2tan (problem 3.3.2)

Average Error: 37.4 → 0.3

Time: 22.3s

Precision: binary64

Cost: 436553

Math

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

↓

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t_0}{t_1}\\ t_3 := 1 + t_2\\ t_4 := t_3 \cdot \frac{\sin x}{\cos x}\\ t_5 := \mathsf{fma}\left(-0.5, t_3, \frac{-t_0}{\frac{t_1}{t_3}}\right) + \mathsf{fma}\left(0.16666666666666666, t_2, 0.16666666666666666\right)\\ \mathbf{if}\;\varepsilon \leq -0.000115 \lor \neg \left(\varepsilon \leq 0.00023\right):\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_3, \mathsf{fma}\left(-1, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t_4, \mathsf{fma}\left(0.16666666666666666, t_4, \frac{t_5}{\frac{\cos x}{\sin x}}\right)\right), {\varepsilon}^{4}, t_5 \cdot {\varepsilon}^{3}\right), \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x \cdot t_3\right)\right)\right)\\ \end{array} \]

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) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ 1.0 t_2))
        (t_4 (* t_3 (/ (sin x) (cos x))))
        (t_5
         (+
          (fma -0.5 t_3 (/ (- t_0) (/ t_1 t_3)))
          (fma 0.16666666666666666 t_2 0.16666666666666666))))
   (if (or (<= eps -0.000115) (not (<= eps 0.00023)))
     (- (/ (- (+ (tan x) (tan eps))) (fma (tan x) (tan eps) -1.0)) (tan x))
     (fma
      eps
      t_3
      (fma
       -1.0
       (fma
        (fma
         -0.5
         t_4
         (fma 0.16666666666666666 t_4 (/ t_5 (/ (cos x) (sin x)))))
        (pow eps 4.0)
        (* t_5 (pow eps 3.0)))
       (* (/ (* eps eps) (cos x)) (* (sin x) t_3)))))))

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), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = 1.0 + t_2;
	double t_4 = t_3 * (sin(x) / cos(x));
	double t_5 = fma(-0.5, t_3, (-t_0 / (t_1 / t_3))) + fma(0.16666666666666666, t_2, 0.16666666666666666);
	double tmp;
	if ((eps <= -0.000115) || !(eps <= 0.00023)) {
		tmp = (-(tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x);
	} else {
		tmp = fma(eps, t_3, fma(-1.0, fma(fma(-0.5, t_4, fma(0.16666666666666666, t_4, (t_5 / (cos(x) / sin(x))))), pow(eps, 4.0), (t_5 * pow(eps, 3.0))), (((eps * eps) / cos(x)) * (sin(x) * t_3))));
	}
	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) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(1.0 + t_2)
	t_4 = Float64(t_3 * Float64(sin(x) / cos(x)))
	t_5 = Float64(fma(-0.5, t_3, Float64(Float64(-t_0) / Float64(t_1 / t_3))) + fma(0.16666666666666666, t_2, 0.16666666666666666))
	tmp = 0.0
	if ((eps <= -0.000115) || !(eps <= 0.00023))
		tmp = Float64(Float64(Float64(-Float64(tan(x) + tan(eps))) / fma(tan(x), tan(eps), -1.0)) - tan(x));
	else
		tmp = fma(eps, t_3, fma(-1.0, fma(fma(-0.5, t_4, fma(0.16666666666666666, t_4, Float64(t_5 / Float64(cos(x) / sin(x))))), (eps ^ 4.0), Float64(t_5 * (eps ^ 3.0))), Float64(Float64(Float64(eps * eps) / cos(x)) * Float64(sin(x) * t_3))));
	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], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-0.5 * t$95$3 + N[((-t$95$0) / N[(t$95$1 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * t$95$2 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -0.000115], N[Not[LessEqual[eps, 0.00023]], $MachinePrecision]], N[(N[((-N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]) / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * t$95$3 + N[(-1.0 * N[(N[(-0.5 * t$95$4 + N[(0.16666666666666666 * t$95$4 + N[(t$95$5 / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision] + N[(t$95$5 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps * eps), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	37.4
Target	15.4
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if eps < -1.15e-4 or 2.3000000000000001e-4 < eps

   1. Initial program 30.4

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

   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

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

   5. Simplified 0.4

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

      [Start] 0.4	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      *-commutative [<=] 0.4	\[ \color{blue}{\frac{-1}{-1 + \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      associate-*l/ [=>] 0.4	\[ \color{blue}{\frac{-1 \cdot \left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      associate-*r/ [<=] 0.4	\[ \color{blue}{-1 \cdot \frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      neg-mul-1 [<=] 0.4	\[ \color{blue}{\left(-\frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}\right)} - \tan x \]
      distribute-neg-frac [=>] 0.4	\[ \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      +-commutative [=>] 0.4	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon + -1}} - \tan x \]
      metadata-eval [<=] 0.4	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\tan x \cdot \tan \varepsilon + \color{blue}{\left(-1\right)}} - \tan x \]
      sub-neg [<=] 0.4	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon - 1}} - \tan x \]
      fma-neg [=>] 0.4	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]
      metadata-eval [=>] 0.4	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, \color{blue}{-1}\right)} - \tan x \]

   if -1.15e-4 < eps < 2.3000000000000001e-4

   1. Initial program 44.5

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

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

      [Start] 0.2

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

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000115 \lor \neg \left(\varepsilon \leq 0.00023\right):\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-1, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}, \mathsf{fma}\left(0.16666666666666666, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}, \frac{\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-{\sin x}^{2}}{\frac{{\cos x}^{2}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666\right)}{\frac{\cos x}{\sin x}}\right)\right), {\varepsilon}^{4}, \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{-{\sin x}^{2}}{\frac{{\cos x}^{2}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666\right)\right) \cdot {\varepsilon}^{3}\right), \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	307081

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_2 := t_1 \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\\ t_3 := \frac{{\sin x}^{3}}{{\cos x}^{3}}\\ \mathbf{if}\;\varepsilon \leq -0.000175 \lor \neg \left(\varepsilon \leq 0.000215\right):\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\frac{\sin x \cdot \left(t_1 + 0.3333333333333333\right)}{\cos x} - \left(\frac{\sin x \cdot t_2}{\cos x} + \left(-0.3333333333333333 \cdot t_3 + t_0 \cdot -0.3333333333333333\right)\right)\right) + \left(\varepsilon \cdot \left(1 + t_1\right) + \left({\varepsilon}^{2} \cdot \left(t_0 + t_3\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(t_2 - t_1\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	78985

\[\begin{array}{l} t_0 := \frac{{\sin x}^{2}}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -0.0004 \lor \neg \left(\varepsilon \leq 0.0003\right):\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + t_0\right) + {\varepsilon}^{3} \cdot \left(\cos x \cdot 0.3333333333333333 + 0.3333333333333333 \cdot t_0\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	72137

\[\begin{array}{l} t_0 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_0, \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x \cdot t_0\right)\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	46089

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array} \]

Alternative 5
Error	0.4
Cost	39305

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.15 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 8
Error	14.7
Cost	26952

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \end{array} \]

Alternative 9
Error	14.9
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 10
Error	14.9
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11
Error	27.3
Cost	6464

\[\tan \varepsilon \]

Alternative 12
Error	44.2
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023041 
(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)))