Average Error: 36.8 → 15.8
Time: 8.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.27808355156448636 \cdot 10^{-22} \lor \neg \left(\varepsilon \le 1.6891714201352877 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6.27808355156448636 \cdot 10^{-22} \lor \neg \left(\varepsilon \le 1.6891714201352877 \cdot 10^{-40}\right):\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\

\mathbf{else}:\\
\;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\

\end{array}
double f(double x, double eps) {
        double r127874 = x;
        double r127875 = eps;
        double r127876 = r127874 + r127875;
        double r127877 = tan(r127876);
        double r127878 = tan(r127874);
        double r127879 = r127877 - r127878;
        return r127879;
}
↓
double f(double x, double eps) {
        double r127880 = eps;
        double r127881 = -6.278083551564486e-22;
        bool r127882 = r127880 <= r127881;
        double r127883 = 1.6891714201352877e-40;
        bool r127884 = r127880 <= r127883;
        double r127885 = !r127884;
        bool r127886 = r127882 || r127885;
        double r127887 = x;
        double r127888 = tan(r127887);
        double r127889 = tan(r127880);
        double r127890 = r127888 + r127889;
        double r127891 = cos(r127887);
        double r127892 = r127890 * r127891;
        double r127893 = 1.0;
        double r127894 = sin(r127887);
        double r127895 = r127894 * r127889;
        double r127896 = r127895 / r127891;
        double r127897 = r127893 - r127896;
        double r127898 = r127897 * r127894;
        double r127899 = r127892 - r127898;
        double r127900 = r127897 * r127891;
        double r127901 = r127899 / r127900;
        double r127902 = r127880 * r127887;
        double r127903 = r127887 + r127880;
        double r127904 = r127902 * r127903;
        double r127905 = r127904 + r127880;
        double r127906 = r127886 ? r127901 : r127905;
        return r127906;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.7
Herbie15.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -6.278083551564486e-22 or 1.6891714201352877e-40 < eps

    1. Initial program 29.4

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

    3. Applied tan-sum2.3

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

    5. Applied tan-quot2.3

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

    6. Applied associate-*l/2.3

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

    8. Applied tan-quot2.3

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

    9. Applied frac-sub2.4

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

    if -6.278083551564486e-22 < eps < 1.6891714201352877e-40

    1. Initial program 45.3

      \[\tan \left(x + \varepsilon\right) - \tan x\]

    2. Taylor expanded around 0 31.7

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

    3. Simplified31.5

      \[\leadsto \color{blue}{\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.27808355156448636 \cdot 10^{-22} \lor \neg \left(\varepsilon \le 1.6891714201352877 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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