Average Error: 0.3 → 0.4
Time: 23.2s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{1 + \frac{\sin x \cdot \tan x}{\cos x}} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

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 div-sub0.4

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

  5. Applied tan-quot0.4

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

  6. Applied associate-*l/0.4

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

  7. Final simplification0.4

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

Runtime

Time bar (total: 23.2s)Debug logProfile

herbie shell --seed 2018254 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))