Error

Target

Derivation

1. Split input into 3 regimes.

2. if eps < -6.041808348837433e-43

   1. Initial program 31.4

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

   3. Applied tan-quot 31.2

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

   4. Applied tan-sum 3.2

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

   5. Applied frac-sub 3.3

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

   if -6.041808348837433e-43 < eps < 3.290445875832927e-111

   1. Initial program 47.6

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

   2. Applied taylor 19.1

      \[\leadsto {\varepsilon}^{4} \cdot {x}^{3} + \left({\varepsilon}^{3} \cdot {x}^2 + \varepsilon\right)\]

   3. Taylor expanded around 0 19.1

      \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot {x}^{3} + \left({\varepsilon}^{3} \cdot {x}^2 + \varepsilon\right)}\]

   4. Applied simplify 19.1

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot {\varepsilon}^3 + {\varepsilon}^{4} \cdot {x}^3\right) + \varepsilon}\]

   if 3.290445875832927e-111 < eps

   1. Initial program 31.2

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

   3. Applied tan-sum 8.4

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

   5. Applied add-cbrt-cube 8.4

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

   6. Applied add-cbrt-cube 8.5

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

   7. Applied cbrt-unprod 8.4

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

   9. Applied flip3-- 8.8

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

   10. Applied simplify 8.8

       \[\leadsto \frac{\color{blue}{{\left(\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan x\right)}^3 \cdot {\left(\tan \varepsilon\right)}^3}}\right)}^3 - {\left(\tan x\right)}^3}}{{\left(\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x\right)}^3 \cdot {\left(\tan \varepsilon\right)}^3}}\right)}^2 + \left({\left(\tan x\right)}^2 + \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x\right)}^3 \cdot {\left(\tan \varepsilon\right)}^3}} \cdot \tan x\right)}\]
3. Recombined 3 regimes into one program.
4. Removed slow pow expressions

Runtime

Total time: 2.0m Debug log

Please include this information when filing a bug report:

herbie --seed '#(1760188614 120530851 1113714957 3291444970 1127176104 4016628637)'
(FPCore (x eps)
  :name "NMSE problem 3.3.2"
  :herbie-expected 28

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

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