Average Error: 37.2 → 0.4
Time: 43.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sin x}{\frac{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}{\sin x}}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{\sin x}{\frac{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}{\sin x}}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}
double f(double x, double eps) {
        double r3562792 = x;
        double r3562793 = eps;
        double r3562794 = r3562792 + r3562793;
        double r3562795 = tan(r3562794);
        double r3562796 = tan(r3562792);
        double r3562797 = r3562795 - r3562796;
        return r3562797;
}
↓
double f(double x, double eps) {
        double r3562798 = eps;
        double r3562799 = sin(r3562798);
        double r3562800 = x;
        double r3562801 = cos(r3562800);
        double r3562802 = r3562799 * r3562801;
        double r3562803 = cos(r3562798);
        double r3562804 = r3562802 / r3562803;
        double r3562805 = sin(r3562800);
        double r3562806 = r3562799 / r3562803;
        double r3562807 = r3562801 / r3562806;
        double r3562808 = r3562807 / r3562805;
        double r3562809 = r3562805 / r3562808;
        double r3562810 = r3562804 + r3562809;
        double r3562811 = 1.0;
        double r3562812 = tan(r3562798);
        double r3562813 = tan(r3562800);
        double r3562814 = r3562812 * r3562813;
        double r3562815 = r3562811 - r3562814;
        double r3562816 = r3562815 * r3562801;
        double r3562817 = r3562810 / r3562816;
        return r3562817;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target14.9
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.2

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

  3. Applied tan-quot37.2

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

  4. Applied tan-sum22.3

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

  5. Applied frac-sub22.3

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

  6. Taylor expanded around inf 0.4

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

  7. Simplified0.5

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

  8. Taylor expanded around -inf 0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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