Average Error: 37.1 → 0.4
Time: 24.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \sin x}{\cos x \cdot \cos \varepsilon} + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \sin x}{\cos x \cdot \cos \varepsilon} + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r102826 = x;
        double r102827 = eps;
        double r102828 = r102826 + r102827;
        double r102829 = tan(r102828);
        double r102830 = tan(r102826);
        double r102831 = r102829 - r102830;
        return r102831;
}
↓
double f(double x, double eps) {
        double r102832 = x;
        double r102833 = sin(r102832);
        double r102834 = eps;
        double r102835 = sin(r102834);
        double r102836 = r102833 * r102835;
        double r102837 = r102836 * r102833;
        double r102838 = cos(r102832);
        double r102839 = cos(r102834);
        double r102840 = r102838 * r102839;
        double r102841 = r102837 / r102840;
        double r102842 = r102835 * r102838;
        double r102843 = r102842 / r102839;
        double r102844 = r102841 + r102843;
        double r102845 = 1.0;
        double r102846 = tan(r102832);
        double r102847 = tan(r102834);
        double r102848 = r102846 * r102847;
        double r102849 = r102845 - r102848;
        double r102850 = r102849 * r102838;
        double r102851 = r102844 / r102850;
        return r102851;
}

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

Derivation

  1. Initial program 37.1

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

  3. Applied tan-quot37.1

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

  4. Applied tan-sum21.9

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

  5. Applied frac-sub21.9

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

  6. Taylor expanded around inf 0.4

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

  8. Applied sqr-pow0.4

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

  9. Applied associate-*l*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 
(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)))