Average Error: 37.3 → 0.6
Time: 13.5s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.110958230728241 \cdot 10^{-05}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.767513643243207 \cdot 10^{-20}:\\ \;\;\;\;\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1.3333333333333333 \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot {\varepsilon}^{3}\right)\right) + \left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left(e^{\tan x}\right)}^{\tan \varepsilon}\right)} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -4.110958230728241 \cdot 10^{-05}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 8.767513643243207 \cdot 10^{-20}:\\
\;\;\;\;\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1.3333333333333333 \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot {\varepsilon}^{3}\right)\right) + \left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\

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

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.110958230728241e-05)
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (if (<= eps 8.767513643243207e-20)
     (+
      (+
       (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
       (*
        1.3333333333333333
        (* (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) (pow eps 3.0))))
      (+
       (+
        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)) (- 1.0 (log (pow (exp (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 <= -4.110958230728241e-05) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 8.767513643243207e-20) {
		tmp = ((eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0))) + (1.3333333333333333 * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) * pow(eps, 3.0)))) + ((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)) / (1.0 - log(pow(exp(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

Original37.3
Target15.4
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -4.11095823072824123e-5

    1. Initial program 29.8

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

    3. Applied tan-sum_binary64_22590.4

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

    if -4.11095823072824123e-5 < eps < 8.76751364324320673e-20

    1. Initial program 45.6

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

    3. Applied tan-sum_binary64_225945.3

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

    4. Taylor expanded around 0 0.2

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

    5. Simplified0.2

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

    if 8.76751364324320673e-20 < eps

    1. Initial program 30.0

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

    3. Applied tan-sum_binary64_22591.3

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

    5. Applied add-log-exp_binary64_21631.5

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

    6. Simplified1.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.110958230728241 \cdot 10^{-05}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.767513643243207 \cdot 10^{-20}:\\ \;\;\;\;\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1.3333333333333333 \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot {\varepsilon}^{3}\right)\right) + \left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left(e^{\tan x}\right)}^{\tan \varepsilon}\right)} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(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)))