2tan (problem 3.3.2)

Average Error: 36.7 → 0.3

Time: 23.1s

Precision: binary64

Cost: 235080

Math

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

↓

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{3}\\ t_2 := {\sin x}^{3}\\ t_3 := 1 - \tan \varepsilon \cdot \tan x\\ t_4 := \frac{\tan \varepsilon}{t_3}\\ t_5 := \frac{{\sin x}^{4}}{{\cos x}^{4}}\\ t_6 := {\cos x}^{2}\\ t_7 := \frac{t_0}{t_6}\\ \mathbf{if}\;\varepsilon \leq -0.00037:\\ \;\;\;\;t_4 + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;t_4 + \left(\left(\frac{\varepsilon}{t_6} \cdot t_0 + \left(\frac{\varepsilon \cdot \varepsilon}{t_1} \cdot t_2 + {\varepsilon}^{3} \cdot \left(t_5 + t_7 \cdot 0.3333333333333333\right)\right)\right) - \mathsf{fma}\left(-0.3333333333333333, \frac{t_2}{t_1}, \left(-0.3333333333333333 \cdot t_7 - t_5\right) \cdot \frac{\sin x}{\cos x}\right) \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_3} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 3.0))
        (t_2 (pow (sin x) 3.0))
        (t_3 (- 1.0 (* (tan eps) (tan x))))
        (t_4 (/ (tan eps) t_3))
        (t_5 (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))
        (t_6 (pow (cos x) 2.0))
        (t_7 (/ t_0 t_6)))
   (if (<= eps -0.00037)
     (+
      t_4
      (- (/ (tan x) (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x)))
     (if (<= eps 8.2e-9)
       (+
        t_4
        (-
         (+
          (* (/ eps t_6) t_0)
          (+
           (* (/ (* eps eps) t_1) t_2)
           (* (pow eps 3.0) (+ t_5 (* t_7 0.3333333333333333)))))
         (*
          (fma
           -0.3333333333333333
           (/ t_2 t_1)
           (* (- (* -0.3333333333333333 t_7) t_5) (/ (sin x) (cos x))))
          (pow eps 4.0))))
       (- (/ (+ (tan eps) (tan x)) t_3) (tan x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 3.0);
	double t_2 = pow(sin(x), 3.0);
	double t_3 = 1.0 - (tan(eps) * tan(x));
	double t_4 = tan(eps) / t_3;
	double t_5 = pow(sin(x), 4.0) / pow(cos(x), 4.0);
	double t_6 = pow(cos(x), 2.0);
	double t_7 = t_0 / t_6;
	double tmp;
	if (eps <= -0.00037) {
		tmp = t_4 + ((tan(x) / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	} else if (eps <= 8.2e-9) {
		tmp = t_4 + ((((eps / t_6) * t_0) + ((((eps * eps) / t_1) * t_2) + (pow(eps, 3.0) * (t_5 + (t_7 * 0.3333333333333333))))) - (fma(-0.3333333333333333, (t_2 / t_1), (((-0.3333333333333333 * t_7) - t_5) * (sin(x) / cos(x)))) * pow(eps, 4.0)));
	} else {
		tmp = ((tan(eps) + tan(x)) / t_3) - tan(x);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 3.0
	t_2 = sin(x) ^ 3.0
	t_3 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	t_4 = Float64(tan(eps) / t_3)
	t_5 = Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))
	t_6 = cos(x) ^ 2.0
	t_7 = Float64(t_0 / t_6)
	tmp = 0.0
	if (eps <= -0.00037)
		tmp = Float64(t_4 + Float64(Float64(tan(x) / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x)));
	elseif (eps <= 8.2e-9)
		tmp = Float64(t_4 + Float64(Float64(Float64(Float64(eps / t_6) * t_0) + Float64(Float64(Float64(Float64(eps * eps) / t_1) * t_2) + Float64((eps ^ 3.0) * Float64(t_5 + Float64(t_7 * 0.3333333333333333))))) - Float64(fma(-0.3333333333333333, Float64(t_2 / t_1), Float64(Float64(Float64(-0.3333333333333333 * t_7) - t_5) * Float64(sin(x) / cos(x)))) * (eps ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / t_3) - 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 3.0], $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[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$0 / t$95$6), $MachinePrecision]}, If[LessEqual[eps, -0.00037], N[(t$95$4 + N[(N[(N[Tan[x], $MachinePrecision] / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 8.2e-9], N[(t$95$4 + N[(N[(N[(N[(eps / t$95$6), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[(N[(eps * eps), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(t$95$5 + N[(t$95$7 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.3333333333333333 * N[(t$95$2 / t$95$1), $MachinePrecision] + N[(N[(N[(-0.3333333333333333 * t$95$7), $MachinePrecision] - t$95$5), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Target

Original	36.7
Target	15.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -3.6999999999999999e-4

   1. Initial program 29.8

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

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

   4. Simplified 0.3

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

      [Start] 0.4	\[ \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) \]
      +-commutative [=>] 0.4	\[ \color{blue}{\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) + \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon}} \]
      associate-+l+ [=>] 0.3	\[ \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\left(-\tan x\right) + \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon}\right)} \]
      +-commutative [<=] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]
      sub-neg [<=] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)} \]

   5. Applied egg-rr 0.3

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

   if -3.6999999999999999e-4 < eps < 8.2000000000000006e-9

   1. Initial program 43.7

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

   2. Applied egg-rr 43.2

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

   3. Applied egg-rr 43.2

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

   4. Simplified 24.9

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

      [Start] 43.2	\[ \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) \]
      +-commutative [=>] 43.2	\[ \color{blue}{\left(\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right) + \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon}} \]
      associate-+l+ [=>] 24.9	\[ \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\left(-\tan x\right) + \frac{\tan x}{1 - \tan x \cdot \tan \varepsilon}\right)} \]
      +-commutative [<=] 24.9	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} + \left(-\tan x\right)\right)} \]
      sub-neg [<=] 24.9	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)} \]

   5. Taylor expanded in eps around 0 0.2

      \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(-1 \cdot \left(\left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) \cdot {\varepsilon}^{4}\right) + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)} \]

   6. Simplified 0.2

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

      [Start] 0.2

      \[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(-1 \cdot \left(\left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) \cdot {\varepsilon}^{4}\right) + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right) \]

      +-commutative [=>] 0.2

      \[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \color{blue}{\left(\left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + -1 \cdot \left(\left(-0.3333333333333333 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) \cdot {\varepsilon}^{4}\right)\right)} \]

   if 8.2000000000000006e-9 < eps

   1. Initial program 30.2

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

   2. Applied egg-rr 0.5

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

   3. Simplified 0.4

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

      [Start] 0.5	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      *-commutative [<=] 0.5	\[ \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      associate-*l/ [=>] 0.4	\[ \color{blue}{\frac{1 \cdot \left(\tan x + \tan \varepsilon\right)}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-lft-identity [=>] 0.4	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00037:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\left(\frac{\varepsilon}{{\cos x}^{2}} \cdot {\sin x}^{2} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.3333333333333333\right)\right)\right) - \mathsf{fma}\left(-0.3333333333333333, \frac{{\sin x}^{3}}{{\cos x}^{3}}, \left(-0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot {\varepsilon}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	72136

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_1, \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_0} - \tan x\\ \end{array} \]

Alternative 2
Error	0.4
Cost	52548

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;t_1 + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.3 \cdot 10^{-9}:\\ \;\;\;\;t_1 + \frac{\varepsilon \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_0} - \tan x\\ \end{array} \]

Alternative 3
Error	0.4
Cost	46020

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \frac{\tan \varepsilon}{t_0}\\ \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;t_1 + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;t_1 + \frac{\varepsilon \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{t_0} - \tan x\\ \end{array} \]

Alternative 4
Error	0.4
Cost	39944

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \frac{\varepsilon \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} - \tan x\\ \end{array} \]

Alternative 5
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 7
Error	14.8
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 8
Error	14.8
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 9
Error	36.2
Cost	13257

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00022 \lor \neg \left(x \leq 280\right):\\ \;\;\;\;\sin x - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x + \tan \left(\varepsilon + x\right)\right) - \left(x + x\right)\\ \end{array} \]

Alternative 10
Error	26.7
Cost	12992

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

Alternative 11
Error	38.2
Cost	6976

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

Alternative 12
Error	38.2
Cost	6720

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

Alternative 13
Error	61.7
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023187 
(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)))