Average Error: 36.5 → 15.3
Time: 10.4s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.418632866442672109740022575154699664032 \cdot 10^{-106} \lor \neg \left(\varepsilon \le 1.137895195985122466489334986457483016272 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -4.418632866442672109740022575154699664032 \cdot 10^{-106} \lor \neg \left(\varepsilon \le 1.137895195985122466489334986457483016272 \cdot 10^{-80}\right):\\
\;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\

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

\end{array}
double f(double x, double eps) {
        double r133612 = x;
        double r133613 = eps;
        double r133614 = r133612 + r133613;
        double r133615 = tan(r133614);
        double r133616 = tan(r133612);
        double r133617 = r133615 - r133616;
        return r133617;
}
↓
double f(double x, double eps) {
        double r133618 = eps;
        double r133619 = -4.418632866442672e-106;
        bool r133620 = r133618 <= r133619;
        double r133621 = 1.1378951959851225e-80;
        bool r133622 = r133618 <= r133621;
        double r133623 = !r133622;
        bool r133624 = r133620 || r133623;
        double r133625 = 1.0;
        double r133626 = x;
        double r133627 = tan(r133626);
        double r133628 = tan(r133618);
        double r133629 = r133627 * r133628;
        double r133630 = r133625 - r133629;
        double r133631 = r133627 + r133628;
        double r133632 = r133630 / r133631;
        double r133633 = r133625 / r133632;
        double r133634 = r133633 - r133627;
        double r133635 = r133626 * r133618;
        double r133636 = r133618 + r133626;
        double r133637 = r133635 * r133636;
        double r133638 = r133637 + r133618;
        double r133639 = r133624 ? r133634 : r133638;
        return r133639;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.5
Target14.8
Herbie15.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -4.418632866442672e-106 or 1.1378951959851225e-80 < eps

    1. Initial program 30.2

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

    3. Applied tan-sum7.2

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

    5. Applied clear-num7.3

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

    if -4.418632866442672e-106 < eps < 1.1378951959851225e-80

    1. Initial program 48.1

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

    3. Applied tan-sum48.1

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

    4. Taylor expanded around 0 30.5

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

    5. Simplified30.3

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.418632866442672109740022575154699664032 \cdot 10^{-106} \lor \neg \left(\varepsilon \le 1.137895195985122466489334986457483016272 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))