Average Error: 36.1 → 13.8
Time: 1.3m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3766066923975014 \cdot 10^{-65}:\\ \;\;\;\;\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x\right) + \frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \le 4.257017227961505 \cdot 10^{-151}:\\ \;\;\;\;\varepsilon + \left(x + \frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x\right) + \frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.3766066923975014 \cdot 10^{-65}:\\
\;\;\;\;\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x\right) + \frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\\

\mathbf{elif}\;\varepsilon \le 4.257017227961505 \cdot 10^{-151}:\\
\;\;\;\;\varepsilon + \left(x + \frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\

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

\end{array}
double f(double x, double eps) {
        double r6143322 = x;
        double r6143323 = eps;
        double r6143324 = r6143322 + r6143323;
        double r6143325 = tan(r6143324);
        double r6143326 = tan(r6143322);
        double r6143327 = r6143325 - r6143326;
        return r6143327;
}
↓
double f(double x, double eps) {
        double r6143328 = eps;
        double r6143329 = -1.3766066923975014e-65;
        bool r6143330 = r6143328 <= r6143329;
        double r6143331 = tan(r6143328);
        double r6143332 = x;
        double r6143333 = tan(r6143332);
        double r6143334 = r6143331 + r6143333;
        double r6143335 = 1.0;
        double r6143336 = r6143333 * r6143331;
        double r6143337 = r6143336 * r6143336;
        double r6143338 = r6143337 * r6143336;
        double r6143339 = r6143335 - r6143338;
        double r6143340 = r6143334 / r6143339;
        double r6143341 = r6143340 - r6143333;
        double r6143342 = r6143336 + r6143337;
        double r6143343 = r6143340 * r6143342;
        double r6143344 = r6143341 + r6143343;
        double r6143345 = 4.257017227961505e-151;
        bool r6143346 = r6143328 <= r6143345;
        double r6143347 = 0.3333333333333333;
        double r6143348 = r6143347 * r6143328;
        double r6143349 = r6143332 + r6143348;
        double r6143350 = r6143328 * r6143328;
        double r6143351 = r6143349 * r6143350;
        double r6143352 = r6143328 + r6143351;
        double r6143353 = r6143346 ? r6143352 : r6143344;
        double r6143354 = r6143330 ? r6143344 : r6143353;
        return r6143354;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.1
Target14.3
Herbie13.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.3766066923975014e-65 or 4.257017227961505e-151 < eps

    1. Initial program 30.0

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

    3. Applied +-commutative30.0

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

    4. Applied tan-sum8.5

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

    6. Applied flip3--8.5

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

    7. Applied associate-/r/8.5

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

    8. Simplified8.5

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

    10. Applied +-commutative8.5

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

    11. Applied distribute-lft-in8.5

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

    12. Applied associate--l+7.4

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

    if -1.3766066923975014e-65 < eps < 4.257017227961505e-151

    1. Initial program 48.4

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

    3. Applied +-commutative48.4

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

    4. Applied tan-sum48.4

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

    5. Taylor expanded around 0 26.8

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

    6. Simplified26.8

      \[\leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \frac{1}{3} + x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3766066923975014 \cdot 10^{-65}:\\ \;\;\;\;\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x\right) + \frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \le 4.257017227961505 \cdot 10^{-151}:\\ \;\;\;\;\varepsilon + \left(x + \frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x\right) + \frac{\tan \varepsilon + \tan x}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))