Average Error: 36.9 → 12.8
Time: 2.5m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sqrt[3]{\left(\frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right) \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sqrt[3]{\left(\frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right) \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r5213836 = x;
        double r5213837 = eps;
        double r5213838 = r5213836 + r5213837;
        double r5213839 = tan(r5213838);
        double r5213840 = tan(r5213836);
        double r5213841 = r5213839 - r5213840;
        return r5213841;
}
↓
double f(double x, double eps) {
        double r5213842 = eps;
        double r5213843 = sin(r5213842);
        double r5213844 = cos(r5213842);
        double r5213845 = r5213843 / r5213844;
        double r5213846 = 1.0;
        double r5213847 = x;
        double r5213848 = sin(r5213847);
        double r5213849 = cos(r5213847);
        double r5213850 = r5213849 / r5213845;
        double r5213851 = r5213848 / r5213850;
        double r5213852 = r5213851 * r5213851;
        double r5213853 = r5213852 * r5213851;
        double r5213854 = cbrt(r5213853);
        double r5213855 = r5213846 - r5213854;
        double r5213856 = r5213845 / r5213855;
        double r5213857 = r5213848 / r5213849;
        double r5213858 = r5213846 - r5213851;
        double r5213859 = r5213857 / r5213858;
        double r5213860 = r5213859 - r5213857;
        double r5213861 = r5213856 + r5213860;
        return r5213861;
}

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
Target15.1
Herbie12.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.8

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

  4. Taylor expanded around inf 21.9

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

  5. Simplified12.8

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

  7. Applied add-cbrt-cube12.8

    \[\leadsto \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\sqrt[3]{\left(\frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right) \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}}}}\]

  8. Final simplification12.8

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sqrt[3]{\left(\frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}} \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right) \cdot \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}} - \frac{\sin x}{\cos x}\right)\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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