2tan (problem 3.3.2)

Percentage Accurate: 42.6% → 98.3%

Time: 19.0s

Precision: binary64

Cost: 248456

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := {\sin x}^{2}\\ t_2 := {\cos x}^{2}\\ t_3 := 1 - \tan \varepsilon \cdot \tan x\\ t_4 := \frac{\tan \varepsilon}{t_3}\\ t_5 := t_0 \cdot -0.3333333333333333\\ t_6 := \frac{\sin x}{\frac{\cos x}{t_5}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\\ \mathbf{if}\;\varepsilon \leq -1.2 \cdot 10^{+15}:\\ \;\;\;\;t_4 - \left(\tan x - \frac{t_0}{1 - t_0 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_4 + \left(\mathsf{fma}\left(-{\varepsilon}^{4}, \frac{t_1}{\frac{t_2}{t_5}} + \frac{\sin x}{\frac{\cos x}{t_6}}, \frac{\varepsilon \cdot t_1}{t_2}\right) + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - t_6 \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_3}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (pow (cos x) 2.0))
        (t_3 (- 1.0 (* (tan eps) (tan x))))
        (t_4 (/ (tan eps) t_3))
        (t_5 (* t_0 -0.3333333333333333))
        (t_6
         (-
          (/ (sin x) (/ (cos x) t_5))
          (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))
   (if (<= eps -1.2e+15)
     (- t_4 (- (tan x) (/ t_0 (- 1.0 (* t_0 (/ (sin eps) (cos eps)))))))
     (if (<= eps 1.6e-27)
       (+
        t_4
        (+
         (fma
          (- (pow eps 4.0))
          (+ (/ t_1 (/ t_2 t_5)) (/ (sin x) (/ (cos x) t_6)))
          (/ (* eps t_1) t_2))
         (-
          (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))
          (* t_6 (pow eps 3.0)))))
       (- (/ 1.0 (/ t_3 (+ (tan eps) (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 = sin(x) / cos(x);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = pow(cos(x), 2.0);
	double t_3 = 1.0 - (tan(eps) * tan(x));
	double t_4 = tan(eps) / t_3;
	double t_5 = t_0 * -0.3333333333333333;
	double t_6 = (sin(x) / (cos(x) / t_5)) - (pow(sin(x), 4.0) / pow(cos(x), 4.0));
	double tmp;
	if (eps <= -1.2e+15) {
		tmp = t_4 - (tan(x) - (t_0 / (1.0 - (t_0 * (sin(eps) / cos(eps))))));
	} else if (eps <= 1.6e-27) {
		tmp = t_4 + (fma(-pow(eps, 4.0), ((t_1 / (t_2 / t_5)) + (sin(x) / (cos(x) / t_6))), ((eps * t_1) / t_2)) + (((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))) - (t_6 * pow(eps, 3.0))));
	} else {
		tmp = (1.0 / (t_3 / (tan(eps) + tan(x)))) - 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(sin(x) / cos(x))
	t_1 = sin(x) ^ 2.0
	t_2 = cos(x) ^ 2.0
	t_3 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_4 = Float64(tan(eps) / t_3)
	t_5 = Float64(t_0 * -0.3333333333333333)
	t_6 = Float64(Float64(sin(x) / Float64(cos(x) / t_5)) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))
	tmp = 0.0
	if (eps <= -1.2e+15)
		tmp = Float64(t_4 - Float64(tan(x) - Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(eps) / cos(eps)))))));
	elseif (eps <= 1.6e-27)
		tmp = Float64(t_4 + Float64(fma(Float64(-(eps ^ 4.0)), Float64(Float64(t_1 / Float64(t_2 / t_5)) + Float64(sin(x) / Float64(cos(x) / t_6))), Float64(Float64(eps * t_1) / t_2)) + Float64(Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) - Float64(t_6 * (eps ^ 3.0)))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_3 / Float64(tan(eps) + tan(x)))) - tan(x));
	end
	return 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[eps], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * -0.3333333333333333), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.2e+15], N[(t$95$4 - N[(N[Tan[x], $MachinePrecision] - N[(t$95$0 / N[(1.0 - N[(t$95$0 * N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.6e-27], N[(t$95$4 + N[(N[((-N[Power[eps, 4.0], $MachinePrecision]) * N[(N[(t$95$1 / N[(t$95$2 / t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] / N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$6 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$3 / N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Target

Original	42.6%
Target	76.6%
Herbie	98.3%

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.2e15

   1. Initial program 49.3%

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

   2. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
      Step-by-step derivation

      [Start] 49.3%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.5%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.5%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.5%	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Simplified 99.5%

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

      [Start] 99.5%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      fma-neg [<=] 99.5%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
      associate-*r/ [=>] 99.5%	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 99.5%	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Taylor expanded in x around inf 99.1%

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

   5. Simplified 99.2%

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

      [Start] 99.1%	\[ \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x} \]
      associate--l+ [=>] 99.2%	\[ \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right)} \]

   6. Applied egg-rr 82.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)} - 1\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      Step-by-step derivation

      [Start] 99.2%	\[ \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-log1p-u [=>] 82.9%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right)\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-udef [=>] 82.7%	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right)} - 1\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      associate-/r* [=>] 82.6%	\[ \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      tan-quot [<=] 82.7%	\[ \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\tan \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      tan-quot [<=] 82.7%	\[ \left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      tan-quot [<=] 82.7%	\[ \left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]

   7. Simplified 99.6%

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

      [Start] 82.7%	\[ \left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-def [=>] 83.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-log1p [=>] 99.6%	\[ \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]

   8. Applied egg-rr 88.0%

      \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan x\right)} - 1\right)}\right) \]
      Step-by-step derivation

      [Start] 99.6%	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      tan-quot [<=] 99.6%	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \color{blue}{\tan x}\right) \]
      expm1-log1p-u [=>] 88.1%	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x\right)\right)}\right) \]
      expm1-udef [=>] 88.0%	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan x\right)} - 1\right)}\right) \]

   9. Simplified 99.6%

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

      [Start] 88.0%	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \left(e^{\mathsf{log1p}\left(\tan x\right)} - 1\right)\right) \]
      expm1-def [=>] 88.1%	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x\right)\right)}\right) \]
      expm1-log1p [=>] 99.6%	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \color{blue}{\tan x}\right) \]

   if -1.2e15 < eps < 1.59999999999999995e-27

   1. Initial program 25.9%

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

   2. Applied egg-rr 27.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
      Step-by-step derivation

      [Start] 25.9%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 27.1%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 27.1%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 27.1%	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Simplified 27.1%

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

      [Start] 27.1%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      fma-neg [<=] 27.1%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
      associate-*r/ [=>] 27.1%	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 27.1%	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Taylor expanded in x around inf 27.1%

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

   5. Simplified 59.6%

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

      [Start] 27.1%	\[ \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x} \]
      associate--l+ [=>] 59.6%	\[ \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right)} \]

   6. Applied egg-rr 8.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)} - 1\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      Step-by-step derivation

      [Start] 59.6%	\[ \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-log1p-u [=>] 59.6%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right)\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-udef [=>] 8.4%	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}\right)} - 1\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      associate-/r* [=>] 8.4%	\[ \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      tan-quot [<=] 8.4%	\[ \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\tan \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      tan-quot [<=] 8.4%	\[ \left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \color{blue}{\tan x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      tan-quot [<=] 8.4%	\[ \left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]

   7. Simplified 59.6%

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

      [Start] 8.4%	\[ \left(e^{\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)} - 1\right) + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-def [=>] 59.6%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right)\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]
      expm1-log1p [=>] 59.6%	\[ \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) \]

   8. Taylor expanded in eps around 0 99.7%

      \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(\frac{{\sin x}^{2} \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{{\cos x}^{2}} + \frac{\sin x \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)}{\cos x}\right)\right) + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right)\right)} \]

   9. Simplified 99.7%

      \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\mathsf{fma}\left(-{\varepsilon}^{4}, \frac{{\sin x}^{2}}{\frac{{\cos x}^{2}}{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}} + \frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}}}, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - {\varepsilon}^{3} \cdot \left(\frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 99.7%

      \[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(\frac{{\sin x}^{2} \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{{\cos x}^{2}} + \frac{\sin x \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)}{\cos x}\right)\right) + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right)\right) \]

      associate-+r+ [=>] 99.7%

      \[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(\frac{{\sin x}^{2} \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{{\cos x}^{2}} + \frac{\sin x \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)}{\cos x}\right)\right) + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right)} \]

   if 1.59999999999999995e-27 < eps

   1. Initial program 45.0%

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

   2. Applied egg-rr 99.3%

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

      [Start] 45.0%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.2%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      clear-num [=>] 99.3%	\[ \color{blue}{\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}} - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} - \left(\tan x - \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\mathsf{fma}\left(-{\varepsilon}^{4}, \frac{{\sin x}^{2}}{\frac{{\cos x}^{2}}{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}} + \frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}}}, \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - \left(\frac{\sin x}{\frac{\cos x}{\frac{\sin x}{\cos x} \cdot -0.3333333333333333}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	248456

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := {\sin x}^{2}\\ t_2 := {\cos x}^{2}\\ t_3 := 1 - \tan \varepsilon \cdot \tan x\\ t_4 := \frac{\tan \varepsilon}{t_3}\\ t_5 := t_0 \cdot -0.3333333333333333\\ t_6 := \frac{\sin x}{\frac{\cos x}{t_5}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\\ \mathbf{if}\;\varepsilon \leq -1.2 \cdot 10^{+15}:\\ \;\;\;\;t_4 - \left(\tan x - \frac{t_0}{1 - t_0 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_4 + \left(\mathsf{fma}\left(-{\varepsilon}^{4}, \frac{t_1}{\frac{t_2}{t_5}} + \frac{\sin x}{\frac{\cos x}{t_6}}, \frac{\varepsilon \cdot t_1}{t_2}\right) + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - t_6 \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_3}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	131080

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ t_2 := \frac{\tan \varepsilon}{t_1}\\ \mathbf{if}\;\varepsilon \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;t_2 + \left(\frac{t_0}{1 - t_0 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_2 + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - \left(\frac{\sin x}{\frac{\cos x}{t_0 \cdot -0.3333333333333333}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 3
Accuracy	99.0%
Cost	72264

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -3.35 \cdot 10^{-6}:\\ \;\;\;\;t_1 + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_1 + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 4
Accuracy	98.9%
Cost	72264

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ t_2 := \frac{\tan \varepsilon}{t_1}\\ \mathbf{if}\;\varepsilon \leq -2.85 \cdot 10^{-6}:\\ \;\;\;\;t_2 + \left(\frac{t_0}{1 - t_0 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_2 + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	65736

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	46024

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;t_1 + \tan x \cdot \left(\frac{1}{t_0} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_1 + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	46024

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;t_1 + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;t_1 + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 8
Accuracy	98.9%
Cost	39748

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \tan x \cdot \left(\frac{1}{t_0} + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 9
Accuracy	98.9%
Cost	39300

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -2.85 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_0}} - \tan x\\ \end{array} \]

Alternative 10
Accuracy	98.9%
Cost	33096

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_0}} - \tan x\\ \end{array} \]

Alternative 11
Accuracy	98.9%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.6 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 12
Accuracy	77.0%
Cost	26696

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) \cdot \frac{1}{\cos \varepsilon - x \cdot \sin \varepsilon} - \tan x\\ \end{array} \]

Alternative 13
Accuracy	77.2%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00011:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 14
Accuracy	58.3%
Cost	6464

\[\tan \varepsilon \]

Alternative 15
Accuracy	30.6%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023165 
(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)))