Average Error: 37.3 → 13.8
Time: 7.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.33565384300228764 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 3.2412061887490125 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{{\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}^{3} - {\left(\tan x\right)}^{3}}{\mathsf{fma}\left(\tan x, \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x, \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.33565384300228764 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 3.2412061887490125 \cdot 10^{-51}\right):\\
\;\;\;\;\frac{{\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}^{3} - {\left(\tan x\right)}^{3}}{\mathsf{fma}\left(\tan x, \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x, \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\\

\end{array}
double code(double x, double eps) {
	return (tan((x + eps)) - tan(x));
}
↓
double code(double x, double eps) {
	double VAR;
	if (((eps <= -1.3356538430022876e-17) || !(eps <= 3.2412061887490125e-51))) {
		VAR = ((pow(((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))), 3.0) - pow(tan(x), 3.0)) / fma(tan(x), (((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) + tan(x)), (((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) * ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))))));
	} else {
		VAR = fma(x, pow(eps, 2.0), fma(0.3333333333333333, pow(eps, 3.0), 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

Original37.3
Target15.1
Herbie13.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.3356538430022876e-17 or 3.2412061887490125e-51 < eps

    1. Initial program 30.1

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    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 flip3--2.8

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

    6. Simplified2.7

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

    if -1.3356538430022876e-17 < eps < 3.2412061887490125e-51

    1. Initial program 46.2

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

    3. Applied tan-sum46.2

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

    4. Taylor expanded around inf 46.4

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

    5. Taylor expanded around 0 27.4

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

    6. Simplified27.4

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.33565384300228764 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 3.2412061887490125 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{{\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}^{3} - {\left(\tan x\right)}^{3}}{\mathsf{fma}\left(\tan x, \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x, \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(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)))