2tan (problem 3.3.2)

Average Error: 37.0 → 0.5

Time: 22.1s

Precision: binary64

Cost: 137352

Math

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

↓

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := {\cos x}^{2}\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;\varepsilon \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - t_0} + \left(\frac{\varepsilon}{\frac{t_2}{t_3}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - {\varepsilon}^{3} \cdot \mathsf{fma}\left(-1, \frac{{\sin x}^{4}}{{\cos x}^{4}}, -0.3333333333333333 \cdot \frac{t_3}{t_2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{t_0 + -1}, -\tan x\right)\\ \end{array} \]

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 (+ (tan x) (tan eps)))
        (t_2 (pow (cos x) 2.0))
        (t_3 (pow (sin x) 2.0)))
   (if (<= eps -6.5e-5)
     (- (/ t_1 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
     (if (<= eps 6.8e-19)
       (+
        (/ (tan eps) (- 1.0 t_0))
        (+
         (/ eps (/ t_2 t_3))
         (-
          (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))
          (*
           (pow eps 3.0)
           (fma
            -1.0
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
            (* -0.3333333333333333 (/ t_3 t_2)))))))
       (fma t_1 (/ -1.0 (+ t_0 -1.0)) (- (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 = tan(x) + tan(eps);
	double t_2 = pow(cos(x), 2.0);
	double t_3 = pow(sin(x), 2.0);
	double tmp;
	if (eps <= -6.5e-5) {
		tmp = (t_1 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else if (eps <= 6.8e-19) {
		tmp = (tan(eps) / (1.0 - t_0)) + ((eps / (t_2 / t_3)) + (((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))) - (pow(eps, 3.0) * fma(-1.0, (pow(sin(x), 4.0) / pow(cos(x), 4.0)), (-0.3333333333333333 * (t_3 / t_2))))));
	} else {
		tmp = fma(t_1, (-1.0 / (t_0 + -1.0)), -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(tan(x) + tan(eps))
	t_2 = cos(x) ^ 2.0
	t_3 = sin(x) ^ 2.0
	tmp = 0.0
	if (eps <= -6.5e-5)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	elseif (eps <= 6.8e-19)
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - t_0)) + Float64(Float64(eps / Float64(t_2 / t_3)) + Float64(Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) - Float64((eps ^ 3.0) * fma(-1.0, Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)), Float64(-0.3333333333333333 * Float64(t_3 / t_2)))))));
	else
		tmp = fma(t_1, Float64(-1.0 / Float64(t_0 + -1.0)), Float64(-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[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[eps, -6.5e-5], N[(N[(t$95$1 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6.8e-19], N[(N[(N[Tan[eps], $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / N[(t$95$2 / t$95$3), $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[(N[Power[eps, 3.0], $MachinePrecision] * N[(-1.0 * N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]]]

Target

Original	37.0
Target	15.1
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if eps < -6.49999999999999943e-5

   1. Initial program 29.5

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

   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

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

   if -6.49999999999999943e-5 < eps < 6.8000000000000004e-19

   1. Initial program 44.7

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

   2. Applied egg-rr 44.3

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

   3. Applied egg-rr 25.7

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

   4. Taylor expanded in eps around 0 0.2

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

   5. Simplified 0.2

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

      [Start] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      associate-/l* [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\color{blue}{\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      associate-/l* [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\color{blue}{\frac{{\varepsilon}^{2}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      unpow2 [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\color{blue}{\varepsilon \cdot \varepsilon}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      mul-1-neg [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \color{blue}{\left(-{\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right)\right) \]
      fma-def [=>] 0.2	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \left(-{\varepsilon}^{3} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{{\sin x}^{4}}{{\cos x}^{4}}, -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right)\right) \]

   if 6.8000000000000004e-19 < eps

   1. Initial program 30.2

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

   2. Applied egg-rr 1.0

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

   3. Simplified 1.0

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

      [Start] 1.0	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 1.0	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 1.0	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{-1 + \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - {\varepsilon}^{3} \cdot \mathsf{fma}\left(-1, \frac{{\sin x}^{4}}{{\cos x}^{4}}, -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\tan x \cdot \tan \varepsilon + -1}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	130952

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := {\sin x}^{2}\\ t_3 := {\cos x}^{2}\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - t_0} + \left(t_2 \cdot \frac{\varepsilon}{t_3} + \left({\sin x}^{3} \cdot \frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + \frac{t_2}{t_3} \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{t_0 + -1}, -\tan x\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	72264

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - t_0} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{t_0 + -1}, -\tan x\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	65736

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 - t_1\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_2} + \mathsf{fma}\left(\tan x, \frac{1}{t_2}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{t_1 + -1}, t_0\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	52484

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 - t_1\\ t_3 := \frac{\tan \varepsilon}{t_2}\\ \mathbf{if}\;\varepsilon \leq -1.62 \cdot 10^{-6}:\\ \;\;\;\;t_3 + \mathsf{fma}\left(\tan x, \frac{1}{t_2}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;t_3 + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{t_1 + -1}, t_0\right)\\ \end{array} \]

Alternative 5
Error	0.5
Cost	46024

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.62 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - t_0} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{t_0 + -1}, -\tan x\right)\\ \end{array} \]

Alternative 6
Error	0.5
Cost	39432

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.65 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\tan x \cdot \tan \varepsilon + -1}, -\tan x\right)\\ \end{array} \]

Alternative 7
Error	0.5
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.66 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 8
Error	0.5
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 9
Error	14.6
Cost	27784

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333\\ t_2 := \frac{x}{t_1}\\ \mathbf{if}\;\varepsilon \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{1 - t_2} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 + \left(-0.3333333333333333 \cdot \frac{{x}^{3}}{t_1} - t_2\right)} - \tan x\\ \end{array} \]

Alternative 10
Error	14.7
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.6 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 6.8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 11
Error	14.7
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 6.8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 12
Error	26.9
Cost	6464

\[\tan \varepsilon \]

Alternative 13
Error	44.0
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023066 
(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)))