Average Error: 28.8 → 0.0
Time: 10.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\frac{-\left(-\left(3 + 1 \cdot \frac{1}{x}\right)\right)}{1 \cdot \frac{1}{x} - x}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\frac{-\left(-\left(3 + 1 \cdot \frac{1}{x}\right)\right)}{1 \cdot \frac{1}{x} - x}
double code(double x) {
	return ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)));
}
↓
double code(double x) {
	return (--(3.0 + (1.0 * (1.0 / x))) / ((1.0 * (1.0 / x)) - x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.8

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

  3. Applied frac-2neg28.8

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

  4. Applied clear-num28.8

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

  5. Applied frac-sub28.6

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

  6. Simplified28.6

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

  7. Taylor expanded around inf 0.1

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

  8. Taylor expanded around 0 0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))