Average Error: 37.5 → 15.3
Time: 47.2s
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4665576405016369 \cdot 10^{-55} \lor \neg \left(\varepsilon \le 7.190644688190906 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{\left(\tan x \cdot \sin x\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \left(\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x\right)}{\cos x - \frac{\cos x \cdot \tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target15.2
Herbie15.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.4665576405016369e-55 or 7.190644688190906e-72 < eps

    1. Initial program 30.9

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

    3. Applied tan-sum5.1

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

    5. Applied tan-quot5.1

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

    6. Applied associate-*r/5.1

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

    8. Applied add-cbrt-cube5.2

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

    9. Applied add-cbrt-cube5.2

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

    10. Applied cbrt-unprod5.1

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

    11. Simplified5.2

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

    13. Applied tan-quot5.2

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

    14. Applied frac-sub5.2

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

    15. Simplified4.8

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

    16. Simplified4.8

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

    if -1.4665576405016369e-55 < eps < 7.190644688190906e-72

    1. Initial program 47.3

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

    3. Applied tan-sum47.3

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

    5. Applied tan-quot47.3

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

    6. Applied associate-*r/47.3

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

    7. Taylor expanded around 0 31.0

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

    8. Simplified30.9

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.4665576405016369 \cdot 10^{-55} \lor \neg \left(\varepsilon \le 7.190644688190906 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{\left(\tan x \cdot \sin x\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + \left(\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \sin x\right)}{\cos x - \frac{\cos x \cdot \tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right)\\ \end{array}\]

Runtime

Time bar (total: 47.2s)Debug logProfile

herbie shell --seed 2018273 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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