2tan (problem 3.3.2)

Average Error: 36.7 → 0.3

Time: 22.2s

Precision: binary64

Cost: 65472

Math

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

↓

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

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

↓

(FPCore (x eps)
 :precision binary64
 (-
  (/
   (/ (sin eps) (cos eps))
   (- 1.0 (* (sin eps) (/ (sin x) (* (cos eps) (cos x))))))
  (/ (tan eps) (/ (+ (tan eps) (/ -1.0 (tan x))) (tan x)))))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

↓

double code(double x, double eps) {
	return ((sin(eps) / cos(eps)) / (1.0 - (sin(eps) * (sin(x) / (cos(eps) * cos(x)))))) - (tan(eps) / ((tan(eps) + (-1.0 / tan(x))) / tan(x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((sin(eps) / cos(eps)) / (1.0d0 - (sin(eps) * (sin(x) / (cos(eps) * cos(x)))))) - (tan(eps) / ((tan(eps) + ((-1.0d0) / tan(x))) / tan(x)))
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

↓

public static double code(double x, double eps) {
	return ((Math.sin(eps) / Math.cos(eps)) / (1.0 - (Math.sin(eps) * (Math.sin(x) / (Math.cos(eps) * Math.cos(x)))))) - (Math.tan(eps) / ((Math.tan(eps) + (-1.0 / Math.tan(x))) / Math.tan(x)));
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

↓

def code(x, eps):
	return ((math.sin(eps) / math.cos(eps)) / (1.0 - (math.sin(eps) * (math.sin(x) / (math.cos(eps) * math.cos(x)))))) - (math.tan(eps) / ((math.tan(eps) + (-1.0 / math.tan(x))) / math.tan(x)))

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

↓

function code(x, eps)
	return Float64(Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - Float64(sin(eps) * Float64(sin(x) / Float64(cos(eps) * cos(x)))))) - Float64(tan(eps) / Float64(Float64(tan(eps) + Float64(-1.0 / tan(x))) / tan(x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

↓

function tmp = code(x, eps)
	tmp = ((sin(eps) / cos(eps)) / (1.0 - (sin(eps) * (sin(x) / (cos(eps) * cos(x)))))) - (tan(eps) / ((tan(eps) + (-1.0 / tan(x))) / tan(x)));
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := N[(N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[eps], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Tan[eps], $MachinePrecision] / N[(N[(N[Tan[eps], $MachinePrecision] + N[(-1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	36.7
Target	14.8
Herbie	0.3

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

1. Initial program 36.7

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

2. Applied egg-rr 21.8

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

3. Simplified 21.8

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

   [Start] 21.8	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
   associate-*r/ [=>] 21.8	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
   *-rgt-identity [=>] 21.8	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

4. Taylor expanded in x around inf 21.9

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

5. Simplified 12.9

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

   [Start] 21.9	\[ \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x} \]
   associate--l+ [=>] 12.9	\[ \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right)} \]

6. Applied egg-rr 12.9

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

7. Simplified 0.3

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

   [Start] 12.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\frac{1}{\tan x} - \frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot 1}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   *-rgt-identity [=>] 12.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\frac{1}{\tan x} - \color{blue}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x}}}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   div-sub [=>] 12.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\frac{1}{\tan x} - \color{blue}{\left(\frac{1}{\tan x} - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x}\right)}}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   associate--r- [=>] 0.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\color{blue}{\left(\frac{1}{\tan x} - \frac{1}{\tan x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x}}}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   associate-/l* [=>] 0.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\left(\frac{1}{\tan x} - \frac{1}{\tan x}\right) + \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\frac{\tan x}{\tan x}}}}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   *-rgt-identity [<=] 0.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\left(\frac{1}{\tan x} - \frac{1}{\tan x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\frac{\color{blue}{\tan x \cdot 1}}{\tan x}}}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   associate-*r/ [<=] 0.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\left(\frac{1}{\tan x} - \frac{1}{\tan x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\color{blue}{\tan x \cdot \frac{1}{\tan x}}}}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   rgt-mult-inverse [=>] 0.4	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\left(\frac{1}{\tan x} - \frac{1}{\tan x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\color{blue}{1}}}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot \frac{1}{\tan x}} \]
   associate-*r/ [=>] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\left(\frac{1}{\tan x} - \frac{1}{\tan x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1}}{\color{blue}{\frac{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x} \cdot 1}{\tan x}}} \]
   *-rgt-identity [=>] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\left(\frac{1}{\tan x} - \frac{1}{\tan x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1}}{\frac{\color{blue}{\frac{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}{\tan x}}}{\tan x}} \]

8. Applied egg-rr 0.3

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

9. Simplified 0.3

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

   [Start] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \left(-\tan \varepsilon\right) \cdot \frac{1}{\frac{\frac{-1}{\tan x} + \tan \varepsilon}{\tan x}} \]
   associate-*r/ [=>] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \color{blue}{\frac{\left(-\tan \varepsilon\right) \cdot 1}{\frac{\frac{-1}{\tan x} + \tan \varepsilon}{\tan x}}} \]
   associate-/l* [=>] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \color{blue}{\frac{-\tan \varepsilon}{\frac{\frac{\frac{-1}{\tan x} + \tan \varepsilon}{\tan x}}{1}}} \]
   associate-/l/ [=>] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{-\tan \varepsilon}{\color{blue}{\frac{\frac{-1}{\tan x} + \tan \varepsilon}{1 \cdot \tan x}}} \]
   *-lft-identity [=>] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{-\tan \varepsilon}{\frac{\frac{-1}{\tan x} + \tan \varepsilon}{\color{blue}{\tan x}}} \]
   +-commutative [=>] 0.3	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{-\tan \varepsilon}{\frac{\color{blue}{\tan \varepsilon + \frac{-1}{\tan x}}}{\tan x}} \]

10. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	65608

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{-1 + \tan \varepsilon \cdot \tan x}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	65472

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

Alternative 3
Error	0.3
Cost	65472

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

Alternative 4
Error	0.4
Cost	39304

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 5
Error	0.4
Cost	39304

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{-1 + \tan \varepsilon \cdot \tan x}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	33224

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -3.65 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t_0} \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)} - \tan x\\ \end{array} \]

Alternative 7
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.04 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 8
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 9
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 10
Error	14.1
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11
Error	14.1
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12
Error	14.1
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 13
Error	26.6
Cost	6464

\[\tan \varepsilon \]

Alternative 14
Error	43.9
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023059 
(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)))