Average Error: 28.8 → 0.1
Time: 3.3m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11487.584036687787 \lor \neg \left(x \le 15792.30916261313\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{\left(x \cdot \left(\left(1 - x\right) + x \cdot x\right)\right) \cdot \left(\left(-1 + {x}^{3}\right) \cdot \left(-1 + x\right)\right) - \left(x \cdot x + \left(1 + x\right)\right) \cdot \left(\left(1 + {x}^{3}\right) \cdot \left(-1 + x \cdot x\right)\right)}{\left(\left(1 + x \cdot x\right) + x\right) \cdot \left(\left(x - 1\right) \cdot \left(\left(1 - x\right) + x \cdot x\right)\right)} \cdot \left({\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}\right)\right)}}{\sqrt[3]{\left(\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \frac{1 + x}{x - 1}\right) + \frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \left(\left(x - 1\right) \cdot \left(1 + x\right)\right)}}\\ \end{array}\]

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 2 regimes

  2. if x < -11487.584036687787 or 15792.30916261313 < x

    1. Initial program 59.2

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

    3. Applied add-cbrt-cube59.2

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

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.0

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

    if -11487.584036687787 < x < 15792.30916261313

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    5. Applied frac-sub0.1

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

    6. Applied flip3--0.1

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

    7. Applied frac-times0.1

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

    8. Applied associate-*l/0.1

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

    9. Applied cbrt-div0.1

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

    11. Applied flip3-+0.1

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

    12. Applied flip-+0.1

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

    13. Applied frac-times0.1

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

    14. Applied flip3--0.1

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

    15. Applied associate-*r/0.1

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

    16. Applied frac-sub0.1

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

    17. Simplified0.1

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

    18. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11487.584036687787 \lor \neg \left(x \le 15792.30916261313\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{x}{1 + x} - \frac{1 + x}{x - 1}\right) \cdot \left(\frac{\left(x \cdot \left(\left(1 - x\right) + x \cdot x\right)\right) \cdot \left(\left(-1 + {x}^{3}\right) \cdot \left(-1 + x\right)\right) - \left(x \cdot x + \left(1 + x\right)\right) \cdot \left(\left(1 + {x}^{3}\right) \cdot \left(-1 + x \cdot x\right)\right)}{\left(\left(1 + x \cdot x\right) + x\right) \cdot \left(\left(x - 1\right) \cdot \left(\left(1 - x\right) + x \cdot x\right)\right)} \cdot \left({\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}\right)\right)}}{\sqrt[3]{\left(\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \frac{1 + x}{x - 1}\right) + \frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot \left(\left(x - 1\right) \cdot \left(1 + x\right)\right)}}\\ \end{array}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.70.10.029.699.9%
herbie shell --seed 2018290 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))