Average Error: 37.1 → 0.6
Time: 31.2s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} - \frac{\sin x}{\cos x}\right) + \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} - \frac{\sin x}{\cos x}\right) + \left(\left(\frac{\sin x}{\cos x} \cdot \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}} \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)}}\right)
double f(double x, double eps) {
        double r3434898 = x;
        double r3434899 = eps;
        double r3434900 = r3434898 + r3434899;
        double r3434901 = tan(r3434900);
        double r3434902 = tan(r3434898);
        double r3434903 = r3434901 - r3434902;
        return r3434903;
}
↓
double f(double x, double eps) {
        double r3434904 = x;
        double r3434905 = sin(r3434904);
        double r3434906 = cos(r3434904);
        double r3434907 = r3434905 / r3434906;
        double r3434908 = 1.0;
        double r3434909 = eps;
        double r3434910 = sin(r3434909);
        double r3434911 = r3434905 * r3434910;
        double r3434912 = r3434911 * r3434911;
        double r3434913 = cos(r3434909);
        double r3434914 = r3434913 * r3434906;
        double r3434915 = r3434914 * r3434914;
        double r3434916 = r3434912 / r3434915;
        double r3434917 = r3434908 - r3434916;
        double r3434918 = r3434907 / r3434917;
        double r3434919 = r3434918 - r3434907;
        double r3434920 = r3434910 / r3434913;
        double r3434921 = r3434920 * r3434920;
        double r3434922 = r3434921 / r3434917;
        double r3434923 = r3434907 * r3434922;
        double r3434924 = r3434920 / r3434917;
        double r3434925 = r3434907 * r3434907;
        double r3434926 = r3434924 * r3434925;
        double r3434927 = r3434923 + r3434926;
        double r3434928 = r3434927 + r3434924;
        double r3434929 = r3434919 + r3434928;
        return r3434929;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.3
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum21.7

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

  5. Applied flip--21.8

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

  6. Applied associate-/r/21.8

    \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x\]
  7. Using strategy rm

  8. Applied tan-quot21.8

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

  9. Applied associate-*l/21.8

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

  10. Applied associate-*r/21.8

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

  11. Taylor expanded around -inf 21.9

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2019141 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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