Average Error: 37.3 → 13.3
Time: 1.2m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right) \le -3.1923979085602207 \cdot 10^{-18}:\\ \;\;\;\;\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}} + \frac{\frac{\frac{{\left(\sin x\right)}^{3}}{\cos \varepsilon \cdot \cos \varepsilon}}{{\left(\cos x\right)}^{3}} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}\right) + \left(\left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + 1\right) \cdot \frac{\frac{\sin x}{\cos x}}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}\right) + \left(\frac{{\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \left(\sin x \cdot \sin x\right)}{\left(1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right) \cdot \left(\cos x \cdot \cos x\right)} - \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{if}\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right) \le 2.4845994639220834 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin \varepsilon}{\cos x \cdot \cos \varepsilon} \cdot \frac{\sin x \cdot \sin x}{\cos x}}{1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}} + \frac{\frac{{\left(\sin x\right)}^{3}}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{{\left(\cos x\right)}^{3} \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)}\right) + \left(\left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + 1\right) \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}}\right) + \left(\frac{\frac{\sin x \cdot \sin x}{{\left(\cos \varepsilon\right)}^{3}} \cdot {\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left(\cos x \cdot \cos x\right)} - \frac{\sin x}{\cos x}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target14.5
Herbie13.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -3.1923979085602207e-18

    1. Initial program 35.7

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

    3. Applied tan-sum9.9

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

    5. Applied flip3--9.9

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

    6. Applied associate-/r/9.9

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

    7. Applied simplify9.9

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

    8. Taylor expanded around -inf 10.1

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

    9. Applied simplify8.7

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

    10. Taylor expanded around inf 8.7

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

    11. Applied simplify8.7

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

    if -3.1923979085602207e-18 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 2.4845994639220834e-49

    1. Initial program 40.7

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

    2. Taylor expanded around 0 16.2

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

    if 2.4845994639220834e-49 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 35.5

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

    3. Applied tan-sum15.0

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

    5. Applied flip3--15.0

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

    6. Applied associate-/r/15.0

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

    7. Applied simplify15.0

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

    8. Taylor expanded around -inf 15.2

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

    9. Applied simplify13.1

      \[\leadsto \color{blue}{\left(\frac{\frac{\sin \varepsilon}{\cos x \cdot \cos \varepsilon} \cdot \frac{\sin x \cdot \sin x}{\cos x}}{1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}} + \frac{\frac{{\left(\sin x\right)}^{3}}{\cos \varepsilon \cdot \cos \varepsilon} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{{\left(\cos x\right)}^{3} \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)}\right) + \left(\left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon} + 1\right) \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}}\right) + \left(\frac{\frac{\sin x \cdot \sin x}{{\left(\cos \varepsilon\right)}^{3}} \cdot {\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left(\cos x \cdot \cos x\right)} - \frac{\sin x}{\cos x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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