Average Error: 36.9 → 0.6
Time: 1.8m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}}\right) + \left(\left(\left((\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\frac{\sin x}{\cos x} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)}\right))_* + \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right) + \frac{\sin \varepsilon}{\left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right))_*\]
\tan \left(x + \varepsilon\right) - \tan x
↓
(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}}\right) + \left(\left(\left((\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\frac{\sin x}{\cos x} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)}\right))_* + \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right) + \frac{\sin \varepsilon}{\left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right))_*
double f(double x, double eps) {
        double r16051846 = x;
        double r16051847 = eps;
        double r16051848 = r16051846 + r16051847;
        double r16051849 = tan(r16051848);
        double r16051850 = tan(r16051846);
        double r16051851 = r16051849 - r16051850;
        return r16051851;
}
↓
double f(double x, double eps) {
        double r16051852 = x;
        double r16051853 = sin(r16051852);
        double r16051854 = cos(r16051852);
        double r16051855 = r16051853 / r16051854;
        double r16051856 = r16051855 * r16051855;
        double r16051857 = eps;
        double r16051858 = sin(r16051857);
        double r16051859 = cos(r16051857);
        double r16051860 = r16051858 / r16051859;
        double r16051861 = 1.0;
        double r16051862 = r16051858 * r16051858;
        double r16051863 = r16051862 * r16051858;
        double r16051864 = r16051855 * r16051856;
        double r16051865 = r16051863 * r16051864;
        double r16051866 = r16051859 * r16051859;
        double r16051867 = r16051866 * r16051859;
        double r16051868 = r16051865 / r16051867;
        double r16051869 = r16051861 - r16051868;
        double r16051870 = r16051860 / r16051869;
        double r16051871 = r16051863 / r16051866;
        double r16051872 = r16051871 / r16051859;
        double r16051873 = r16051864 * r16051872;
        double r16051874 = r16051861 - r16051873;
        double r16051875 = r16051872 / r16051874;
        double r16051876 = r16051855 * r16051862;
        double r16051877 = r16051866 * r16051874;
        double r16051878 = r16051876 / r16051877;
        double r16051879 = fma(r16051856, r16051875, r16051878);
        double r16051880 = r16051855 / r16051874;
        double r16051881 = r16051879 + r16051880;
        double r16051882 = r16051864 * r16051862;
        double r16051883 = r16051882 / r16051877;
        double r16051884 = r16051883 - r16051855;
        double r16051885 = r16051881 + r16051884;
        double r16051886 = r16051874 * r16051859;
        double r16051887 = r16051858 / r16051886;
        double r16051888 = r16051885 + r16051887;
        double r16051889 = fma(r16051856, r16051870, r16051888);
        return r16051889;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.1
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.7

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

  5. Applied flip3--21.7

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

  6. Applied associate-/r/21.7

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

  7. Applied fma-neg21.7

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

  8. Simplified21.7

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

  9. Taylor expanded around inf 21.8

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

  10. Simplified19.6

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

  11. Taylor expanded around inf 19.6

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

  12. Simplified0.6

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

  13. Final simplification0.6

    \[\leadsto (\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \cos \varepsilon}}\right) + \left(\left(\left((\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\frac{\sin x}{\cos x} \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)}\right))_* + \frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}}\right) + \left(\frac{\left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\sin \varepsilon \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos \varepsilon\right) \cdot \left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right)} - \frac{\sin x}{\cos x}\right)\right) + \frac{\sin \varepsilon}{\left(1 - \left(\frac{\sin x}{\cos x} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \frac{\frac{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{\cos \varepsilon}\right) \cdot \cos \varepsilon}\right))_*\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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