Average Error: 36.5 → 0.3
Time: 13.9s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.846200722042183 \cdot 10^{-05}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.2886471627852177 \cdot 10^{-05}:\\ \;\;\;\;\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + 1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}}\right) + \left(\varepsilon + \left({\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -5.846200722042183 \cdot 10^{-05}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 2.2886471627852177 \cdot 10^{-05}:\\
\;\;\;\;\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + 1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}}\right) + \left(\varepsilon + \left({\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\right)\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -5.846200722042183e-05)
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (/ (* (tan eps) (sin x)) (cos x))))
    (tan x))
   (if (<= eps 2.2886471627852177e-05)
     (+
      (+
       (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))
       (*
        1.3333333333333333
        (/ (* (pow (sin x) 2.0) (pow eps 3.0)) (pow (cos x) 2.0))))
      (+
       eps
       (+
        (*
         (pow eps 3.0)
         (+ (/ (pow (sin x) 4.0) (pow (cos x) 4.0)) 0.3333333333333333))
        (*
         (* eps eps)
         (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x)))))))
     (/
      (-
       (* (+ (tan x) (tan eps)) (cos x))
       (* (sin x) (- 1.0 (* (tan x) (tan eps)))))
      (* (cos x) (- 1.0 (* (tan x) (tan eps))))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -5.846200722042183e-05) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	} else if (eps <= 2.2886471627852177e-05) {
		tmp = (((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)) + (1.3333333333333333 * ((pow(sin(x), 2.0) * pow(eps, 3.0)) / pow(cos(x), 2.0)))) + (eps + ((pow(eps, 3.0) * ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) + 0.3333333333333333)) + ((eps * eps) * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))))));
	} else {
		tmp = (((tan(x) + tan(eps)) * cos(x)) - (sin(x) * (1.0 - (tan(x) * tan(eps))))) / (cos(x) * (1.0 - (tan(x) * tan(eps))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.5
Target15.2
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -5.84620072204218282e-5

    1. Initial program 29.9

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

    3. Applied tan-sum_binary64_15770.4

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

    5. Applied tan-quot_binary64_16010.4

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

    6. Applied associate-*l/_binary64_13850.4

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

    if -5.84620072204218282e-5 < eps < 2.28864716278521768e-5

    1. Initial program 44.2

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\frac{\sin x \cdot {\varepsilon}^{2}}{\cos x} + \left(1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2} \cdot \varepsilon}{{\cos x}^{2}} + \left(0.3333333333333333 \cdot {\varepsilon}^{3} + \left(\frac{{\sin x}^{3} \cdot {\varepsilon}^{2}}{{\cos x}^{3}} + \left(\varepsilon + \frac{{\sin x}^{4} \cdot {\varepsilon}^{3}}{{\cos x}^{4}}\right)\right)\right)\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + 1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}}\right) + \left(\varepsilon + \left({\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\right)}\]

    if 2.28864716278521768e-5 < eps

    1. Initial program 28.8

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

    3. Applied tan-quot_binary64_160128.7

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

    4. Applied tan-sum_binary64_15770.4

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

    5. Applied frac-sub_binary64_14510.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.846200722042183 \cdot 10^{-05}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.2886471627852177 \cdot 10^{-05}:\\ \;\;\;\;\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + 1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}}\right) + \left(\varepsilon + \left({\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021032 
(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)))