor \neg \left(x \le 8870.39112568465316\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(\frac{x}{x \cdot x - 1 \cdot 1} \cdot \left({x}^{3} - {1}^{3}\right)\right) \cdot \left(x - 1\right) - x \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\right) - \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right) \cdot 1}{\left(x - 1\right) \cdot \left(1 \cdot \left(1 + x\right) + {x}^{2}\right)}\\

\end{array}
double code(double x) {
	return ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)));
}
↓
double code(double x) {
	double temp;
	if (((x <= -9937.586096843137) || !(x <= 8870.391125684653))) {
		temp = (((-1.0 / pow(x, 2.0)) - (3.0 / x)) - (3.0 / pow(x, 3.0)));
	} else {
		temp = ((((((x / ((x * x) - (1.0 * 1.0))) * (pow(x, 3.0) - pow(1.0, 3.0))) * (x - 1.0)) - (x * ((x * x) + ((1.0 * 1.0) + (x * 1.0))))) - (((x * x) + ((1.0 * 1.0) + (x * 1.0))) * 1.0)) / ((x - 1.0) * ((1.0 * (1.0 + x)) + pow(x, 2.0))));
	}
	return temp;
}

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 < -9937.586096843137 or 8870.391125684653 < x

    1. Initial program 59.3

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

    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. Simplified0.0

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

    if -9937.586096843137 < x < 8870.391125684653

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied associate-/r/0.1

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

    6. Applied flip3--0.1

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

    7. Applied associate-*r/0.1

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

    8. Applied frac-sub0.1

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

    9. Simplified0.1

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

    11. Applied distribute-lft-in0.1

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

    12. Applied associate--r+0.1

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

    13. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9937.5860968431371 \lor \neg \left(x \le 8870.39112568465316\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{x}{x \cdot x - 1 \cdot 1} \cdot \left({x}^{3} - {1}^{3}\right)\right) \cdot \left(x - 1\right) - x \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\right) - \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right) \cdot 1}{\left(x - 1\right) \cdot \left(1 \cdot \left(1 + x\right) + {x}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))