Average Error: 36.9 → 15.3
Time: 10.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.52409292857505093 \cdot 10^{-57} \lor \neg \left(\varepsilon \le 3.02977531859297208 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left(e^{\tan x}\right)}^{\left(\tan \varepsilon\right)}\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.52409292857505093 \cdot 10^{-57} \lor \neg \left(\varepsilon \le 3.02977531859297208 \cdot 10^{-50}\right):\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left(e^{\tan x}\right)}^{\left(\tan \varepsilon\right)}\right)} - \tan x\\

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

\end{array}
double f(double x, double eps) {
        double r178434 = x;
        double r178435 = eps;
        double r178436 = r178434 + r178435;
        double r178437 = tan(r178436);
        double r178438 = tan(r178434);
        double r178439 = r178437 - r178438;
        return r178439;
}
↓
double f(double x, double eps) {
        double r178440 = eps;
        double r178441 = -1.524092928575051e-57;
        bool r178442 = r178440 <= r178441;
        double r178443 = 3.029775318592972e-50;
        bool r178444 = r178440 <= r178443;
        double r178445 = !r178444;
        bool r178446 = r178442 || r178445;
        double r178447 = x;
        double r178448 = tan(r178447);
        double r178449 = tan(r178440);
        double r178450 = r178448 + r178449;
        double r178451 = 1.0;
        double r178452 = exp(r178448);
        double r178453 = pow(r178452, r178449);
        double r178454 = log(r178453);
        double r178455 = r178451 - r178454;
        double r178456 = r178450 / r178455;
        double r178457 = r178456 - r178448;
        double r178458 = r178447 * r178440;
        double r178459 = r178440 + r178447;
        double r178460 = r178458 * r178459;
        double r178461 = r178460 + r178440;
        double r178462 = r178446 ? r178457 : r178461;
        return r178462;
}

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
Target15.2
Herbie15.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.524092928575051e-57 or 3.029775318592972e-50 < eps

    1. Initial program 30.2

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

    3. Applied tan-sum4.4

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

    5. Applied add-log-exp4.4

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

    7. Applied add-log-exp4.5

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

    8. Applied exp-to-pow4.5

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

    if -1.524092928575051e-57 < eps < 3.029775318592972e-50

    1. Initial program 46.4

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

    3. Applied tan-sum46.4

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

    4. Taylor expanded around 0 30.9

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

    5. Simplified30.7

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.52409292857505093 \cdot 10^{-57} \lor \neg \left(\varepsilon \le 3.02977531859297208 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left(e^{\tan x}\right)}^{\left(\tan \varepsilon\right)}\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))