Average Error: 29.4 → 0.1
Time: 4.4s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 6.486197660082738 \cdot 10^{-07}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x \cdot -3}{-1 + x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 6.486197660082738 \cdot 10^{-07}:\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 + x \cdot -3}{-1 + x \cdot x}\\

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 6.486197660082738e-07)
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (/ (+ -1.0 (* x -3.0)) (+ -1.0 (* x x)))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 6.486197660082738e-07) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = (-1.0 + (x * -3.0)) / (-1.0 + (x * x));
	}
	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 (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 6.48619766008e-7

    1. Initial program 59.0

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.3

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

    if 6.48619766008e-7 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

    1. Initial program 0.1

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

    3. Applied div-inv_binary64_28030.1

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

    5. Applied flip-+_binary64_27800.2

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

    6. Applied frac-times_binary64_28160.1

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

    7. Applied frac-sub_binary64_28150.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

    10. Taylor expanded around 0 0.0

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

    11. Simplified0.0

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

    13. Applied associate-/r*_binary64_27500.0

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

    14. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 6.486197660082738 \cdot 10^{-07}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x \cdot -3}{-1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020295 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))