Average Error: 36.8 → 15.1
Time: 12.6s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.944935459056774 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{\tan x + \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x \cdot \tan x}{\tan x + \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}}\\ \mathbf{elif}\;\varepsilon \leq 7.958520989490561 \cdot 10^{-111}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -8.944935459056774 \cdot 10^{-23}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{\tan x + \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x \cdot \tan x}{\tan x + \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}}\\

\mathbf{elif}\;\varepsilon \leq 7.958520989490561 \cdot 10^{-111}:\\
\;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} - \tan 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 <= -8.944935459056774e-23)) {
		VAR = (((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps))))))))))) - ((double) (((double) tan(x)) * ((double) tan(x)))))) / ((double) (((double) tan(x)) + (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps))))))))));
	} else {
		double VAR_1;
		if ((eps <= 7.958520989490561e-111)) {
			VAR_1 = ((double) (eps + ((double) (x * ((double) (eps * ((double) (eps + x))))))));
		} else {
			VAR_1 = ((double) ((((double) (((double) (((double) sin(x)) * ((double) cos(eps)))) + ((double) (((double) cos(x)) * ((double) sin(eps)))))) / ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))) * ((double) (((double) cos(eps)) * ((double) cos(x))))))) - ((double) tan(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

Original36.8
Target15.5
Herbie15.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -8.9449354590567743e-23

    1. Initial program 31.3

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

    3. Applied tan-sum1.6

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

    5. Applied flip--1.7

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

    6. Simplified1.8

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

    7. Simplified1.8

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

    if -8.9449354590567743e-23 < eps < 7.9585209894905614e-111

    1. Initial program 46.1

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

    2. Taylor expanded around 0 30.4

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

    3. Simplified30.2

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

    if 7.9585209894905614e-111 < eps

    1. Initial program 30.6

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

    3. Applied tan-sum8.2

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

    5. Applied tan-quot8.3

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

    6. Applied tan-quot8.3

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

    7. Applied frac-add8.3

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

    8. Applied associate-/l/8.3

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.944935459056774 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \frac{\tan x + \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x \cdot \tan x}{\tan x + \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}}\\ \mathbf{elif}\;\varepsilon \leq 7.958520989490561 \cdot 10^{-111}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))