Error

Target

Derivation

1. Split input into 2 regimes

2. if eps < -6.34294204433499e-13 or 3.0147826755181865e-19 < eps

   1. Initial program 28.7

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

   3. Applied tan-sum0.8

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

   5. Applied add-sqr-sqrt32.3

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\sqrt{\tan x} \cdot \sqrt{\tan x}}\]

   6. Applied flip3--32.3

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

   7. Applied associate-/r/32.3

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

   8. Applied prod-diff32.3

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

   9. Simplified32.2

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

   10. Simplified0.9

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right) + \color{blue}{0}\]
   11. Using strategy rm

   12. Applied cube-mult0.9

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - \color{blue}{\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)}} - \tan x\right) + 0\]

   if -6.34294204433499e-13 < eps < 3.0147826755181865e-19

   1. Initial program 45.4

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

   3. Applied tan-sum45.3

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

   5. Applied add-sqr-sqrt54.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\sqrt{\tan x} \cdot \sqrt{\tan x}}\]

   6. Applied flip3--54.7

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

   7. Applied associate-/r/54.7

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

   8. Applied prod-diff54.8

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

   9. Simplified54.8

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

   10. Simplified45.3

       \[\leadsto \left(\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan x + \tan \varepsilon\right)}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right) + \color{blue}{0}\]

   11. Taylor expanded around 0 28.2

       \[\leadsto \color{blue}{\left(x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)\right)} + 0\]

   12. Simplified28.2

       \[\leadsto \color{blue}{(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*} + 0\]
3. Recombined 2 regimes into one program.

4. Final simplification13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.34294204433499 \cdot 10^{-13} \lor \neg \left(\varepsilon \le 3.0147826755181865 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* \cdot \left(\tan \varepsilon + \tan x\right)}{1 - \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018255 +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)))