Average Error: 0.3 → 0.4
Time: 30.3s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\frac{1 - {\left(\tan x \cdot \tan x\right)}^{3}}{1 + \left(\tan x \cdot \tan x + \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)}}{1 + \tan x \cdot \tan x}\]

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 flip3--0.4

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

  4. Final simplification0.4

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

Runtime

Time bar (total: 30.3s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))