Average Error: 29.6 → 0.1
Time: 3.6m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le -3.058877297278178 \cdot 10^{-05} \lor \neg \left(\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le 6.01037603715776 \cdot 10^{-09}\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x \cdot x - 1}\right)}^{3} \cdot {\left(x - 1\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \left(\frac{1 + x}{x - 1} + \frac{x}{1 + x}\right) \cdot \frac{1 + x}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \end{array}\]

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 (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < -3.058877297278178e-05 or 6.01037603715776e-09 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x)))

    1. Initial program 0.2

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

    3. Applied flip3--0.3

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

    4. Applied simplify0.2

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

    6. Applied flip-+0.3

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

    7. Applied associate-/r/0.3

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

    8. Applied unpow-prod-down0.3

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

    9. Applied simplify0.3

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

    if -3.058877297278178e-05 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < 6.01037603715776e-09

    1. Initial program 59.8

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

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

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le -3.058877297278178 \cdot 10^{-05} \lor \neg \left(\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le 6.01037603715776 \cdot 10^{-09}\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x \cdot x - 1}\right)}^{3} \cdot {\left(x - 1\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}{\frac{x}{1 + x} \cdot \frac{x}{1 + x} + \left(\frac{1 + x}{x - 1} + \frac{x}{1 + x}\right) \cdot \frac{1 + x}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \end{array}}\]

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018167 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))