Average Error: 0.3 → 0.4
Time: 25.5s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\tan x}{\frac{\cos x}{\sin x}}}{1 + \frac{1}{\frac{\cos x}{\sin x \cdot \tan x}}}\]

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. Using strategy rm

  3. Applied tan-quot0.4

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

  4. Applied associate-*r/0.4

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

  6. Applied tan-quot0.4

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

  7. Applied associate-*r/0.3

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

  9. Applied associate-/l*0.4

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

  11. Applied clear-num0.4

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

  12. Final simplification0.4

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

Runtime

Time bar (total: 25.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.40.40.10.30%
herbie shell --seed 2018297 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))