Average Error: 36.6 → 15.3
Time: 7.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.17742371337321258 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 8.3497052348174857 \cdot 10^{-111}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \cos x - \left(\tan x - \tan \varepsilon\right) \cdot \sin x}{\left(\tan x - \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.17742371337321258 \cdot 10^{-51}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\

\mathbf{elif}\;\varepsilon \le 8.3497052348174857 \cdot 10^{-111}:\\
\;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\

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

\end{array}
double code(double x, double eps) {
	return ((double) (((double) tan(((double) (x + eps)))) - ((double) tan(x))));
}
↓
double code(double x, double eps) {
	double VAR;
	if ((eps <= -5.1774237133732126e-51)) {
		VAR = ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) cbrt(((double) pow(((double) (((double) tan(x)) * ((double) tan(eps)))), 3.0)))))))) - ((double) tan(x))));
	} else {
		double VAR_1;
		if ((eps <= 8.349705234817486e-111)) {
			VAR_1 = ((double) (((double) (((double) (eps * x)) * ((double) (x + eps)))) + eps));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (((double) (((double) tan(x)) * ((double) tan(x)))) - ((double) (((double) tan(eps)) * ((double) tan(eps)))))) * ((double) (1.0 / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))))))) * ((double) cos(x)))) - ((double) (((double) (((double) tan(x)) - ((double) tan(eps)))) * ((double) sin(x)))))) / ((double) (((double) (((double) tan(x)) - ((double) tan(eps)))) * ((double) cos(x))))));
		}
		VAR = VAR_1;
	}
	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

Original36.6
Target15.2
Herbie15.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -5.1774237133732126e-51

    1. Initial program 29.1

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

    3. Applied tan-sum3.4

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

    5. Applied add-cbrt-cube3.4

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

    6. Applied add-cbrt-cube3.5

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

    7. Applied cbrt-unprod3.4

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

    8. Simplified3.4

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

    if -5.1774237133732126e-51 < eps < 8.349705234817486e-111

    1. Initial program 47.5

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

    2. Taylor expanded around 0 31.0

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

    3. Simplified30.8

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

    if 8.349705234817486e-111 < eps

    1. Initial program 31.7

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

    3. Applied tan-sum9.5

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

    5. Applied div-inv9.5

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

    7. Applied tan-quot9.5

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

    8. Applied flip-+9.6

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

    9. Applied associate-*l/9.6

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

    10. Applied frac-sub9.6

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.17742371337321258 \cdot 10^{-51}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 8.3497052348174857 \cdot 10^{-111}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \cos x - \left(\tan x - \tan \varepsilon\right) \cdot \sin x}{\left(\tan x - \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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)))