2tan (problem 3.3.2)

Average Error: 36.3 → 0.3

Time: 23.1s

Precision: binary64

Cost: 72392

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -0.000108:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.000105:\\ \;\;\;\;\tan x \cdot \frac{t_1}{1 - t_1} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\varepsilon \cdot \sin x}{\cos x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (/ (* (sin eps) (tan x)) (cos eps))))
   (if (<= eps -0.000108)
     (- (/ t_0 (- 1.0 (/ (tan eps) (/ (cos x) (sin x))))) (tan x))
     (if (<= eps 0.000105)
       (+
        (* (tan x) (/ t_1 (- 1.0 t_1)))
        (/ (/ (sin eps) (cos eps)) (- 1.0 (/ (* eps (sin x)) (cos x)))))
       (- (/ t_0 (- 1.0 (/ (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) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = (sin(eps) * tan(x)) / cos(eps);
	double tmp;
	if (eps <= -0.000108) {
		tmp = (t_0 / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	} else if (eps <= 0.000105) {
		tmp = (tan(x) * (t_1 / (1.0 - t_1))) + ((sin(eps) / cos(eps)) / (1.0 - ((eps * sin(x)) / cos(x))));
	} else {
		tmp = (t_0 / (1.0 - (tan(eps) / (1.0 / tan(x))))) - tan(x);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    t_1 = (sin(eps) * tan(x)) / cos(eps)
    if (eps <= (-0.000108d0)) then
        tmp = (t_0 / (1.0d0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x)
    else if (eps <= 0.000105d0) then
        tmp = (tan(x) * (t_1 / (1.0d0 - t_1))) + ((sin(eps) / cos(eps)) / (1.0d0 - ((eps * sin(x)) / cos(x))))
    else
        tmp = (t_0 / (1.0d0 - (tan(eps) / (1.0d0 / tan(x))))) - tan(x)
    end if
    code = tmp
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) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double t_1 = (Math.sin(eps) * Math.tan(x)) / Math.cos(eps);
	double tmp;
	if (eps <= -0.000108) {
		tmp = (t_0 / (1.0 - (Math.tan(eps) / (Math.cos(x) / Math.sin(x))))) - Math.tan(x);
	} else if (eps <= 0.000105) {
		tmp = (Math.tan(x) * (t_1 / (1.0 - t_1))) + ((Math.sin(eps) / Math.cos(eps)) / (1.0 - ((eps * Math.sin(x)) / Math.cos(x))));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(eps) / (1.0 / Math.tan(x))))) - Math.tan(x);
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	t_1 = (math.sin(eps) * math.tan(x)) / math.cos(eps)
	tmp = 0
	if eps <= -0.000108:
		tmp = (t_0 / (1.0 - (math.tan(eps) / (math.cos(x) / math.sin(x))))) - math.tan(x)
	elif eps <= 0.000105:
		tmp = (math.tan(x) * (t_1 / (1.0 - t_1))) + ((math.sin(eps) / math.cos(eps)) / (1.0 - ((eps * math.sin(x)) / math.cos(x))))
	else:
		tmp = (t_0 / (1.0 - (math.tan(eps) / (1.0 / math.tan(x))))) - math.tan(x)
	return tmp

Julia

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

↓

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(Float64(sin(eps) * tan(x)) / cos(eps))
	tmp = 0.0
	if (eps <= -0.000108)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(eps) / Float64(cos(x) / sin(x))))) - tan(x));
	elseif (eps <= 0.000105)
		tmp = Float64(Float64(tan(x) * Float64(t_1 / Float64(1.0 - t_1))) + Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - Float64(Float64(eps * sin(x)) / cos(x)))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(eps) / Float64(1.0 / tan(x))))) - tan(x));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	t_1 = (sin(eps) * tan(x)) / cos(eps);
	tmp = 0.0;
	if (eps <= -0.000108)
		tmp = (t_0 / (1.0 - (tan(eps) / (cos(x) / sin(x))))) - tan(x);
	elseif (eps <= 0.000105)
		tmp = (tan(x) * (t_1 / (1.0 - t_1))) + ((sin(eps) / cos(eps)) / (1.0 - ((eps * sin(x)) / cos(x))));
	else
		tmp = (t_0 / (1.0 - (tan(eps) / (1.0 / tan(x))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.000108], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.000105], N[(N[(N[Tan[x], $MachinePrecision] * N[(t$95$1 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	36.3
Target	15.1
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.08e-4

   1. Initial program 28.7

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

   2. Applied egg-rr 0.4

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

   3. Simplified 0.3

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

      [Start] 0.4	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.3	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.3	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 0.3

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

   5. Taylor expanded in x around inf 0.4

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

   if -1.08e-4 < eps < 1.05e-4

   1. Initial program 43.8

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

   2. Applied egg-rr 43.0

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

   3. Simplified 43.0

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

      [Start] 43.0	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 43.0	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 43.0	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Taylor expanded in x around inf 43.0

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

   5. Simplified 25.2

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

      [Start] 43.0	\[ \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x} \]
      associate--l+ [=>] 25.2	\[ \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right)} \]

   6. Applied egg-rr 27.9

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

   7. Simplified 0.2

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

      [Start] 27.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\tan x \cdot \frac{1}{\tan x} - \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x\right)}{1} \cdot \frac{\tan x}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x} \]
      associate-*r/ [=>] 27.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \color{blue}{\frac{\frac{\tan x \cdot \frac{1}{\tan x} - \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x\right)}{1} \cdot \tan x}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}} \]
      /-rgt-identity [=>] 27.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\color{blue}{\left(\tan x \cdot \frac{1}{\tan x} - \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x\right)\right)} \cdot \tan x}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x} \]
      associate-*l/ [<=] 27.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \color{blue}{\frac{\tan x \cdot \frac{1}{\tan x} - \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x\right)}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x} \cdot \tan x} \]
      *-commutative [=>] 27.9	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \color{blue}{\tan x \cdot \frac{\tan x \cdot \frac{1}{\tan x} - \left(1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x\right)}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \tan x}} \]

   8. Taylor expanded in eps around 0 0.2

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

   if 1.05e-4 < eps

   1. Initial program 29.6

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

   2. Applied egg-rr 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.4	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.4	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000108:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.000105:\\ \;\;\;\;\tan x \cdot \frac{\frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\varepsilon \cdot \sin x}{\cos x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	78528

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}\\ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} + \tan x \cdot \frac{t_0}{1 - t_0} \end{array} \]

Alternative 2
Error	0.4
Cost	65736

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \end{array} \]

Alternative 3
Error	0.5
Cost	65608

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + {\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \cos x + \sin x \cdot \left(t_1 + -1\right)}{\cos x \cdot \left(1 - t_1\right)}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	65608

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \cos x + \sin x \cdot \left(t_1 + -1\right)}{\cos x \cdot \left(1 - t_1\right)}\\ \end{array} \]

Alternative 5
Error	0.5
Cost	59208

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \cos x + \sin x \cdot \left(t_0 + -1\right)}{\cos x \cdot \left(1 - t_0\right)}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	39364

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \end{array} \]

Alternative 7
Error	0.4
Cost	39364

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \end{array} \]

Alternative 8
Error	0.4
Cost	33097

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 9
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 10
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 11
Error	14.2
Cost	26697

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00025 \lor \neg \left(\varepsilon \leq 5.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{x}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 12
Error	15.2
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1050000:\\ \;\;\;\;\frac{1}{\frac{1}{\tan \left(\varepsilon + x\right)}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.8:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 13
Error	15.1
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1050000:\\ \;\;\;\;\frac{1}{\frac{1}{\tan \left(\varepsilon + x\right)}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.8:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 14
Error	26.6
Cost	6464

\[\tan \varepsilon \]

Alternative 15
Error	44.2
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023057 
(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)))