2tan (problem 3.3.2)

Average Error: 37.4 → 0.4

Time: 36.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r3345831 = x;
        double r3345832 = eps;
        double r3345833 = r3345831 + r3345832;
        double r3345834 = tan(r3345833);
        double r3345835 = tan(r3345831);
        double r3345836 = r3345834 - r3345835;
        return r3345836;
}

↓

double f(double x, double eps) {
        double r3345837 = x;
        double r3345838 = sin(r3345837);
        double r3345839 = cos(r3345837);
        double r3345840 = r3345838 / r3345839;
        double r3345841 = r3345840 * r3345840;
        double r3345842 = eps;
        double r3345843 = sin(r3345842);
        double r3345844 = cos(r3345842);
        double r3345845 = 1.0;
        double r3345846 = r3345843 * r3345838;
        double r3345847 = r3345844 * r3345839;
        double r3345848 = r3345847 * r3345847;
        double r3345849 = r3345846 / r3345848;
        double r3345850 = r3345846 * r3345846;
        double r3345851 = r3345850 / r3345847;
        double r3345852 = r3345849 * r3345851;
        double r3345853 = r3345845 - r3345852;
        double r3345854 = r3345844 * r3345853;
        double r3345855 = r3345843 / r3345854;
        double r3345856 = r3345840 / r3345853;
        double r3345857 = r3345856 - r3345840;
        double r3345858 = r3345843 * r3345843;
        double r3345859 = r3345844 * r3345844;
        double r3345860 = r3345858 / r3345859;
        double r3345861 = r3345860 / r3345853;
        double r3345862 = fma(r3345841, r3345840, r3345840);
        double r3345863 = r3345850 * r3345843;
        double r3345864 = r3345848 * r3345844;
        double r3345865 = r3345863 / r3345864;
        double r3345866 = r3345865 / r3345853;
        double r3345867 = r3345866 + r3345855;
        double r3345868 = fma(r3345861, r3345862, r3345867);
        double r3345869 = r3345857 + r3345868;
        double r3345870 = fma(r3345841, r3345855, r3345869);
        return r3345870;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.6
Herbie	0.4

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

Derivation

1. Initial program 37.4

   \[\tan \left(x + \varepsilon\right) - \tan x\]
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-- 21.8

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

6. Applied associate-/r/ 21.8

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

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 21.9

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

9. Simplified 19.6

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

11. Applied associate--l+ 0.4

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

12. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019162 +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)))