Average Error: 29.3 → 0.1
Time: 3.1m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\left(-\frac{18}{x}\right) - \frac{51}{x \cdot x}\right) - \frac{\frac{348}{x}}{x \cdot x}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}}{\frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} \cdot \frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} + \left({\left(\frac{x}{1 + x}\right)}^{\left(1 + 3\right)} + {\left(\frac{1 + x}{x - 1}\right)}^{\left(1 + 3\right)}\right)} \le -4.1007362573475046 \cdot 10^{-07}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(1 + x\right)\right)\\ \mathbf{if}\;\frac{\frac{\left(\left(-\frac{18}{x}\right) - \frac{51}{x \cdot x}\right) - \frac{\frac{348}{x}}{x \cdot x}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}}{\frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} \cdot \frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} + \left({\left(\frac{x}{1 + x}\right)}^{\left(1 + 3\right)} + {\left(\frac{1 + x}{x - 1}\right)}^{\left(1 + 3\right)}\right)} \le 0.0002815692338335252:\\ \;\;\;\;\left(-\frac{3}{x}\right) - \frac{\frac{3}{x} + 1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3} \cdot {\left(\frac{1 + x}{x - 1}\right)}^{3}}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}\\ \end{array}\]

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 (/ (/ (- (- (- (/ 18 x)) (/ 51 (* x x))) (/ (/ 348 x) (* x x))) (+ (/ x (+ 1 x)) (/ (+ 1 x) (- x 1)))) (+ (* (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x))) (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x)))) (+ (pow (/ (+ 1 x) (- x 1)) (+ 3 1)) (pow (/ x (+ 1 x)) (+ 3 1))))) < -4.1007362573475046e-07

    1. Initial program 0.4

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

    3. Applied flip3--0.4

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

    4. Applied associate-/r/0.4

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

    5. Applied simplify0.4

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

    if -4.1007362573475046e-07 < (/ (/ (- (- (- (/ 18 x)) (/ 51 (* x x))) (/ (/ 348 x) (* x x))) (+ (/ x (+ 1 x)) (/ (+ 1 x) (- x 1)))) (+ (* (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x))) (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x)))) (+ (pow (/ (+ 1 x) (- x 1)) (+ 3 1)) (pow (/ x (+ 1 x)) (+ 3 1))))) < 0.0002815692338335252

    1. Initial program 59.4

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

    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 0.0002815692338335252 < (/ (/ (- (- (- (/ 18 x)) (/ 51 (* x x))) (/ (/ 348 x) (* x x))) (+ (/ x (+ 1 x)) (/ (+ 1 x) (- x 1)))) (+ (* (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x))) (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x)))) (+ (pow (/ (+ 1 x) (- x 1)) (+ 3 1)) (pow (/ x (+ 1 x)) (+ 3 1)))))

    1. Initial program 0.0

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

    3. Applied flip--0.1

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

    5. Applied add-cbrt-cube0.1

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

    6. Applied add-cbrt-cube0.1

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

    7. Applied cbrt-undiv0.1

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

    8. Applied add-cbrt-cube0.1

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

    9. Applied add-cbrt-cube0.1

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

    10. Applied cbrt-undiv0.1

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

    11. Applied cbrt-unprod0.1

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

    12. Applied simplify0.1

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

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\left(-\frac{18}{x}\right) - \frac{51}{x \cdot x}\right) - \frac{\frac{348}{x}}{x \cdot x}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}}{\frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} \cdot \frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} + \left({\left(\frac{x}{1 + x}\right)}^{\left(1 + 3\right)} + {\left(\frac{1 + x}{x - 1}\right)}^{\left(1 + 3\right)}\right)} \le -4.1007362573475046 \cdot 10^{-07}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1 + x}{{x}^{3} - 1} \cdot \left(x \cdot x + \left(1 + x\right)\right)\\ \mathbf{if}\;\frac{\frac{\left(\left(-\frac{18}{x}\right) - \frac{51}{x \cdot x}\right) - \frac{\frac{348}{x}}{x \cdot x}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}}{\frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} \cdot \frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} + \left({\left(\frac{x}{1 + x}\right)}^{\left(1 + 3\right)} + {\left(\frac{1 + x}{x - 1}\right)}^{\left(1 + 3\right)}\right)} \le 0.0002815692338335252:\\ \;\;\;\;\left(-\frac{3}{x}\right) - \frac{\frac{3}{x} + 1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + x} \cdot \frac{x}{1 + x} - \sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3} \cdot {\left(\frac{1 + x}{x - 1}\right)}^{3}}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}\\ \end{array}}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))