Average Error: 28.6 → 0.0
Time: 7.8s
Precision: binary64
Cost: 14658
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -10487.592747220726:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{elif}\;x \leq 1977.3153917547227:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -10487.592747220726:\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{elif}\;x \leq 1977.3153917547227:\\
\;\;\;\;\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}}\\

\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 -10487.592747220726)
   (- (- (/ -3.0 x) (/ 1.0 (* x x))) (/ 3.0 (pow x 3.0)))
   (if (<= x 1977.3153917547227)
     (/
      (-
       (* (/ x (+ x 1.0)) (/ x (+ x 1.0)))
       (* (/ (+ x 1.0) (- x 1.0)) (/ (+ x 1.0) (- x 1.0))))
      (+ (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
     (-
      (- (/ -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 <= -10487.592747220726) {
		tmp = ((-3.0 / x) - (1.0 / (x * x))) - (3.0 / pow(x, 3.0));
	} else if (x <= 1977.3153917547227) {
		tmp = (((x / (x + 1.0)) * (x / (x + 1.0))) - (((x + 1.0) / (x - 1.0)) * ((x + 1.0) / (x - 1.0)))) / ((x / (x + 1.0)) + ((x + 1.0) / (x - 1.0)));
	} 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

Alternatives

Alternative 1
Error0.1
Cost7624
\[\begin{array}{l} \mathbf{if}\;x \leq -10487.592747220726 \lor \neg \left(x \leq 13635.28794537155\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]
Alternative 2
Error0.1
Cost2114
\[\begin{array}{l} \mathbf{if}\;x \leq -515943.379764342:\\ \;\;\;\;\frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{elif}\;x \leq 540498.6414826551:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{x \cdot x + -1} - \frac{x + 1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-6}{x} + \frac{-5}{x \cdot x}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]
Alternative 3
Error0.1
Cost1544
\[\begin{array}{l} \mathbf{if}\;x \leq -515943.379764342 \lor \neg \left(x \leq 303456.6825641965\right):\\ \;\;\;\;\frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{x \cdot x + -1} - \frac{x + 1}{x - 1}\\ \end{array}\]
Alternative 4
Error0.1
Cost1288
\[\begin{array}{l} \mathbf{if}\;x \leq -515943.379764342 \lor \neg \left(x \leq 303456.6825641965\right):\\ \;\;\;\;\frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x}} - \frac{x + 1}{x - 1}\\ \end{array}\]
Alternative 5
Error0.1
Cost1160
\[\begin{array}{l} \mathbf{if}\;x \leq -515943.379764342 \lor \neg \left(x \leq 384257.84897965455\right):\\ \;\;\;\;\frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array}\]
Alternative 6
Error0.6
Cost904
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0247328982238428 \lor \neg \left(x \leq 0.992455561826402\right):\\ \;\;\;\;\frac{-3}{x} - \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array}\]
Alternative 7
Error0.9
Cost776
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0247328982238428 \lor \neg \left(x \leq 0.992455561826402\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x + 3\right)\\ \end{array}\]
Alternative 8
Error1.0
Cost648
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0247328982238428 \lor \neg \left(x \leq 0.992455561826402\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array}\]
Alternative 9
Error1.4
Cost520
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0247328982238428 \lor \neg \left(x \leq 0.992455561826402\right):\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 10
Error30.9
Cost64
\[1\]

Error

Derivation

  1. Split input into 3 regimes

  2. if x < -10487.592747220726

    1. Initial program 59.2

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

    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. Simplified0.0

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

    if -10487.592747220726 < x < 1977.31539175472267

    1. Initial program 0.1

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

    3. Applied flip--_binary64_34630.1

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

    4. Simplified0.1

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

    if 1977.31539175472267 < 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

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

    4. 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 simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10487.592747220726:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{elif}\;x \leq 1977.3153917547227:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))