Average Error: 36.8 → 15.6
Time: 7.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.79464116829470944 \cdot 10^{-99} \lor \neg \left(\varepsilon \le 6.98350813945844421 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.79464116829470944 \cdot 10^{-99} \lor \neg \left(\varepsilon \le 6.98350813945844421 \cdot 10^{-29}\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(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\

\end{array}
double code(double x, double eps) {
	return (tan((x + eps)) - tan(x));
}
↓
double code(double x, double eps) {
	double VAR;
	if (((eps <= -2.7946411682947094e-99) || !(eps <= 6.983508139458444e-29))) {
		VAR = ((((tan(x) + tan(eps)) * cos(x)) - ((1.0 - (tan(x) * tan(eps))) * sin(x))) / ((1.0 - (tan(x) * tan(eps))) * cos(x)));
	} else {
		VAR = (((eps * x) * (x + eps)) + eps);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.9
Herbie15.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -2.7946411682947094e-99 or 6.983508139458444e-29 < eps

    1. Initial program 30.2

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

    3. Applied tan-quot30.1

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

    4. Applied tan-sum5.4

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

    5. Applied frac-sub5.4

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

    if -2.7946411682947094e-99 < eps < 6.983508139458444e-29

    1. Initial program 46.6

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

    2. Taylor expanded around 0 31.0

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

    3. Simplified30.8

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

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.79464116829470944 \cdot 10^{-99} \lor \neg \left(\varepsilon \le 6.98350813945844421 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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)))