Average Error: 29.1 → 0.0
Time: 3.8s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2698086.9467811254 \lor \neg \left(x \leq 683394.7402826605\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x \cdot 3}{-1 + {x}^{6}} \cdot \left(1 + \left(x \cdot x + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2698086.9467811254 \lor \neg \left(x \leq 683394.7402826605\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 - x \cdot 3}{-1 + {x}^{6}} \cdot \left(1 + \left(x \cdot x + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -2698086.9467811254) (not (<= x 683394.7402826605)))
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (*
    (/ (- -1.0 (* x 3.0)) (+ -1.0 (pow x 6.0)))
    (+ 1.0 (+ (* x x) (* (* x x) (* 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 <= -2698086.9467811254) || !(x <= 683394.7402826605)) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = ((-1.0 - (x * 3.0)) / (-1.0 + pow(x, 6.0))) * (1.0 + ((x * x) + ((x * x) * (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 x < -2698086.94678112539 or 683394.740282660467 < x

    1. Initial program 59.6

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

    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

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

    if -2698086.94678112539 < x < 683394.740282660467

    1. Initial program 0.1

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

    3. Applied frac-sub_binary64_55430.1

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

    4. Simplified0.1

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

    5. Taylor expanded around 0 0.0

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

    6. Simplified0.0

      \[\leadsto \frac{\color{blue}{-1 - x \cdot 3}}{-1 + x \cdot x}\]
    7. Using strategy rm

    8. Applied flip3-+_binary64_55370.0

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

    9. Applied associate-/r/_binary64_54800.0

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

    10. Simplified0.0

      \[\leadsto \color{blue}{\frac{-1 - x \cdot 3}{-1 + {x}^{6}}} \cdot \left(-1 \cdot -1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - -1 \cdot \left(x \cdot x\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2698086.9467811254 \lor \neg \left(x \leq 683394.7402826605\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x \cdot 3}{-1 + {x}^{6}} \cdot \left(1 + \left(x \cdot x + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020352 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))