Average Error: 29.6 → 0.0
Time: 3.8s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -5946613085329.435 \lor \neg \left(x \leq 358991.0367498862\right):\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x \cdot -3}{x \cdot x + -1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -5946613085329.435 \lor \neg \left(x \leq 358991.0367498862\right):\\
\;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 + x \cdot -3}{x \cdot 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 -5946613085329.435) (not (<= x 358991.0367498862)))
   (- (- (/ -3.0 x) (pow x -2.0)) (+ (/ 3.0 (pow x 3.0)) (/ 1.0 (pow x 4.0))))
   (/ (+ -1.0 (* x -3.0)) (+ (* x 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 <= -5946613085329.435) || !(x <= 358991.0367498862)) {
		tmp = ((-3.0 / x) - pow(x, -2.0)) - ((3.0 / pow(x, 3.0)) + (1.0 / pow(x, 4.0)));
	} else {
		tmp = (-1.0 + (x * -3.0)) / ((x * 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 < -5946613085329.43457 or 358991.036749886174 < x

    1. Initial program 60.0

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

    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. Using strategy rm

    5. Applied pow2_binary640.0

      \[\leadsto \left(\frac{-3}{x} - \frac{1}{\color{blue}{{x}^{2}}}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\]

    6. Applied pow-flip_binary640.0

      \[\leadsto \left(\frac{-3}{x} - \color{blue}{{x}^{\left(-2\right)}}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\]

    7. Simplified0.0

      \[\leadsto \left(\frac{-3}{x} - {x}^{\color{blue}{-2}}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\]

    if -5946613085329.43457 < x < 358991.036749886174

    1. Initial program 0.5

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

    3. Applied frac-sub_binary640.5

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

    4. Simplified0.5

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

    5. Simplified0.5

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

    6. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5946613085329.435 \lor \neg \left(x \leq 358991.0367498862\right):\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + x \cdot -3}{x \cdot x + -1}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))