Average Error: 29.4 → 0.1
Time: 3.5m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le -3.954960597368781 \cdot 10^{-05}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{{x}^{3} + {1}^{3}}{\left(\left(1 - x\right) + x \cdot x\right) \cdot \left(x - 1\right)}\\ \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le 5.347362044394717 \cdot 10^{-08}:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{1 + x} - \frac{\left(\left(1 + x\right) + x \cdot x\right) \cdot \left(1 + x\right)}{{x}^{3} - 1}\right)}^{3}}\\ \end{array}\]

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))) < -3.954960597368781e-05

    1. Initial program 0.1

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

    3. Applied flip3-+0.1

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

    4. Applied associate-/l/0.1

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

    5. Applied simplify0.1

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

    if -3.954960597368781e-05 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < 5.347362044394717e-08

    1. Initial program 59.7

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

    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 5.347362044394717e-08 < (- (/ (- 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--0.2

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

    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 add-cbrt-cube0.2

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

    8. Applied simplify0.2

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed 2018201 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))