Average Error: 37.0 → 14.4
Time: 2.6m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -1.2455945272111048 \cdot 10^{-13} \lor \neg \left(\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 1.3012109729505754 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\tan \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}} \cdot \left(1 + \tan \varepsilon \cdot \tan x\right) - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.4
Herbie14.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -1.2455945272111048e-13 or 1.3012109729505754e-42 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 35.8

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

    3. Applied tan-sum13.3

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

    5. Applied flip--13.3

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

    6. Applied associate-/r/13.3

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

    7. Applied simplify13.3

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

    9. Applied tan-quot13.3

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

    10. Applied associate-*r/13.3

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

    11. Applied tan-quot13.3

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

    12. Applied tan-quot13.3

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

    13. Applied frac-times13.3

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

    14. Applied frac-times13.3

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

    if -1.2455945272111048e-13 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 1.3012109729505754e-42

    1. Initial program 39.7

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

    2. Taylor expanded around 0 16.8

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

  4. Applied simplify14.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -1.2455945272111048 \cdot 10^{-13} \lor \neg \left(\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 1.3012109729505754 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\tan \varepsilon \cdot \sin x\right)}{\cos x \cdot \left(\cos x \cdot \cos \varepsilon\right)}} \cdot \left(1 + \tan \varepsilon \cdot \tan x\right) - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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