Average Error: 36.6 → 0.4
Time: 26.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon} + \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x \cdot \sin x}{\cos \varepsilon}}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon} + \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x \cdot \sin x}{\cos \varepsilon}}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}
double f(double x, double eps) {
        double r1306748 = x;
        double r1306749 = eps;
        double r1306750 = r1306748 + r1306749;
        double r1306751 = tan(r1306750);
        double r1306752 = tan(r1306748);
        double r1306753 = r1306751 - r1306752;
        return r1306753;
}
↓
double f(double x, double eps) {
        double r1306754 = x;
        double r1306755 = cos(r1306754);
        double r1306756 = eps;
        double r1306757 = sin(r1306756);
        double r1306758 = r1306755 * r1306757;
        double r1306759 = cos(r1306756);
        double r1306760 = r1306758 / r1306759;
        double r1306761 = r1306757 / r1306755;
        double r1306762 = sin(r1306754);
        double r1306763 = r1306762 * r1306762;
        double r1306764 = r1306763 / r1306759;
        double r1306765 = r1306761 * r1306764;
        double r1306766 = r1306760 + r1306765;
        double r1306767 = 1.0;
        double r1306768 = tan(r1306756);
        double r1306769 = tan(r1306754);
        double r1306770 = r1306768 * r1306769;
        double r1306771 = r1306767 - r1306770;
        double r1306772 = r1306755 * r1306771;
        double r1306773 = r1306766 / r1306772;
        return r1306773;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target14.6
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.6

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

  3. Applied tan-quot36.7

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

  4. Applied tan-sum22.1

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

  5. Applied frac-sub22.1

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

  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. Simplified0.4

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

  9. Applied associate-*l/0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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