Average Error: 37.0 → 15.0
Time: 31.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.686281805292481 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} + \tan x\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)}{\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{elif}\;\varepsilon \le 7.203202361914666 \cdot 10^{-37}:\\ \;\;\;\;\varepsilon + \left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.686281805292481 \cdot 10^{-23}:\\
\;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} + \tan x\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)}{\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\

\mathbf{elif}\;\varepsilon \le 7.203202361914666 \cdot 10^{-37}:\\
\;\;\;\;\varepsilon + \left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)\\

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

\end{array}
double f(double x, double eps) {
        double r2500008 = x;
        double r2500009 = eps;
        double r2500010 = r2500008 + r2500009;
        double r2500011 = tan(r2500010);
        double r2500012 = tan(r2500008);
        double r2500013 = r2500011 - r2500012;
        return r2500013;
}
↓
double f(double x, double eps) {
        double r2500014 = eps;
        double r2500015 = -2.686281805292481e-23;
        bool r2500016 = r2500014 <= r2500015;
        double r2500017 = tan(r2500014);
        double r2500018 = x;
        double r2500019 = tan(r2500018);
        double r2500020 = r2500017 + r2500019;
        double r2500021 = 1.0;
        double r2500022 = r2500019 * r2500017;
        double r2500023 = r2500021 - r2500022;
        double r2500024 = r2500020 / r2500023;
        double r2500025 = r2500024 + r2500019;
        double r2500026 = r2500024 - r2500019;
        double r2500027 = r2500025 * r2500026;
        double r2500028 = r2500027 / r2500025;
        double r2500029 = 7.203202361914666e-37;
        bool r2500030 = r2500014 <= r2500029;
        double r2500031 = r2500018 + r2500014;
        double r2500032 = r2500018 * r2500014;
        double r2500033 = r2500031 * r2500032;
        double r2500034 = r2500014 + r2500033;
        double r2500035 = r2500021 + r2500022;
        double r2500036 = r2500022 * r2500022;
        double r2500037 = r2500021 - r2500036;
        double r2500038 = r2500020 / r2500037;
        double r2500039 = r2500035 * r2500038;
        double r2500040 = r2500039 - r2500019;
        double r2500041 = r2500030 ? r2500034 : r2500040;
        double r2500042 = r2500016 ? r2500028 : r2500041;
        return r2500042;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.4
Herbie15.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.686281805292481e-23

    1. Initial program 30.1

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

    3. Applied tan-sum1.5

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

    5. Applied flip--1.6

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

    6. Simplified1.5

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

    if -2.686281805292481e-23 < eps < 7.203202361914666e-37

    1. Initial program 45.7

      \[\tan \left(x + \varepsilon\right) - \tan x\]

    2. Taylor expanded around 0 31.0

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

    3. Simplified31.0

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

    if 7.203202361914666e-37 < eps

    1. Initial program 30.0

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

    3. Applied tan-sum2.5

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

    5. Applied flip--2.6

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

    6. Applied associate-/r/2.6

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

    7. Simplified2.6

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

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.686281805292481 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} + \tan x\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)}{\frac{\tan \varepsilon + \tan x}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{elif}\;\varepsilon \le 7.203202361914666 \cdot 10^{-37}:\\ \;\;\;\;\varepsilon + \left(x + \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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