Average Error: 37.3 → 13.0
Time: 34.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x}{\frac{\cos \varepsilon}{\sin \varepsilon}}}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\frac{\cos \varepsilon}{\sin \varepsilon}}}{\cos x}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x}{\frac{\cos \varepsilon}{\sin \varepsilon}}}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\frac{\cos \varepsilon}{\sin \varepsilon}}}{\cos x}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r3957885 = x;
        double r3957886 = eps;
        double r3957887 = r3957885 + r3957886;
        double r3957888 = tan(r3957887);
        double r3957889 = tan(r3957885);
        double r3957890 = r3957888 - r3957889;
        return r3957890;
}
↓
double f(double x, double eps) {
        double r3957891 = eps;
        double r3957892 = sin(r3957891);
        double r3957893 = cos(r3957891);
        double r3957894 = r3957892 / r3957893;
        double r3957895 = 1.0;
        double r3957896 = x;
        double r3957897 = sin(r3957896);
        double r3957898 = r3957893 / r3957892;
        double r3957899 = r3957897 / r3957898;
        double r3957900 = cos(r3957896);
        double r3957901 = r3957899 / r3957900;
        double r3957902 = r3957895 - r3957901;
        double r3957903 = r3957894 / r3957902;
        double r3957904 = r3957897 / r3957900;
        double r3957905 = r3957904 / r3957902;
        double r3957906 = r3957905 - r3957904;
        double r3957907 = r3957903 + r3957906;
        return r3957907;
}

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
Herbie13.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

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. Applied fma-neg21.9

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

  8. Applied add-log-exp22.0

    \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)}}, -\tan x\right)\]

  9. Taylor expanded around -inf 22.0

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

  10. Simplified13.0

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

  11. Final simplification13.0

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

Reproduce

herbie shell --seed 2019151 +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)))