Average Error: 36.6 → 15.8
Time: 10.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.2637252434053613 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.2637252434053613 \cdot 10^{-90}:\\
\;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\

\mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\

\mathbf{else}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\

\end{array}
double f(double x, double eps) {
        double r121869 = x;
        double r121870 = eps;
        double r121871 = r121869 + r121870;
        double r121872 = tan(r121871);
        double r121873 = tan(r121869);
        double r121874 = r121872 - r121873;
        return r121874;
}
↓
double f(double x, double eps) {
        double r121875 = eps;
        double r121876 = -2.2637252434053613e-90;
        bool r121877 = r121875 <= r121876;
        double r121878 = 1.0;
        double r121879 = x;
        double r121880 = tan(r121879);
        double r121881 = tan(r121875);
        double r121882 = r121880 * r121881;
        double r121883 = r121878 - r121882;
        double r121884 = r121880 + r121881;
        double r121885 = r121883 / r121884;
        double r121886 = r121878 / r121885;
        double r121887 = r121886 - r121880;
        double r121888 = 5.748527172026931e-167;
        bool r121889 = r121875 <= r121888;
        double r121890 = r121879 * r121875;
        double r121891 = r121875 + r121879;
        double r121892 = r121890 * r121891;
        double r121893 = r121892 + r121875;
        double r121894 = 3.0;
        double r121895 = pow(r121882, r121894);
        double r121896 = r121878 - r121895;
        double r121897 = r121884 / r121896;
        double r121898 = r121882 * r121882;
        double r121899 = r121878 * r121882;
        double r121900 = r121898 + r121899;
        double r121901 = r121878 + r121900;
        double r121902 = r121897 * r121901;
        double r121903 = r121902 - r121880;
        double r121904 = r121889 ? r121893 : r121903;
        double r121905 = r121877 ? r121887 : r121904;
        return r121905;
}

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
Target15.4
Herbie15.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.2637252434053613e-90

    1. Initial program 30.9

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

    3. Applied tan-sum6.8

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

    5. Applied clear-num6.9

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

    if -2.2637252434053613e-90 < eps < 5.748527172026931e-167

    1. Initial program 48.9

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

    3. Applied tan-sum48.9

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

    5. Applied tan-quot48.9

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

    6. Applied associate-*r/48.9

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

    7. Taylor expanded around 0 30.0

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

    8. Simplified29.7

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

    if 5.748527172026931e-167 < eps

    1. Initial program 32.4

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

    3. Applied tan-sum13.0

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

    5. Applied flip3--13.1

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

    6. Applied associate-/r/13.1

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

    7. Simplified13.1

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.2637252434053613 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 5.7485271720269307 \cdot 10^{-167}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))