Average Error: 36.7 → 15.4
Time: 10.1s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8367694074004902 \cdot 10^{-68} \lor \neg \left(\varepsilon \le 1.476728953358803 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.8367694074004902 \cdot 10^{-68} \lor \neg \left(\varepsilon \le 1.476728953358803 \cdot 10^{-117}\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 r118302 = x;
        double r118303 = eps;
        double r118304 = r118302 + r118303;
        double r118305 = tan(r118304);
        double r118306 = tan(r118302);
        double r118307 = r118305 - r118306;
        return r118307;
}
↓
double f(double x, double eps) {
        double r118308 = eps;
        double r118309 = -1.8367694074004902e-68;
        bool r118310 = r118308 <= r118309;
        double r118311 = 1.4767289533588029e-117;
        bool r118312 = r118308 <= r118311;
        double r118313 = !r118312;
        bool r118314 = r118310 || r118313;
        double r118315 = x;
        double r118316 = tan(r118315);
        double r118317 = tan(r118308);
        double r118318 = r118316 + r118317;
        double r118319 = cos(r118315);
        double r118320 = r118318 * r118319;
        double r118321 = 1.0;
        double r118322 = r118316 * r118317;
        double r118323 = r118321 - r118322;
        double r118324 = sin(r118315);
        double r118325 = r118323 * r118324;
        double r118326 = r118320 - r118325;
        double r118327 = r118323 * r118319;
        double r118328 = r118326 / r118327;
        double r118329 = r118315 * r118308;
        double r118330 = r118308 + r118315;
        double r118331 = r118329 * r118330;
        double r118332 = r118331 + r118308;
        double r118333 = r118314 ? r118328 : r118332;
        return r118333;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target15.1
Herbie15.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.8367694074004902e-68 or 1.4767289533588029e-117 < eps

    1. Initial program 30.4

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

    3. Applied tan-quot30.3

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

    4. Applied tan-sum7.2

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

    5. Applied frac-sub7.2

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

    if -1.8367694074004902e-68 < eps < 1.4767289533588029e-117

    1. Initial program 48.6

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

    3. Applied tan-sum48.6

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

    4. Taylor expanded around 0 31.1

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

    5. Simplified30.9

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8367694074004902 \cdot 10^{-68} \lor \neg \left(\varepsilon \le 1.476728953358803 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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)))