Average Error: 36.7 → 0.3
Time: 2.3min
Precision: binary64
Cost: 66114
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7473206182972273 \cdot 10^{-07}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6655339994443544 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -1.7473206182972273 \cdot 10^{-07}:\\
\;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 3.6655339994443544 \cdot 10^{-07}:\\
\;\;\;\;\varepsilon + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \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 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.7473206182972273e-07)
   (- (* (+ (tan x) (tan eps)) (/ 1.0 (- 1.0 (* (tan x) (tan eps))))) (tan x))
   (if (<= eps 3.6655339994443544e-07)
     (+
      eps
      (+
       (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))
       (*
        (* eps eps)
        (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x))))))
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan x) (/ (cos eps) (sin 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 <= -1.7473206182972273e-07) {
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	} else if (eps <= 3.6655339994443544e-07) {
		tmp = eps + (((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)) + ((eps * eps) * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x)))));
	} else {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) / (cos(eps) / sin(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

Original36.7
Target15.4
Herbie0.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Alternatives

Alternative 1
Error0.4
Cost39874
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4524649511233764 \cdot 10^{-09}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4205365137533804 \cdot 10^{-09}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array}\]
Alternative 2
Error0.4
Cost33346
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.983762250734975 \cdot 10^{-09}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.0131377431129065 \cdot 10^{-09}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]
Alternative 3
Error0.4
Cost33032
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6483423041732394 \cdot 10^{-09} \lor \neg \left(\varepsilon \leq 4.3750894905222405 \cdot 10^{-09}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array}\]
Alternative 4
Error14.8
Cost26818
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.2598987238706772 \cdot 10^{-05}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0028560446610936562:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array}\]
Alternative 5
Error26.7
Cost6464
\[\tan \varepsilon\]
Alternative 6
Error42.0
Cost1218
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4967221658712426:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 0.09455824190685944:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 7
Error42.1
Cost706
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -24.817807291206645:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 64304119.503859706:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 8
Error43.4
Cost385
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6383273519735958 \cdot 10^{+75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array}\]
Alternative 9
Error44.3
Cost64
\[\varepsilon\]
Alternative 10
Error61.3
Cost64
\[0\]

Error

Time

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.74732061829722733e-7

    1. Initial program 29.5

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    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 pow1_binary64_15030.4

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

    7. Applied div-inv_binary64_14390.5

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

    8. Simplified0.5

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

    if -1.74732061829722733e-7 < eps < 3.6655339994443544e-7

    1. Initial program 44.5

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    4. Simplified0.2

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

    if 3.6655339994443544e-7 < eps

    1. Initial program 29.7

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

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

    6. Applied associate-*r/_binary64_13840.4

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

    8. Applied associate-/l*_binary64_13870.4

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

    9. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7473206182972273 \cdot 10^{-07}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6655339994443544 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(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)))