Average Error: 0.3 → 0.4
Time: 21.4s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(\tan x \cdot \tan x - 1\right) \cdot \frac{1 - \tan x \cdot \tan x}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - 1}\]

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 flip-+0.4

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

  5. Applied associate-/r/0.4

    \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - 1 \cdot 1} \cdot \left(\tan x \cdot \tan x - 1\right)}\]
  6. Using strategy rm

  7. Applied *-commutative0.4

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

  8. Final simplification0.4

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

Runtime

Time bar (total: 21.4s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))