Average Error: 29.0 → 0.2
Time: 3.9s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -29361.18241715501:\\ \;\;\;\;\frac{-\left(\frac{12}{x \cdot x} + \left(\frac{9}{x} + \frac{48}{{x}^{3}}\right)\right)}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \mathbf{elif}\;x \leq 14875.083937536368:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{{x}^{3} - {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} \cdot \left(\frac{1}{x} + 3\right) - \frac{3}{{x}^{3}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -29361.18241715501:\\
\;\;\;\;\frac{-\left(\frac{12}{x \cdot x} + \left(\frac{9}{x} + \frac{48}{{x}^{3}}\right)\right)}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\

\mathbf{elif}\;x \leq 14875.083937536368:\\
\;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{{x}^{3} - {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -29361.18241715501)
   (/
    (- (+ (/ 12.0 (* x x)) (+ (/ 9.0 x) (/ 48.0 (pow x 3.0)))))
    (+
     (* (/ x (+ x 1.0)) (/ x (+ x 1.0)))
     (/ (+ x (* (+ x 1.0) (/ (+ x 1.0) (- x 1.0)))) (- x 1.0))))
   (if (<= x 14875.083937536368)
     (-
      (/ x (+ x 1.0))
      (*
       (/ (+ x 1.0) (- (pow x 3.0) (pow 1.0 3.0)))
       (+ (* x x) (+ (* 1.0 1.0) (* x 1.0)))))
     (- (* (/ -1.0 x) (+ (/ 1.0 x) 3.0)) (/ 3.0 (pow x 3.0))))))
double code(double x) {
	return ((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -29361.18241715501)) {
		tmp = (((double) -(((double) ((12.0 / ((double) (x * x))) + ((double) ((9.0 / x) + (48.0 / ((double) pow(x, 3.0))))))))) / ((double) (((double) ((x / ((double) (x + 1.0))) * (x / ((double) (x + 1.0))))) + (((double) (x + ((double) (((double) (x + 1.0)) * (((double) (x + 1.0)) / ((double) (x - 1.0))))))) / ((double) (x - 1.0))))));
	} else {
		double tmp_1;
		if ((x <= 14875.083937536368)) {
			tmp_1 = ((double) ((x / ((double) (x + 1.0))) - ((double) ((((double) (x + 1.0)) / ((double) (((double) pow(x, 3.0)) - ((double) pow(1.0, 3.0))))) * ((double) (((double) (x * x)) + ((double) (((double) (1.0 * 1.0)) + ((double) (x * 1.0))))))))));
		} else {
			tmp_1 = ((double) (((double) ((-1.0 / x) * ((double) ((1.0 / x) + 3.0)))) - (3.0 / ((double) pow(x, 3.0)))));
		}
		tmp = tmp_1;
	}
	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 < -29361.18241715501

    1. Initial program 59.4

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

    3. Applied flip3--_binary6459.4

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

    4. Simplified59.4

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

    5. Taylor expanded around inf 0.5

      \[\leadsto \frac{\color{blue}{-\left(12 \cdot \frac{1}{{x}^{2}} + \left(9 \cdot \frac{1}{x} + 48 \cdot \frac{1}{{x}^{3}}\right)\right)}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\]

    6. Simplified0.4

      \[\leadsto \frac{\color{blue}{-\left(\frac{12}{x \cdot x} + \left(\frac{9}{x} + \frac{48}{{x}^{3}}\right)\right)}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\]

    if -29361.18241715501 < x < 14875.083937536368

    1. Initial program 0.1

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

    3. Applied flip3--_binary640.1

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

    4. Applied associate-/r/_binary640.1

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

    if 14875.083937536368 < 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. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -29361.18241715501:\\ \;\;\;\;\frac{-\left(\frac{12}{x \cdot x} + \left(\frac{9}{x} + \frac{48}{{x}^{3}}\right)\right)}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \mathbf{elif}\;x \leq 14875.083937536368:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{{x}^{3} - {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} \cdot \left(\frac{1}{x} + 3\right) - \frac{3}{{x}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))