2tan (problem 3.3.2)

Average Error: 36.8 → 0.5

Time: 10.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r162796 = x;
        double r162797 = eps;
        double r162798 = r162796 + r162797;
        double r162799 = tan(r162798);
        double r162800 = tan(r162796);
        double r162801 = r162799 - r162800;
        return r162801;
}

↓

double f(double x, double eps) {
        double r162802 = x;
        double r162803 = sin(r162802);
        double r162804 = 2.0;
        double r162805 = pow(r162803, r162804);
        double r162806 = eps;
        double r162807 = cos(r162806);
        double r162808 = r162805 / r162807;
        double r162809 = sin(r162806);
        double r162810 = 1.0;
        double r162811 = 3.0;
        double r162812 = pow(r162803, r162811);
        double r162813 = pow(r162809, r162811);
        double r162814 = r162812 * r162813;
        double r162815 = cos(r162802);
        double r162816 = pow(r162815, r162811);
        double r162817 = pow(r162807, r162811);
        double r162818 = r162816 * r162817;
        double r162819 = r162814 / r162818;
        double r162820 = r162810 - r162819;
        double r162821 = pow(r162815, r162804);
        double r162822 = r162820 * r162821;
        double r162823 = r162809 / r162822;
        double r162824 = r162820 * r162807;
        double r162825 = r162809 / r162824;
        double r162826 = fma(r162808, r162823, r162825);
        double r162827 = r162803 / r162815;
        double r162828 = pow(r162827, r162811);
        double r162829 = pow(r162809, r162804);
        double r162830 = pow(r162807, r162804);
        double r162831 = r162820 * r162830;
        double r162832 = r162829 / r162831;
        double r162833 = r162815 * r162820;
        double r162834 = r162803 / r162833;
        double r162835 = fma(r162828, r162832, r162834);
        double r162836 = r162805 / r162821;
        double r162837 = r162820 * r162817;
        double r162838 = r162813 / r162837;
        double r162839 = r162803 * r162829;
        double r162840 = r162830 * r162833;
        double r162841 = r162839 / r162840;
        double r162842 = fma(r162836, r162838, r162841);
        double r162843 = r162842 - r162827;
        double r162844 = r162835 + r162843;
        double r162845 = r162826 + r162844;
        double r162846 = tan(r162802);
        double r162847 = -r162846;
        double r162848 = fma(r162847, r162810, r162846);
        double r162849 = r162845 + r162848;
        return r162849;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	14.7
Herbie	0.5

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

Derivation

1. Initial program 36.8

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

3. Applied tan-sum 22.1

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

5. Applied add-cube-cbrt 22.6

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

6. Applied flip3-- 22.6

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

7. Applied associate-/r/ 22.6

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

8. Applied prod-diff 22.7

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

9. Simplified 22.5

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

10. Simplified 22.1

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

11. Taylor expanded around -inf 22.2

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

12. Simplified 0.5

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

13. Taylor expanded around inf 0.5

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

14. Simplified 0.5

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

15. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020059 +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)))