Average Error: 37.4 → 15.5
Time: 11.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.950013469467184 \cdot 10^{-38} \lor \neg \left(\varepsilon \le 2.8556285663292881 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.950013469467184 \cdot 10^{-38} \lor \neg \left(\varepsilon \le 2.8556285663292881 \cdot 10^{-18}\right):\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\

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

\end{array}
double f(double x, double eps) {
        double r130279 = x;
        double r130280 = eps;
        double r130281 = r130279 + r130280;
        double r130282 = tan(r130281);
        double r130283 = tan(r130279);
        double r130284 = r130282 - r130283;
        return r130284;
}
↓
double f(double x, double eps) {
        double r130285 = eps;
        double r130286 = -1.9500134694671836e-38;
        bool r130287 = r130285 <= r130286;
        double r130288 = 2.855628566329288e-18;
        bool r130289 = r130285 <= r130288;
        double r130290 = !r130289;
        bool r130291 = r130287 || r130290;
        double r130292 = x;
        double r130293 = tan(r130292);
        double r130294 = tan(r130285);
        double r130295 = r130293 + r130294;
        double r130296 = cos(r130292);
        double r130297 = r130295 * r130296;
        double r130298 = 1.0;
        double r130299 = r130293 * r130294;
        double r130300 = r130298 - r130299;
        double r130301 = sin(r130292);
        double r130302 = r130300 * r130301;
        double r130303 = r130297 - r130302;
        double r130304 = r130300 * r130296;
        double r130305 = r130303 / r130304;
        double r130306 = r130292 * r130285;
        double r130307 = r130285 + r130292;
        double r130308 = r130306 * r130307;
        double r130309 = r130308 + r130285;
        double r130310 = r130291 ? r130305 : r130309;
        return r130310;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.2
Herbie15.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.9500134694671836e-38 or 2.855628566329288e-18 < eps

    1. Initial program 30.2

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

    3. Applied tan-quot30.0

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

    4. Applied tan-sum2.2

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

    5. Applied frac-sub2.3

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

    if -1.9500134694671836e-38 < eps < 2.855628566329288e-18

    1. Initial program 45.9

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

    3. Applied tan-sum45.9

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

    4. Taylor expanded around 0 31.4

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

    5. Simplified31.2

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.950013469467184 \cdot 10^{-38} \lor \neg \left(\varepsilon \le 2.8556285663292881 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))