Average Error: 37.3 → 0.4
Time: 23.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\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}
double f(double x, double eps) {
        double r127796 = x;
        double r127797 = eps;
        double r127798 = r127796 + r127797;
        double r127799 = tan(r127798);
        double r127800 = tan(r127796);
        double r127801 = r127799 - r127800;
        return r127801;
}
↓
double f(double x, double eps) {
        double r127802 = eps;
        double r127803 = sin(r127802);
        double r127804 = x;
        double r127805 = cos(r127804);
        double r127806 = r127803 * r127805;
        double r127807 = cos(r127802);
        double r127808 = r127806 / r127807;
        double r127809 = sin(r127804);
        double r127810 = 2.0;
        double r127811 = pow(r127809, r127810);
        double r127812 = r127811 * r127803;
        double r127813 = r127805 * r127807;
        double r127814 = r127812 / r127813;
        double r127815 = r127808 + r127814;
        double r127816 = 1.0;
        double r127817 = tan(r127804);
        double r127818 = tan(r127802);
        double r127819 = r127817 * r127818;
        double r127820 = r127816 - r127819;
        double r127821 = r127820 * r127805;
        double r127822 = r127815 / r127821;
        return r127822;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.1
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.3

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

  3. Applied tan-quot37.3

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

  4. Applied tan-sum22.2

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

  5. Applied frac-sub22.2

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019212 
(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)))