Average Error: 0.3 → 0.5
Time: 27.0s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{1 + \sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)}} - \frac{\frac{\tan x \cdot \sin x}{\cos x}}{\tan x \cdot \tan x + 1}\]

Error

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Results

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Derivation

  1. Initial program 0.3

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

  2. Initial simplification0.3

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{\tan x \cdot \tan x + 1}\]
  3. Using strategy rm

  4. Applied tan-quot0.4

    \[\leadsto \frac{1 - \tan x \cdot \color{blue}{\frac{\sin x}{\cos x}}}{\tan x \cdot \tan x + 1}\]

  5. Applied associate-*r/0.4

    \[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{\tan x \cdot \tan x + 1}\]
  6. Using strategy rm

  7. Applied div-sub0.5

    \[\leadsto \color{blue}{\frac{1}{\tan x \cdot \tan x + 1} - \frac{\frac{\tan x \cdot \sin x}{\cos x}}{\tan x \cdot \tan x + 1}}\]
  8. Using strategy rm

  9. Applied add-cbrt-cube0.5

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

  10. Final simplification0.5

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

Runtime

Time bar (total: 27.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.50.50.10.40%
herbie shell --seed 2018295 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))