2tan (problem 3.3.2)

Average Error: 37.2 → 14.4

Time: 45.0s

Precision: binary64

Cost: 157384

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_2 := t_1 - -1\\ \mathbf{if}\;\varepsilon \leq -0.0092:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \sin \varepsilon \cdot \left(-x\right)} - t_0\\ \mathbf{elif}\;\varepsilon \leq 7400:\\ \;\;\;\;\varepsilon \cdot t_2 + \left(t_0 \cdot \left(t_2 \cdot {\varepsilon}^{2}\right) + {\varepsilon}^{3} \cdot \left(-\left(0.16666666666666666 \cdot t_1 + \left(0.16666666666666666 + t_2 \cdot \left(-0.5 + \left(-t_1\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_2 (- t_1 -1.0)))
   (if (<= eps -0.0092)
     (- (/ (sin (+ eps x)) (+ (cos eps) (* (sin eps) (- x)))) t_0)
     (if (<= eps 7400.0)
       (+
        (* eps t_2)
        (+
         (* t_0 (* t_2 (pow eps 2.0)))
         (*
          (pow eps 3.0)
          (-
           (+
            (* 0.16666666666666666 t_1)
            (+ 0.16666666666666666 (* t_2 (+ -0.5 (- t_1)))))))))
       (/ (sin eps) (cos eps))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_2 = t_1 - -1.0;
	double tmp;
	if (eps <= -0.0092) {
		tmp = (sin((eps + x)) / (cos(eps) + (sin(eps) * -x))) - t_0;
	} else if (eps <= 7400.0) {
		tmp = (eps * t_2) + ((t_0 * (t_2 * pow(eps, 2.0))) + (pow(eps, 3.0) * -((0.16666666666666666 * t_1) + (0.16666666666666666 + (t_2 * (-0.5 + -t_1))))));
	} else {
		tmp = sin(eps) / cos(eps);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = sin(x) / cos(x)
    t_1 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    t_2 = t_1 - (-1.0d0)
    if (eps <= (-0.0092d0)) then
        tmp = (sin((eps + x)) / (cos(eps) + (sin(eps) * -x))) - t_0
    else if (eps <= 7400.0d0) then
        tmp = (eps * t_2) + ((t_0 * (t_2 * (eps ** 2.0d0))) + ((eps ** 3.0d0) * -((0.16666666666666666d0 * t_1) + (0.16666666666666666d0 + (t_2 * ((-0.5d0) + -t_1))))))
    else
        tmp = sin(eps) / cos(eps)
    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.sin(x) / Math.cos(x);
	double t_1 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	double t_2 = t_1 - -1.0;
	double tmp;
	if (eps <= -0.0092) {
		tmp = (Math.sin((eps + x)) / (Math.cos(eps) + (Math.sin(eps) * -x))) - t_0;
	} else if (eps <= 7400.0) {
		tmp = (eps * t_2) + ((t_0 * (t_2 * Math.pow(eps, 2.0))) + (Math.pow(eps, 3.0) * -((0.16666666666666666 * t_1) + (0.16666666666666666 + (t_2 * (-0.5 + -t_1))))));
	} else {
		tmp = Math.sin(eps) / Math.cos(eps);
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = math.sin(x) / math.cos(x)
	t_1 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	t_2 = t_1 - -1.0
	tmp = 0
	if eps <= -0.0092:
		tmp = (math.sin((eps + x)) / (math.cos(eps) + (math.sin(eps) * -x))) - t_0
	elif eps <= 7400.0:
		tmp = (eps * t_2) + ((t_0 * (t_2 * math.pow(eps, 2.0))) + (math.pow(eps, 3.0) * -((0.16666666666666666 * t_1) + (0.16666666666666666 + (t_2 * (-0.5 + -t_1))))))
	else:
		tmp = math.sin(eps) / math.cos(eps)
	return tmp

Julia

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

↓

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	t_1 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_2 = Float64(t_1 - -1.0)
	tmp = 0.0
	if (eps <= -0.0092)
		tmp = Float64(Float64(sin(Float64(eps + x)) / Float64(cos(eps) + Float64(sin(eps) * Float64(-x)))) - t_0);
	elseif (eps <= 7400.0)
		tmp = Float64(Float64(eps * t_2) + Float64(Float64(t_0 * Float64(t_2 * (eps ^ 2.0))) + Float64((eps ^ 3.0) * Float64(-Float64(Float64(0.16666666666666666 * t_1) + Float64(0.16666666666666666 + Float64(t_2 * Float64(-0.5 + Float64(-t_1)))))))));
	else
		tmp = Float64(sin(eps) / cos(eps));
	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 = sin(x) / cos(x);
	t_1 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	t_2 = t_1 - -1.0;
	tmp = 0.0;
	if (eps <= -0.0092)
		tmp = (sin((eps + x)) / (cos(eps) + (sin(eps) * -x))) - t_0;
	elseif (eps <= 7400.0)
		tmp = (eps * t_2) + ((t_0 * (t_2 * (eps ^ 2.0))) + ((eps ^ 3.0) * -((0.16666666666666666 * t_1) + (0.16666666666666666 + (t_2 * (-0.5 + -t_1))))));
	else
		tmp = sin(eps) / cos(eps);
	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[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -1.0), $MachinePrecision]}, If[LessEqual[eps, -0.0092], N[(N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[eps, 7400.0], N[(N[(eps * t$95$2), $MachinePrecision] + N[(N[(t$95$0 * N[(t$95$2 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * (-N[(N[(0.16666666666666666 * t$95$1), $MachinePrecision] + N[(0.16666666666666666 + N[(t$95$2 * N[(-0.5 + (-t$95$1)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	37.2
Target	15.2
Herbie	14.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -0.0091999999999999998

   1. Initial program 30.1

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

   2. Taylor expanded in x around inf 30.3

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

   3. Taylor expanded in x around 0 28.4

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

   4. Simplified 28.4

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

      [Start] 28.4	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + -1 \cdot \left(x \cdot \sin \varepsilon\right)} - \frac{\sin x}{\cos x} \]
      rational.json-simplify-2 [=>] 28.4	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + -1 \cdot \color{blue}{\left(\sin \varepsilon \cdot x\right)}} - \frac{\sin x}{\cos x} \]
      rational.json-simplify-43 [=>] 28.4	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \color{blue}{\sin \varepsilon \cdot \left(x \cdot -1\right)}} - \frac{\sin x}{\cos x} \]
      rational.json-simplify-8 [<=] 28.4	\[ \frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \sin \varepsilon \cdot \color{blue}{\left(-x\right)}} - \frac{\sin x}{\cos x} \]

   if -0.0091999999999999998 < eps < 7400

   1. Initial program 44.5

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

   2. Taylor expanded in eps around 0 0.9

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right)} \]

   3. Simplified 0.9

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) + \left(\frac{\sin x}{\cos x} \cdot \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot {\varepsilon}^{2}\right) + {\varepsilon}^{3} \cdot \left(-\left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(-0.5 + \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right)} \]
      Proof

      [Start] 0.9	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \]
      rational.json-simplify-2 [=>] 0.9	\[ \varepsilon \cdot \left(1 - \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -1}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \]
      rational.json-simplify-9 [=>] 0.9	\[ \varepsilon \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \]
      rational.json-simplify-12 [=>] 0.9	\[ \varepsilon \cdot \left(1 - \color{blue}{\left(0 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \]
      rational.json-simplify-45 [=>] 0.9	\[ \varepsilon \cdot \color{blue}{\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(0 - 1\right)\right)} + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \]
      metadata-eval [=>] 0.9	\[ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \color{blue}{-1}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \]

   if 7400 < eps

   1. Initial program 29.4

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

   2. Taylor expanded in x around 0 31.5

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

   3. Taylor expanded in x around 0 27.7

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0092:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \sin \varepsilon \cdot \left(-x\right)} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 7400:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) + \left(\frac{\sin x}{\cos x} \cdot \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot {\varepsilon}^{2}\right) + {\varepsilon}^{3} \cdot \left(-\left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right) \cdot \left(-0.5 + \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.4
Cost	105544

\[\begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := 1 - \sin x \cdot \frac{-\sin x}{t_0}\\ \mathbf{if}\;\varepsilon \leq -0.0155:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \sin \varepsilon \cdot \left(-x\right)} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 7400:\\ \;\;\;\;t_1 \cdot \left(\varepsilon + \left(-{\varepsilon}^{2} \cdot \frac{\sin x}{-\cos x}\right)\right) + \left(0.16666666666666666 + t_1 \cdot \left(-0.5 + \left(-\frac{{\sin x}^{2}}{t_0}\right)\right)\right) \cdot \left(-{\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 2
Error	14.5
Cost	72200

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\\ \mathbf{if}\;\varepsilon \leq -0.017:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \sin \varepsilon \cdot \left(-x\right)} - t_0\\ \mathbf{elif}\;\varepsilon \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot t_1 + t_0 \cdot \left(t_1 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 3
Error	14.5
Cost	46344

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00021:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \sin \varepsilon \cdot \left(-x\right)} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - \sin x \cdot \frac{-\sin x}{{\cos x}^{2}}\right) \cdot \left(\varepsilon + \left(-{\varepsilon}^{2} \cdot \frac{\sin x}{-\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 4
Error	14.6
Cost	33028

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000112:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon + \sin \varepsilon \cdot \left(-x\right)} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{{\cos x}^{2}} \cdot \frac{1}{\frac{1}{{\sin x}^{2}}} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 5
Error	14.6
Cost	26824

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{{\cos x}^{2}} \cdot \frac{1}{\frac{1}{{\sin x}^{2}}} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 6
Error	14.6
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}}\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 7
Error	27.0
Cost	12992

\[\frac{\sin \varepsilon}{\cos \varepsilon} \]

Alternative 8
Error	28.5
Cost	7112

\[\begin{array}{l} t_0 := 1 + \left(\tan \left(x + \varepsilon\right) + -1\right)\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7400:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	29.6
Cost	6984

\[\begin{array}{l} t_0 := \tan \left(x + \varepsilon\right) - x\\ \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	41.6
Cost	6464

\[\sin \varepsilon \]

Alternative 11
Error	44.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023074 
(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)))