Average Error: 28.8 → 0.0
Time: 23.6s
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x} \le -9.35362369942047 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(x + 1\right)\right) - \frac{x + 1}{x}}{\frac{x + 1}{x} \cdot \left(x - 1\right)}\\ \mathbf{elif}\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x} \le 1.4968623894584635 \cdot 10^{-08}:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(x + 1\right)\right) - \frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}{\frac{x}{\sqrt[3]{x + 1}}}}{\frac{x + 1}{x} \cdot \left(x - 1\right)}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < -9.35362369942047e-11

    1. Initial program 0.4

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

    3. Applied clear-num0.4

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1}\]
    4. Using strategy rm

    5. Applied frac-sub0.4

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

    6. Simplified0.0

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

    if -9.35362369942047e-11 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < 1.4968623894584635e-08

    1. Initial program 60.1

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if 1.4968623894584635e-08 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x)))

    1. Initial program 0.3

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

    3. Applied clear-num0.3

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1}\]
    4. Using strategy rm

    5. Applied frac-sub0.3

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

    6. Simplified0.0

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied associate-/l*0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x} \le -9.35362369942047 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(x + 1\right)\right) - \frac{x + 1}{x}}{\frac{x + 1}{x} \cdot \left(x - 1\right)}\\ \mathbf{elif}\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x} \le 1.4968623894584635 \cdot 10^{-08}:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x - 1\right) - \left(x + 1\right)\right) - \frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}{\frac{x}{\sqrt[3]{x + 1}}}}{\frac{x + 1}{x} \cdot \left(x - 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 23.6s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))