Input Error: 36.3b
Output Error: 24.4b
Time: 45.3s
Precision: 64b
Ground Truth: 128b
\[\tan \left(x + \varepsilon\right) - \tan x\]
⬇
\[\begin{cases} \left(1 - \frac{\sqrt[3]{{\left(\cot \left(\varepsilon + x\right)\right)}^3}}{\frac{\cos x}{\sin x}}\right) \cdot \tan \left(x + \varepsilon\right) & \text{when } \varepsilon \le -3.1929114467955524 \cdot 10^{-61} \\ \left(\left(x \cdot x\right) \cdot {\varepsilon}^3 + {\varepsilon}^{4} \cdot {x}^3\right) + \varepsilon & \text{when } \varepsilon \le 0.4439134345824553 \\ \left(1 - {\left(\frac{\sqrt[3]{\cot \left(\varepsilon + x\right)}}{\sqrt[3]{\frac{\cos x}{\sin x}}}\right)}^3\right) \cdot \tan \left(x + \varepsilon\right) & \text{otherwise} \end{cases}\]

Error

Bits error versus x
Bits error versus eps

Derivation

    if eps < -3.1929114467955524e-61

    1. Initial program 30.1b

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

    3. Applied tan-quot 29.9b

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

    4. Applied tan-cotan 30.0b

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

    5. Applied frac-sub 29.9b

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

    6. Applied simplify 29.9b

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

    8. Applied cotan-tan 29.9b

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

    9. Applied associate-*l/ 29.9b

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

    10. Applied associate-/r/ 29.9b

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

    11. Applied simplify 29.9b

      \[\leadsto \color{blue}{\left(1 - \frac{\cot \left(\varepsilon + x\right)}{\frac{\cos x}{\sin x}}\right)} \cdot \tan \left(x + \varepsilon\right)\]
    12. Using strategy rm

    13. Applied add-cbrt-cube 29.8b

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

    if -3.1929114467955524e-61 < eps < 0.4439134345824553

    1. Initial program 44.7b

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

    2. Applied taylor 17.1b

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

    3. Taylor expanded around 0 17.1b

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

    4. Applied simplify 17.1b

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

    if 0.4439134345824553 < eps

    1. Initial program 29.9b

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

    3. Applied tan-quot 29.8b

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

    4. Applied tan-cotan 29.8b

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

    5. Applied frac-sub 29.8b

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

    6. Applied simplify 29.8b

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

    8. Applied cotan-tan 29.8b

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

    9. Applied associate-*l/ 29.8b

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

    10. Applied associate-/r/ 29.8b

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

    11. Applied simplify 29.8b

      \[\leadsto \color{blue}{\left(1 - \frac{\cot \left(\varepsilon + x\right)}{\frac{\cos x}{\sin x}}\right)} \cdot \tan \left(x + \varepsilon\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt 29.5b

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

    14. Applied add-cube-cbrt 29.9b

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

    15. Applied cube-undiv 29.9b

      \[\leadsto \left(1 - \color{blue}{{\left(\frac{\sqrt[3]{\cot \left(\varepsilon + x\right)}}{\sqrt[3]{\frac{\cos x}{\sin x}}}\right)}^3}\right) \cdot \tan \left(x + \varepsilon\right)\]
  1. Removed slow pow expressions

Runtime

Total time: 45.3s Debug log

herbie --seed '#(3145657944 3435231854 1845925313 1249348575 978218321 2513120512)'
(FPCore (x eps)
  :name "NMSE problem 3.3.2"
  
  :target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))(- (tan (+ x eps)) (tan x)))