2tan (problem 3.3.2)

Percentage Accurate: 41.7% → 99.5%

Time: 26.4s

Precision: binary64

Cost: 91848

Math

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

↓

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 + \left(1 + \left(-1 - t_0\right)\right)}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 3.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - t_0}, t_1\right)\\ \end{array} \]

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)))
        (t_2 (+ (tan x) (tan eps))))
   (if (<= eps -2.1e-6)
     (fma t_2 (/ 1.0 (+ 1.0 (+ 1.0 (- -1.0 t_0)))) t_1)
     (if (<= eps 3.15e-6)
       (+
        (/
         (/ (sin eps) (cos eps))
         (- 1.0 (/ (sin x) (/ (* (cos eps) (cos x)) (sin eps)))))
        (+
         (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))
         (/ eps (/ (pow (cos x) 2.0) (pow (sin x) 2.0)))))
       (fma t_2 (/ 1.0 (- 1.0 t_0)) t_1)))))

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);
	double t_2 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2.1e-6) {
		tmp = fma(t_2, (1.0 / (1.0 + (1.0 + (-1.0 - t_0)))), t_1);
	} else if (eps <= 3.15e-6) {
		tmp = ((sin(eps) / cos(eps)) / (1.0 - (sin(x) / ((cos(eps) * cos(x)) / sin(eps))))) + (((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))) + (eps / (pow(cos(x), 2.0) / pow(sin(x), 2.0))));
	} else {
		tmp = fma(t_2, (1.0 / (1.0 - t_0)), t_1);
	}
	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))
	t_2 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2.1e-6)
		tmp = fma(t_2, Float64(1.0 / Float64(1.0 + Float64(1.0 + Float64(-1.0 - t_0)))), t_1);
	elseif (eps <= 3.15e-6)
		tmp = Float64(Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - Float64(sin(x) / Float64(Float64(cos(eps) * cos(x)) / sin(eps))))) + Float64(Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0))) + Float64(eps / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0)))));
	else
		tmp = fma(t_2, Float64(1.0 / Float64(1.0 - t_0)), t_1);
	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[Tan[x], $MachinePrecision])}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.1e-6], N[(t$95$2 * N[(1.0 / N[(1.0 + N[(1.0 + N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 3.15e-6], N[(N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $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[(eps / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(1.0 / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Target

Original	41.7%
Target	75.9%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.0999999999999998e-6

   1. Initial program 53.3%

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

   2. Applied egg-rr 99.5%

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

      [Start] 53.3%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.5%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.4%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.5%	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Applied egg-rr 99.5%

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

      [Start] 99.5%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      expm1-log1p-u [=>] 80.8%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan \varepsilon\right)\right)}}, -\tan x\right) \]
      expm1-udef [=>] 80.8%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan x \cdot \tan \varepsilon\right)} - 1\right)}}, -\tan x\right) \]
      log1p-udef [=>] 80.8%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(e^{\color{blue}{\log \left(1 + \tan x \cdot \tan \varepsilon\right)}} - 1\right)}, -\tan x\right) \]
      add-exp-log [<=] 99.5%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\color{blue}{\left(1 + \tan x \cdot \tan \varepsilon\right)} - 1\right)}, -\tan x\right) \]
      add-cube-cbrt [=>] 99.2%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\left(1 + \color{blue}{\left(\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\right)} \cdot \tan \varepsilon\right) - 1\right)}, -\tan x\right) \]
      unpow3 [<=] 99.2%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\left(1 + \color{blue}{{\left(\sqrt[3]{\tan x}\right)}^{3}} \cdot \tan \varepsilon\right) - 1\right)}, -\tan x\right) \]
      *-commutative [=>] 99.2%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\left(1 + \color{blue}{\tan \varepsilon \cdot {\left(\sqrt[3]{\tan x}\right)}^{3}}\right) - 1\right)}, -\tan x\right) \]
      unpow3 [=>] 99.2%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\left(1 + \tan \varepsilon \cdot \color{blue}{\left(\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\right)}\right) - 1\right)}, -\tan x\right) \]
      add-cube-cbrt [<=] 99.5%	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \left(\left(1 + \tan \varepsilon \cdot \color{blue}{\tan x}\right) - 1\right)}, -\tan x\right) \]

   if -2.0999999999999998e-6 < eps < 3.14999999999999991e-6

   1. Initial program 21.3%

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

   2. Applied egg-rr 22.9%

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

      [Start] 21.3%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 22.9%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 22.9%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 22.9%	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Taylor expanded in x around inf 22.9%

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

   4. Simplified 56.6%

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

      [Start] 22.9%	\[ \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x} \]
      associate--l+ [=>] 56.5%	\[ \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right)} \]
      associate-/r* [=>] 56.5%	\[ \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]
      associate-/l* [=>] 56.5%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}}} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right)} - \frac{\sin x}{\cos x}\right) \]

   5. Taylor expanded in eps around 0 99.6%

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

   6. Simplified 99.6%

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

      [Start] 99.6%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right) \]
      associate-*r/ [<=] 99.6%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\color{blue}{\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}} + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right) \]
      +-commutative [=>] 99.6%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \color{blue}{\left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
      associate-/l* [=>] 99.6%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\color{blue}{\frac{{\varepsilon}^{2}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}} + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      unpow2 [=>] 99.6%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\frac{\color{blue}{\varepsilon \cdot \varepsilon}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      associate-*r/ [=>] 99.6%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \color{blue}{\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}}\right) \]
      associate-/l* [=>] 99.6%	\[ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \color{blue}{\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}}\right) \]

   if 3.14999999999999991e-6 < eps

   1. Initial program 58.6%

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

   2. Applied egg-rr 99.6%

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

      [Start] 58.6%	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.5%	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.5%	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.6%	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	91848

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 + \left(1 + \left(-1 - t_0\right)\right)}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 3.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - t_0}, t_1\right)\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	72008

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 + \left(1 + \left(-1 - t_0\right)\right)}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - t_0}, t_1\right)\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	65608

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 + \left(1 + \left(-1 - t_0\right)\right)}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin \varepsilon}}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - t_0}, t_1\right)\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	39556

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 + \left(1 + \left(-1 - t_0\right)\right)}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - t_0}, t_1\right)\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	39433

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.25 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6
Accuracy	99.3%
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

Alternative 7
Accuracy	99.3%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.25 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 8
Accuracy	76.9%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 33.5\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 9
Accuracy	76.9%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 33.5\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 10
Accuracy	76.9%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 33.5\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 11
Accuracy	57.7%
Cost	12992

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

Alternative 12
Accuracy	50.8%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]

Alternative 13
Accuracy	27.0%
Cost	448

\[\varepsilon + x \cdot \left(\varepsilon \cdot x\right) \]

Alternative 14
Accuracy	3.6%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023263 
(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)))