Average Error: 37.1 → 0.5
Time: 1.5m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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(\mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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)\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)\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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(\mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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)\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)\right)
double f(double x, double eps) {
        double r13926171 = x;
        double r13926172 = eps;
        double r13926173 = r13926171 + r13926172;
        double r13926174 = tan(r13926173);
        double r13926175 = tan(r13926171);
        double r13926176 = r13926174 - r13926175;
        return r13926176;
}
↓
double f(double x, double eps) {
        double r13926177 = x;
        double r13926178 = sin(r13926177);
        double r13926179 = cos(r13926177);
        double r13926180 = r13926178 / r13926179;
        double r13926181 = r13926180 * r13926180;
        double r13926182 = eps;
        double r13926183 = sin(r13926182);
        double r13926184 = cos(r13926182);
        double r13926185 = r13926183 / r13926184;
        double r13926186 = 1.0;
        double r13926187 = r13926183 * r13926183;
        double r13926188 = r13926187 * r13926183;
        double r13926189 = r13926180 * r13926181;
        double r13926190 = r13926188 * r13926189;
        double r13926191 = r13926184 * r13926184;
        double r13926192 = r13926191 * r13926184;
        double r13926193 = r13926190 / r13926192;
        double r13926194 = r13926186 - r13926193;
        double r13926195 = r13926185 / r13926194;
        double r13926196 = r13926188 / r13926191;
        double r13926197 = r13926196 / r13926184;
        double r13926198 = r13926189 * r13926197;
        double r13926199 = r13926186 - r13926198;
        double r13926200 = r13926197 / r13926199;
        double r13926201 = r13926180 * r13926187;
        double r13926202 = r13926191 * r13926199;
        double r13926203 = r13926201 / r13926202;
        double r13926204 = fma(r13926181, r13926200, r13926203);
        double r13926205 = r13926180 / r13926199;
        double r13926206 = r13926204 + r13926205;
        double r13926207 = r13926189 * r13926187;
        double r13926208 = r13926207 / r13926202;
        double r13926209 = r13926208 - r13926180;
        double r13926210 = r13926206 + r13926209;
        double r13926211 = r13926199 * r13926184;
        double r13926212 = r13926183 / r13926211;
        double r13926213 = r13926210 + r13926212;
        double r13926214 = fma(r13926181, r13926195, r13926213);
        return r13926214;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.2
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum21.9

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

  5. Applied flip3--21.9

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

  6. Applied associate-/r/21.9

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

  7. Applied fma-neg21.9

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

  8. Simplified21.9

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

  9. Taylor expanded around -inf 22.0

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

  10. Simplified19.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\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), \left(\mathsf{fma}\left(\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin x}{\cos x}\right), \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(\mathsf{fma}\left(\left(\frac{\sin x}{\cos x}\right), \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(\mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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)\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)\right)\right) - \frac{\sin x}{\cos x}\right)\right)}\]

  11. Taylor expanded around inf 19.7

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

  12. Simplified0.5

    \[\leadsto \mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\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), \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)} + \mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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)\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)}\right)\]

  13. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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(\mathsf{fma}\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right), \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)\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)\right)\]

Reproduce

herbie shell --seed 2019121 +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)))