2tan (problem 3.3.2)

Average Error: 37.3 → 0.5

Time: 28.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r95015 = x;
        double r95016 = eps;
        double r95017 = r95015 + r95016;
        double r95018 = tan(r95017);
        double r95019 = tan(r95015);
        double r95020 = r95018 - r95019;
        return r95020;
}

↓

double f(double x, double eps) {
        double r95021 = eps;
        double r95022 = sin(r95021);
        double r95023 = 2.0;
        double r95024 = pow(r95022, r95023);
        double r95025 = cos(r95021);
        double r95026 = pow(r95025, r95023);
        double r95027 = 1.0;
        double r95028 = x;
        double r95029 = sin(r95028);
        double r95030 = r95029 * r95022;
        double r95031 = 3.0;
        double r95032 = pow(r95030, r95031);
        double r95033 = cos(r95028);
        double r95034 = r95033 * r95025;
        double r95035 = pow(r95034, r95031);
        double r95036 = r95032 / r95035;
        double r95037 = r95027 - r95036;
        double r95038 = r95026 * r95037;
        double r95039 = r95024 / r95038;
        double r95040 = r95029 / r95033;
        double r95041 = pow(r95040, r95031);
        double r95042 = r95041 + r95040;
        double r95043 = pow(r95029, r95023);
        double r95044 = r95043 * r95022;
        double r95045 = pow(r95033, r95023);
        double r95046 = r95037 * r95045;
        double r95047 = r95046 * r95025;
        double r95048 = r95044 / r95047;
        double r95049 = fma(r95039, r95042, r95048);
        double r95050 = pow(r95022, r95031);
        double r95051 = r95050 / r95046;
        double r95052 = pow(r95025, r95031);
        double r95053 = r95043 / r95052;
        double r95054 = r95022 / r95037;
        double r95055 = r95054 / r95025;
        double r95056 = fma(r95051, r95053, r95055);
        double r95057 = r95049 + r95056;
        double r95058 = r95036 * r95036;
        double r95059 = r95027 - r95058;
        double r95060 = r95033 * r95059;
        double r95061 = r95029 / r95060;
        double r95062 = r95027 + r95036;
        double r95063 = -r95029;
        double r95064 = r95063 / r95033;
        double r95065 = fma(r95061, r95062, r95064);
        double r95066 = r95057 + r95065;
        return r95066;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.3
Target	15.5
Herbie	0.5

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

1. Initial program 37.3

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

3. Applied tan-sum 21.8

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
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-neg 21.8

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

8. Taylor expanded around inf 22.0

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

9. Simplified 0.5

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

11. Applied flip-- 0.5

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

12. Applied associate-*l/ 0.5

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

13. Applied associate-/r/ 0.5

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

14. Applied fma-neg 0.5

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

15. Simplified 0.5

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

16. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019322 +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)))