Average Error: 0.3 → 0.4
Time: 36.2s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \sin x \cdot \frac{\tan x}{\cos x}}{1 + \tan x \cdot \tan x}\]

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 tan-quot0.4

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

  4. Applied associate-*l/0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied times-frac0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Runtime

Time bar (total: 36.2s)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))