Average Error: 37.3 → 14.1
Time: 1.6m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -2.0619556141735675 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(\tan \varepsilon + \tan x\right) + \left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon + \tan x\right)\right)}{1 - \left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} - \tan x\\ \mathbf{elif}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 2.0540401203514266 \cdot 10^{-09}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right)}^{3}} \cdot \left(1 + \left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.5
Herbie14.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -2.0619556141735675e-25

    1. Initial program 36.5

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

    3. Applied tan-sum9.4

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

    5. Applied flip3--9.4

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

    6. Applied associate-/r/9.4

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

    7. Applied simplify9.4

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

    9. Applied add-cbrt-cube9.5

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

    10. Applied add-cbrt-cube9.5

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

    11. Applied cbrt-unprod9.5

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

    12. Applied rem-cube-cbrt9.4

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

    13. Taylor expanded around inf 9.4

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

    14. Applied simplify9.4

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

    if -2.0619556141735675e-25 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 2.0540401203514266e-09

    1. Initial program 38.6

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

    2. Taylor expanded around 0 15.7

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

    if 2.0540401203514266e-09 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 36.7

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

    3. Applied tan-sum14.7

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

    5. Applied flip3--14.7

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

    6. Applied associate-/r/14.7

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

    7. Applied simplify14.7

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

    9. Applied add-cbrt-cube14.8

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

    10. Applied add-cbrt-cube14.8

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

    11. Applied cbrt-unprod14.8

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

    12. Applied simplify14.8

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

  4. Applied simplify14.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -2.0619556141735675 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(\tan \varepsilon + \tan x\right) + \left(1 + \tan \varepsilon \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon + \tan x\right)\right)}{1 - \left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)} - \tan x\\ \mathbf{elif}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 2.0540401203514266 \cdot 10^{-09}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right)}^{3}} \cdot \left(1 + \left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) - \tan x\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018208 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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