Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -1.6718721488586065e-18

   1. Initial program 29.4

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum0.9

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

   5. Applied div-inv0.9

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

   6. Applied fma-neg0.9

      \[\leadsto \color{blue}{(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right))_*}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube1.0

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

   if -1.6718721488586065e-18 < eps < 1.0288762116509696e-29

   1. Initial program 45.7

      \[\tan \left(x + \varepsilon\right) - \tan x\]

   2. Taylor expanded around 0 31.4

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

   3. Simplified31.3

      \[\leadsto \color{blue}{(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon)_*}\]

   if 1.0288762116509696e-29 < eps

   1. Initial program 29.5

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum2.0

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

   5. Applied flip--2.1

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

   6. Applied associate-/r/2.1

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

   7. Applied fma-neg2.1

      \[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\tan x\right))_*}\]

   8. Simplified2.1

      \[\leadsto (\color{blue}{\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\tan x\right))_*\]
3. Recombined 3 regimes into one program.

4. Final simplification15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.6718721488586065 \cdot 10^{-18}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\sqrt[3]{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right)}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 1.0288762116509696 \cdot 10^{-29}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\tan x\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))

Details

Time bar (total: 43.1s)Debug log