Average Error: 0.3 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\frac{1 - {\left(\tan x\right)}^{3} \cdot {\left(\tan x\right)}^{3}}{\left(\tan x \cdot \tan x + 1\right) + {\left(\tan x\right)}^{\left(1 + 3\right)}}}{1 + \tan x \cdot \tan x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Applied simplify0.4

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

  5. Applied simplify0.5

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

  7. Applied unpow-prod-down0.5

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018195 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))