Average Error: 36.6 → 0.4
Time: 8.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\left(\left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)\right) + \mathsf{fma}\left(\frac{\sin x}{\left(1 \cdot 1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}} \cdot \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x}, 1 + \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}, -\frac{\sin x}{\cos x}\right)\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\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} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\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 {\left(\cos \varepsilon\right)}^{3}}, \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\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 {\left(\cos x\right)}^{2}\right)}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{\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 {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)\right) + \mathsf{fma}\left(\frac{\sin x}{\left(1 \cdot 1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}} \cdot \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 x}, 1 + \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}, -\frac{\sin x}{\cos x}\right)\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)
double code(double x, double eps) {
	return (tan((x + eps)) - tan(x));
}
↓
double code(double x, double eps) {
	return (((((sin(eps) / ((1.0 - ((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0)))) * cos(eps))) + fma((pow(sin(x), 2.0) / pow(cos(x), 2.0)), (pow(sin(eps), 3.0) / ((1.0 - ((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0)))) * pow(cos(eps), 3.0))), ((pow(sin(x), 2.0) * sin(eps)) / (cos(eps) * ((1.0 - ((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0)))) * pow(cos(x), 2.0)))))) + ((pow(sin(eps), 2.0) / ((1.0 - ((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0)))) * pow(cos(eps), 2.0))) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))))) + fma((sin(x) / (((1.0 * 1.0) - (((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0))) * ((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0))))) * cos(x))), (1.0 + ((pow(sin(x), 3.0) * pow(sin(eps), 3.0)) / (pow(cos(x), 3.0) * pow(cos(eps), 3.0)))), -(sin(x) / cos(x)))) + fma(-tan(x), 1.0, tan(x)));
}

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
Target15.1
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.6

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

  3. Applied tan-sum21.5

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

  5. Applied add-cube-cbrt22.0

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

  6. Applied flip3--22.0

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

  7. Applied associate-/r/22.0

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

  8. Applied prod-diff22.1

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

  9. Simplified21.9

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

  10. Simplified21.5

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

  11. Taylor expanded around inf 21.6

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

  12. Simplified0.4

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

  14. Applied flip--0.4

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

  15. Applied associate-*l/0.4

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

  16. Applied associate-/r/0.4

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

  17. Applied fma-neg0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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