Average Error: 29.2 → 0.5
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le -5.6229983398801587 \cdot 10^{-08}:\\ \;\;\;\;\frac{x}{1 + {x}^{3}} \cdot \left(x \cdot x + \left(1 - x\right)\right) - \frac{1 + x}{x - 1}\\ \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le 9.253383191458749 \cdot 10^{-20}:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot x - \left(x - 1\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}{\left(1 + x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < -5.6229983398801587e-08

    1. Initial program 0.3

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

    3. Applied flip3-+0.3

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

    4. Applied associate-/r/0.3

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

    5. Applied simplify0.3

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

    if -5.6229983398801587e-08 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < 9.253383191458749e-20

    1. Initial program 60.2

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

    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

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

    if 9.253383191458749e-20 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x)))

    1. Initial program 1.7

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

    3. Applied div-inv1.7

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

    5. Applied flip-+1.7

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

    6. Applied frac-times1.7

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

    7. Applied frac-sub1.7

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

    8. Applied simplify1.7

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

  4. Applied simplify0.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le -5.6229983398801587 \cdot 10^{-08}:\\ \;\;\;\;\frac{x}{1 + {x}^{3}} \cdot \left(x \cdot x + \left(1 - x\right)\right) - \frac{1 + x}{x - 1}\\ \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le 9.253383191458749 \cdot 10^{-20}:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) \cdot \left(x - 1\right)\right) \cdot x - \left(x - 1\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}{\left(1 + x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018166 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))