Average Error: 36.6 → 13.8
Time: 1.8m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right) \le -7.416621346817067 \cdot 10^{-14}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\sin x \cdot \frac{\sin x \cdot \sin x}{{\left(\cos x\right)}^{3}}}{{\left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}\right)}^{3}}} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right) \le 2.4921719870633997 \cdot 10^{-37}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target14.9
Herbie13.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -7.416621346817067e-14

    1. Initial program 36.1

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

    3. Applied tan-sum9.8

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

    4. Taylor expanded around inf 9.9

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

    6. Applied add-cbrt-cube10.0

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

    7. Applied add-cbrt-cube10.0

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

    8. Applied cbrt-unprod10.0

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

    9. Applied add-cbrt-cube10.0

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

    10. Applied cbrt-undiv9.9

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

    11. Applied simplify9.9

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

    if -7.416621346817067e-14 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 2.4921719870633997e-37

    1. Initial program 39.5

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

    2. Taylor expanded around 0 15.7

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

    if 2.4921719870633997e-37 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 34.9

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

    3. Applied tan-sum13.9

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

    5. Applied add-log-exp14.0

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018206 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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