Average Error: 36.9 → 12.7
Time: 29.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\tan \varepsilon}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x}\right) + \frac{\tan \varepsilon}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}
double f(double x, double eps) {
        double r6142747 = x;
        double r6142748 = eps;
        double r6142749 = r6142747 + r6142748;
        double r6142750 = tan(r6142749);
        double r6142751 = tan(r6142747);
        double r6142752 = r6142750 - r6142751;
        return r6142752;
}
↓
double f(double x, double eps) {
        double r6142753 = x;
        double r6142754 = sin(r6142753);
        double r6142755 = cos(r6142753);
        double r6142756 = r6142754 / r6142755;
        double r6142757 = 1.0;
        double r6142758 = eps;
        double r6142759 = sin(r6142758);
        double r6142760 = r6142754 * r6142759;
        double r6142761 = cos(r6142758);
        double r6142762 = r6142760 / r6142761;
        double r6142763 = r6142762 / r6142755;
        double r6142764 = r6142757 - r6142763;
        double r6142765 = r6142756 / r6142764;
        double r6142766 = r6142765 - r6142756;
        double r6142767 = tan(r6142758);
        double r6142768 = r6142759 / r6142761;
        double r6142769 = r6142768 * r6142754;
        double r6142770 = r6142769 / r6142755;
        double r6142771 = r6142757 - r6142770;
        double r6142772 = r6142767 / r6142771;
        double r6142773 = r6142766 + r6142772;
        return r6142773;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.3
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.5

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

  4. Taylor expanded around inf 21.6

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

  5. Simplified12.7

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

  7. Applied associate-*r/12.7

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

  9. Applied quot-tan12.7

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

  10. Final simplification12.7

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

Reproduce

herbie shell --seed 2019169 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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