Average Error: 36.9 → 15.2
Time: 11.2s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.99806522766741552 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 7.73798919875330722 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.99806522766741552 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 7.73798919875330722 \cdot 10^{-19}\right):\\
\;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan 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 temp;
	if (((eps <= -3.9980652276674155e-18) || !(eps <= 7.737989198753307e-19))) {
		temp = ((((tan(x) * tan(x)) - (tan(eps) * tan(eps))) / ((1.0 - (tan(x) * tan(eps))) * (tan(x) - tan(eps)))) - tan(x));
	} else {
		temp = (((eps * x) * (x + eps)) + eps);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target14.8
Herbie15.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -3.9980652276674155e-18 or 7.737989198753307e-19 < eps

    1. Initial program 29.4

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

    3. Applied tan-sum1.0

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

    5. Applied flip-+1.1

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

    6. Applied associate-/l/1.1

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

    if -3.9980652276674155e-18 < eps < 7.737989198753307e-19

    1. Initial program 44.9

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

    2. Taylor expanded around 0 30.6

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

    3. Simplified30.4

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.99806522766741552 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 7.73798919875330722 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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)))