Average Error: 0.3 → 0.5
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{\sqrt[3]{{\left(\sin x\right)}^{3} \cdot {\left(\sin x\right)}^{3}}}{{\left(\cos x\right)}^{2}}}\]

Error

Bits error versus x

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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. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.4

    \[\leadsto \frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{\color{blue}{\sqrt[3]{\left({\left(\sin x\right)}^{2} \cdot {\left(\sin x\right)}^{2}\right) \cdot {\left(\sin x\right)}^{2}}}}{{\left(\cos x\right)}^{2}}}\]

  5. Applied simplify0.5

    \[\leadsto \frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{\sqrt[3]{\color{blue}{{\left(\sin x\right)}^{3} \cdot {\left(\sin x\right)}^{3}}}}{{\left(\cos x\right)}^{2}}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)' 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))