2tan (problem 3.3.2)

Average Error: 36.9 → 0.4

Time: 31.3s

Precision: binary64

Cost: 65736

Math

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

↓

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{1 - {t_0}^{2}} \cdot \left(1 + t_0\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

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))))
   (if (<= eps -4.8e-7)
     (- (* (/ t_1 (- 1.0 (pow t_0 2.0))) (+ 1.0 t_0)) (tan x))
     (if (<= eps 4.1e-7)
       (+
        (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (* eps eps)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (fma t_1 (/ -1.0 (fma (tan x) (tan eps) -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 tmp;
	if (eps <= -4.8e-7) {
		tmp = ((t_1 / (1.0 - pow(t_0, 2.0))) * (1.0 + t_0)) - tan(x);
	} else if (eps <= 4.1e-7) {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = fma(t_1, (-1.0 / fma(tan(x), tan(eps), -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))
	tmp = 0.0
	if (eps <= -4.8e-7)
		tmp = Float64(Float64(Float64(t_1 / Float64(1.0 - (t_0 ^ 2.0))) * Float64(1.0 + t_0)) - tan(x));
	elseif (eps <= 4.1e-7)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = fma(t_1, Float64(-1.0 / fma(tan(x), tan(eps), -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]}, If[LessEqual[eps, -4.8e-7], N[(N[(N[(t$95$1 / N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.1e-7], N[(N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Target

Original	36.9
Target	14.9
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -4.79999999999999957e-7

   1. Initial program 29.2

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

   2. Applied egg-rr 0.5

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

   3. Applied egg-rr 0.5

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

   4. Simplified 0.5

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      Proof

      [Start] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(1 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} - 1\right)\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      associate-+r- [=>] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - 1\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      +-commutative [=>] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\color{blue}{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} + 1\right)} - 1\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      associate--l+ [=>] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} + \left(1 - 1\right)\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      metadata-eval [=>] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} + \color{blue}{0}\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      +-rgt-identity [=>] 0.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   if -4.79999999999999957e-7 < eps < 4.0999999999999999e-7

   1. Initial program 45.3

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

   2. Applied egg-rr 44.9

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

   3. Simplified 44.9

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

      [Start] 44.9	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      *-commutative [<=] 44.9	\[ \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      associate-*l/ [=>] 44.9	\[ \color{blue}{\frac{1 \cdot \left(\tan x + \tan \varepsilon\right)}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-lft-identity [=>] 44.9	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Taylor expanded in eps around 0 0.3

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

   5. Simplified 0.3

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

      [Start] 0.3	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + -1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right) \]
      mul-1-neg [=>] 0.3	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \color{blue}{\left(-{\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right)} \]
      unsub-neg [=>] 0.3	\[ \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)} \]
      sub-neg [=>] 0.3	\[ \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right) \]
      mul-1-neg [=>] 0.3	\[ \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right) \]
      remove-double-neg [=>] 0.3	\[ \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) - {\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right) \]
      distribute-lft-out [=>] 0.3	\[ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - {\varepsilon}^{2} \cdot \color{blue}{\left(-1 \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)} \]

   if 4.0999999999999999e-7 < eps

   1. Initial program 28.9

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

   2. Applied egg-rr 0.5

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

   3. Applied egg-rr 1.5

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x}\right)}^{3}} \]

   4. Applied egg-rr 0.5

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

   5. Simplified 0.4

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

      [Start] 0.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      sub-neg [=>] 0.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{1 + \left(-\tan x \cdot \tan \varepsilon\right)}}, -\tan x\right) \]
      +-commutative [=>] 0.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{\left(-\tan x \cdot \tan \varepsilon\right) + 1}}, -\tan x\right) \]
      metadata-eval [<=] 0.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\left(-\tan x \cdot \tan \varepsilon\right) + \color{blue}{\left(--1\right)}}, -\tan x\right) \]
      distribute-neg-in [<=] 0.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{\color{blue}{-\left(\tan x \cdot \tan \varepsilon + -1\right)}}, -\tan x\right) \]
      fma-def [=>] 0.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{-\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}}, -\tan x\right) \]

   6. Applied egg-rr 0.4

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

   7. Simplified 0.4

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

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

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	52484

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_1}{1 - {t_0}^{2}} \cdot \left(1 + t_0\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	45704

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	39304

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 4
Error	0.4
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \end{array} \]

Alternative 5
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.9 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 7
Error	14.2
Cost	26696

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.04:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\cos \varepsilon - x \cdot \sin \varepsilon} \cdot \sin \left(\varepsilon + x\right) - \tan x\\ \end{array} \]

Alternative 8
Error	14.3
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.029:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 9
Error	14.3
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.029:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 10
Error	14.3
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.029:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 11
Error	26.5
Cost	12992

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

Alternative 12
Error	38.3
Cost	6720

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

Alternative 13
Error	61.7
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023038 
(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)))