Average Error: 29.3 → 0.1
Time: 6.2s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -12791.804318799803 \lor \neg \left(x \leq 14978.088350211741\right):\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -12791.804318799803 \lor \neg \left(x \leq 14978.088350211741\right):\\
\;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-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 -12791.804318799803) (not (<= x 14978.088350211741)))
   (- (- (/ -3.0 x) (pow x -2.0)) (/ 3.0 (pow x 3.0)))
   (+ (/ x (+ x 1.0)) (* (+ x 1.0) (/ -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 <= -12791.804318799803) || !(x <= 14978.088350211741)) {
		tmp = ((-3.0 / x) - pow(x, -2.0)) - (3.0 / pow(x, 3.0));
	} else {
		tmp = (x / (x + 1.0)) + ((x + 1.0) * (-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 < -12791.8043187998028 or 14978.0883502117413 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    5. Applied pow2_binary64_15230.0

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

    6. Applied pow-flip_binary64_15160.0

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

    7. Simplified0.0

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

    if -12791.8043187998028 < x < 14978.0883502117413

    1. Initial program 0.1

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

    3. Applied div-inv_binary64_14390.1

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

    4. Applied cancel-sign-sub-inv_binary64_14080.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12791.804318799803 \lor \neg \left(x \leq 14978.088350211741\right):\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))