2tan (problem 3.3.2)

Percentage Accurate: 42.7% → 99.6%

Time: 22.1s

Precision: binary64

Cost: 150665

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{\sin x}{\frac{\cos x}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}}\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))))
   (if (or (<= eps -5.8e-5) (not (<= eps 6.2e-5)))
     (- (- (tan x)) (/ (+ (tan x) (tan eps)) (fma (tan x) (tan eps) -1.0)))
     (+
      (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
      (+
       (/ eps (/ (pow (cos x) 2.0) (pow (sin x) 2.0)))
       (+
        (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))
        (*
         (pow eps 3.0)
         (-
          (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
          (/
           (sin x)
           (/ (cos x) (/ (* (sin x) -0.3333333333333333) (cos x))))))))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double tmp;
	if ((eps <= -5.8e-5) || !(eps <= 6.2e-5)) {
		tmp = -tan(x) - ((tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0));
	} else {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + ((eps / (pow(cos(x), 2.0) / pow(sin(x), 2.0))) + (((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))) + (pow(eps, 3.0) * ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (sin(x) / (cos(x) / ((sin(x) * -0.3333333333333333) / cos(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(eps) / cos(eps))
	tmp = 0.0
	if ((eps <= -5.8e-5) || !(eps <= 6.2e-5))
		tmp = Float64(Float64(-tan(x)) - Float64(Float64(tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(Float64(eps / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0))) + Float64(Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) + Float64((eps ^ 3.0) * Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(sin(x) / Float64(cos(x) / Float64(Float64(sin(x) * -0.3333333333333333) / cos(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[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -5.8e-5], N[Not[LessEqual[eps, 6.2e-5]], $MachinePrecision]], N[((-N[Tan[x], $MachinePrecision]) - N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $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[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[(N[(N[Sin[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	42.7%
Target	76.6%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if eps < -5.8e-5 or 6.20000000000000027e-5 < eps

   1. Initial program 50.6%

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

   2. Applied egg-rr 27.6%

      \[\leadsto \color{blue}{e^{\log \tan \left(x + \varepsilon\right)}} - \tan x \]
      Step-by-step derivation

      [Start] 50.6%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      add-exp-log [=>] 27.6%	\[ \color{blue}{e^{\log \tan \left(x + \varepsilon\right)}} - \tan x \]

   3. Applied egg-rr 99.4%

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

      [Start] 27.6%	\[ e^{\log \tan \left(x + \varepsilon\right)} - \tan x \]
      add-exp-log [<=] 50.6%	\[ \color{blue}{\tan \left(x + \varepsilon\right)} - \tan x \]
      tan-sum [=>] 99.5%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      frac-2neg [=>] 99.5%	\[ \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x \]
      div-inv [=>] 99.4%	\[ \color{blue}{\left(-\left(\tan x + \tan \varepsilon\right)\right) \cdot \frac{1}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x \]

   4. Simplified 99.5%

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

      [Start] 99.4%	\[ \left(-\left(\tan x + \tan \varepsilon\right)\right) \cdot \frac{1}{-\left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x \]
      associate-*r/ [=>] 99.5%	\[ \color{blue}{\frac{\left(-\left(\tan x + \tan \varepsilon\right)\right) \cdot 1}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x \]
      *-rgt-identity [=>] 99.5%	\[ \frac{\color{blue}{-\left(\tan x + \tan \varepsilon\right)}}{-\left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x \]
      distribute-neg-in [=>] 99.5%	\[ \frac{\color{blue}{\left(-\tan x\right) + \left(-\tan \varepsilon\right)}}{-\left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x \]
      unsub-neg [=>] 99.5%	\[ \frac{\color{blue}{\left(-\tan x\right) - \tan \varepsilon}}{-\left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x \]
      sub-neg [=>] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{-\color{blue}{\left(1 + \left(-\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]
      distribute-rgt-neg-out [<=] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{-\left(1 + \color{blue}{\tan x \cdot \left(-\tan \varepsilon\right)}\right)} - \tan x \]
      +-commutative [<=] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{-\color{blue}{\left(\tan x \cdot \left(-\tan \varepsilon\right) + 1\right)}} - \tan x \]
      distribute-neg-in [=>] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{\color{blue}{\left(-\tan x \cdot \left(-\tan \varepsilon\right)\right) + \left(-1\right)}} - \tan x \]
      distribute-rgt-neg-out [=>] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{\left(-\color{blue}{\left(-\tan x \cdot \tan \varepsilon\right)}\right) + \left(-1\right)} - \tan x \]
      remove-double-neg [=>] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{\color{blue}{\tan x \cdot \tan \varepsilon} + \left(-1\right)} - \tan x \]
      sub-neg [<=] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{\color{blue}{\tan x \cdot \tan \varepsilon - 1}} - \tan x \]
      fma-neg [=>] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]
      metadata-eval [=>] 99.5%	\[ \frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, \color{blue}{-1}\right)} - \tan x \]

   if -5.8e-5 < eps < 6.20000000000000027e-5

   1. Initial program 30.5%

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

   2. Applied egg-rr 32.1%

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

      [Start] 30.5%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 32.1%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 32.1%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 32.1%	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Simplified 32.1%

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

      [Start] 32.1%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      fma-neg [<=] 32.1%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]
      associate-*r/ [=>] 32.1%	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 32.1%	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Taylor expanded in x around inf 32.1%

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

   5. Simplified 63.1%

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

      [Start] 32.1%	\[ \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x} \]
      associate--l+ [=>] 63.0%	\[ \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right)} \]
      associate-/r* [=>] 63.0%	\[ \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]
      *-commutative [=>] 63.0%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\color{blue}{\sin \varepsilon \cdot \sin x}}{\cos \varepsilon \cdot \cos x}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]
      times-frac [=>] 63.0%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]

   6. Taylor expanded in eps around 0 99.7%

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

   7. Simplified 99.7%

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

      [Start] 99.7%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right) \]
      associate-/l* [=>] 99.7%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\color{blue}{\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right) \]
      mul-1-neg [=>] 99.7%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \color{blue}{\left(-{\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)}\right)\right) \]
      unsub-neg [=>] 99.7%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \color{blue}{\left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} - {\varepsilon}^{3} \cdot \left(\frac{\sin x \cdot \left(-0.5 \cdot \frac{\sin x}{\cos x} - -0.16666666666666666 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + -1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{\sin x}{\frac{\cos x}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	150665

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{\sin x}{\frac{\cos x}{\frac{\sin x \cdot -0.3333333333333333}{\cos x}}}\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	65737

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.5 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	65609

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -9.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	39305

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.65 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6
Accuracy	77.6%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.18:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7
Accuracy	77.7%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.18:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8
Accuracy	58.2%
Cost	6464

\[\tan \varepsilon \]

Alternative 9
Accuracy	31.2%
Cost	576

\[\frac{1}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333} \]

Alternative 10
Accuracy	30.9%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023271 
(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)))