Average Error: 36.6 → 0.3
Time: 17.7s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015670222871752272:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.0002144194778667246:\\ \;\;\;\;1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.00015670222871752272:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 0.0002144194778667246:\\
\;\;\;\;1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right)\right)\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00015670222871752272)
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (if (<= eps 0.0002144194778667246)
     (+
      (*
       1.3333333333333333
       (/ (* (pow (sin x) 2.0) (pow eps 3.0)) (pow (cos x) 2.0)))
      (+
       (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))
       (+
        (/ (* (pow (sin x) 5.0) (pow eps 4.0)) (pow (cos x) 5.0))
        (+
         (*
          1.6666666666666667
          (/ (* (pow eps 4.0) (pow (sin x) 3.0)) (pow (cos x) 3.0)))
         (+
          (+
           (+
            eps
            (*
             (pow eps 3.0)
             (+ (/ (pow (sin x) 4.0) (pow (cos x) 4.0)) 0.3333333333333333)))
           (* 0.6666666666666666 (* (pow eps 4.0) (/ (sin x) (cos x)))))
          (*
           (* eps eps)
           (+
            (/ (sin x) (cos x))
            (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))))))
     (-
      (* (+ (tan x) (tan eps)) (/ 1.0 (- 1.0 (* (tan x) (tan eps)))))
      (tan x)))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00015670222871752272) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 0.0002144194778667246) {
		tmp = (1.3333333333333333 * ((pow(sin(x), 2.0) * pow(eps, 3.0)) / pow(cos(x), 2.0))) + (((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)) + (((pow(sin(x), 5.0) * pow(eps, 4.0)) / pow(cos(x), 5.0)) + ((1.6666666666666667 * ((pow(eps, 4.0) * pow(sin(x), 3.0)) / pow(cos(x), 3.0))) + (((eps + (pow(eps, 3.0) * ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) + 0.3333333333333333))) + (0.6666666666666666 * (pow(eps, 4.0) * (sin(x) / cos(x))))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))))))));
	} else {
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	}
	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

Original36.6
Target14.8
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.5670222871752272e-4

    1. Initial program 28.1

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

    3. Applied tan-sum_binary64_19180.3

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

    if -1.5670222871752272e-4 < eps < 2.14419477866724606e-4

    1. Initial program 44.3

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2} \cdot \varepsilon}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\frac{\sin x \cdot {\varepsilon}^{2}}{\cos x} + \left(\frac{{\sin x}^{3} \cdot {\varepsilon}^{2}}{{\cos x}^{3}} + \left(0.3333333333333333 \cdot {\varepsilon}^{3} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + \left(\varepsilon + \frac{{\sin x}^{4} \cdot {\varepsilon}^{3}}{{\cos x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + 0.6666666666666666 \cdot \left(\frac{\sin x}{\cos x} \cdot {\varepsilon}^{4}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)}\]

    if 2.14419477866724606e-4 < eps

    1. Initial program 29.8

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

    3. Applied tan-sum_binary64_19180.3

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

    5. Applied div-inv_binary64_17800.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015670222871752272:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.0002144194778667246:\\ \;\;\;\;1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))