Average Error: 15.3 → 15.3
Time: 25.3s
Precision: 64
Internal Precision: 1344
\[{\left(\tan^{-1} x\right)}^{2} - {\left(\tan^{-1} \left(x + 1\right)\right)}^{2}\]
\[\frac{\sqrt[3]{{\left({\left(\tan^{-1} x\right)}^{3}\right)}^{\left(3 + 1\right)}} - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right) + \tan^{-1} x \cdot \tan^{-1} x}\]

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 15.3

    \[{\left(\tan^{-1} x\right)}^{2} - {\left(\tan^{-1} \left(x + 1\right)\right)}^{2}\]
  2. Initial simplification15.3

    \[\leadsto \tan^{-1} x \cdot \tan^{-1} x - \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\]
  3. Using strategy rm
  4. Applied flip--15.3

    \[\leadsto \color{blue}{\frac{\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \left(\tan^{-1} x \cdot \tan^{-1} x\right) - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} x \cdot \tan^{-1} x + \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)}}\]
  5. Using strategy rm
  6. Applied add-cbrt-cube30.5

    \[\leadsto \frac{\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \left(\tan^{-1} x \cdot \color{blue}{\sqrt[3]{\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x}}\right) - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} x \cdot \tan^{-1} x + \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)}\]
  7. Applied add-cbrt-cube30.5

    \[\leadsto \frac{\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x}} \cdot \sqrt[3]{\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x}\right) - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} x \cdot \tan^{-1} x + \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)}\]
  8. Applied cbrt-unprod30.5

    \[\leadsto \frac{\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x\right) \cdot \left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x\right)}} - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} x \cdot \tan^{-1} x + \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)}\]
  9. Applied add-cbrt-cube15.3

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \left(\tan^{-1} x \cdot \tan^{-1} x\right)\right) \cdot \left(\tan^{-1} x \cdot \tan^{-1} x\right)}} \cdot \sqrt[3]{\left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x\right) \cdot \left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x\right)} - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} x \cdot \tan^{-1} x + \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)}\]
  10. Applied cbrt-unprod15.3

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \left(\tan^{-1} x \cdot \tan^{-1} x\right)\right) \cdot \left(\tan^{-1} x \cdot \tan^{-1} x\right)\right) \cdot \left(\left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x\right) \cdot \left(\left(\tan^{-1} x \cdot \tan^{-1} x\right) \cdot \tan^{-1} x\right)\right)}} - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} x \cdot \tan^{-1} x + \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)}\]
  11. Simplified15.3

    \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left({\left(\tan^{-1} x\right)}^{3}\right)}^{\left(3 + 1\right)}}} - \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right) \cdot \left(\tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)\right)}{\tan^{-1} x \cdot \tan^{-1} x + \tan^{-1} \left(x + 1\right) \cdot \tan^{-1} \left(x + 1\right)}\]
  12. Final simplification15.3

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

Runtime

Time bar (total: 25.3s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "atan(x)^2-atan(x+1)^2"
  (- (pow (atan x) 2) (pow (atan (+ x 1)) 2)))