Error

Target

Derivation

1. Split input into 3 regimes.

2. if eps < -2.9959312647223546e-24

   1. Initial program 29.7

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

   3. Applied tan-quot 29.6

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

   4. Applied tan-cotan 29.6

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

   5. Applied frac-sub 29.6

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

   6. Applied simplify 29.6

      \[\leadsto \frac{\color{blue}{\cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 29.4

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

   9. Applied add-cube-cbrt 29.5

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

   10. Applied cube-unprod 29.5

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

   if -2.9959312647223546e-24 < eps < 2.9920447415528193e-87

   1. Initial program 46.8

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

   2. Applied taylor 19.4

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

   3. Taylor expanded around 0 19.4

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

   4. Applied simplify 19.4

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

   if 2.9920447415528193e-87 < eps

   1. Initial program 30.6

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

   3. Applied tan-quot 30.5

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

   4. Applied tan-cotan 30.5

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

   5. Applied frac-sub 30.5

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

   6. Applied simplify 30.5

      \[\leadsto \frac{\color{blue}{\cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 30.4

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

Runtime

Total time: 47.5s Debug log

Please report a bug with the following info:

herbie --seed '#(4087287089 436356386 1312402636 2028493862 2625229117 1407192970)'
(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)))