Average Error: 29.3 → 0.2
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.215708945304996 \cdot 10^{-08}:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot x - \left(x + 1\right)\right) - \frac{\left(x + 1\right) \cdot \left(x + 1\right)}{{x}^{3} - 1} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(x \cdot x - \left(1 + x\right)\right) \cdot \left(1 + x\right)}\\ \end{array}\]

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 (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))) < 1.215708945304996e-08

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.5

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

    3. Applied simplify0.2

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}}\]

    if 1.215708945304996e-08 < (- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))

    1. Initial program 0.2

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

    3. Applied flip3--0.2

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

    4. Applied associate-/r/0.2

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

    5. Applied simplify0.2

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

    7. Applied flip-+0.2

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

    8. Applied associate-*r/0.2

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

    9. Applied frac-sub0.2

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

    10. Applied simplify0.2

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

    11. Applied simplify0.2

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018193 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))