Average Error: 37.3 → 0.5
Time: 52.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\cos x}{\frac{\cos \varepsilon}{\sin \varepsilon}} + \frac{\sin x \cdot \sin x}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\cos x}{\frac{\cos \varepsilon}{\sin \varepsilon}} + \frac{\sin x \cdot \sin x}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}
double f(double x, double eps) {
        double r4068750 = x;
        double r4068751 = eps;
        double r4068752 = r4068750 + r4068751;
        double r4068753 = tan(r4068752);
        double r4068754 = tan(r4068750);
        double r4068755 = r4068753 - r4068754;
        return r4068755;
}
↓
double f(double x, double eps) {
        double r4068756 = x;
        double r4068757 = cos(r4068756);
        double r4068758 = eps;
        double r4068759 = cos(r4068758);
        double r4068760 = sin(r4068758);
        double r4068761 = r4068759 / r4068760;
        double r4068762 = r4068757 / r4068761;
        double r4068763 = sin(r4068756);
        double r4068764 = r4068763 * r4068763;
        double r4068765 = r4068757 * r4068759;
        double r4068766 = r4068765 / r4068760;
        double r4068767 = r4068764 / r4068766;
        double r4068768 = r4068762 + r4068767;
        double r4068769 = 1.0;
        double r4068770 = tan(r4068758);
        double r4068771 = tan(r4068756);
        double r4068772 = r4068770 * r4068771;
        double r4068773 = r4068769 - r4068772;
        double r4068774 = r4068757 * r4068773;
        double r4068775 = r4068768 / r4068774;
        return r4068775;
}

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
Target15.2
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.3

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

  3. Applied tan-sum21.9

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

  5. Applied div-inv21.9

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

  7. Applied tan-quot22.0

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

  8. Applied un-div-inv22.0

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

  9. Applied frac-sub22.1

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

  10. Taylor expanded around -inf 0.4

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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