2tan (problem 3.3.2)

Average Error: 36.9 → 0.4

Time: 22.7s

Precision: binary64

Cost: 137352

Math

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

↓

\[\begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 - t_1\\ t_3 := \frac{\tan \varepsilon}{t_2}\\ t_4 := {\sin x}^{2}\\ \mathbf{if}\;\varepsilon \leq -42:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_2}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-5}:\\ \;\;\;\;t_3 + \left(\frac{\varepsilon}{\frac{t_0}{t_4}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - {\varepsilon}^{3} \cdot \mathsf{fma}\left(-1, \frac{{\sin x}^{4}}{{\cos x}^{4}}, -0.3333333333333333 \cdot \frac{t_4}{t_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\frac{\tan x}{1 + \left(1 - \left(1 + t_1\right)\right)} - \tan x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (* (tan x) (tan eps)))
        (t_2 (- 1.0 t_1))
        (t_3 (/ (tan eps) t_2))
        (t_4 (pow (sin x) 2.0)))
   (if (<= eps -42.0)
     (fma (/ 1.0 t_2) (+ (tan x) (tan eps)) (- (tan x)))
     (if (<= eps 6e-5)
       (+
        t_3
        (+
         (/ eps (/ t_0 t_4))
         (-
          (/ (* 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_4 t_0)))))))
       (+ t_3 (- (/ (tan x) (+ 1.0 (- 1.0 (+ 1.0 t_1)))) (tan x)))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = tan(x) * tan(eps);
	double t_2 = 1.0 - t_1;
	double t_3 = tan(eps) / t_2;
	double t_4 = pow(sin(x), 2.0);
	double tmp;
	if (eps <= -42.0) {
		tmp = fma((1.0 / t_2), (tan(x) + tan(eps)), -tan(x));
	} else if (eps <= 6e-5) {
		tmp = t_3 + ((eps / (t_0 / t_4)) + (((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_4 / t_0))))));
	} else {
		tmp = t_3 + ((tan(x) / (1.0 + (1.0 - (1.0 + t_1)))) - tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

↓

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64(tan(x) * tan(eps))
	t_2 = Float64(1.0 - t_1)
	t_3 = Float64(tan(eps) / t_2)
	t_4 = sin(x) ^ 2.0
	tmp = 0.0
	if (eps <= -42.0)
		tmp = fma(Float64(1.0 / t_2), Float64(tan(x) + tan(eps)), Float64(-tan(x)));
	elseif (eps <= 6e-5)
		tmp = Float64(t_3 + Float64(Float64(eps / Float64(t_0 / t_4)) + 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_4 / t_0)))))));
	else
		tmp = Float64(t_3 + Float64(Float64(tan(x) / Float64(1.0 + Float64(1.0 - Float64(1.0 + t_1)))) - 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[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[eps], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[eps, -42.0], N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 6e-5], N[(t$95$3 + N[(N[(eps / N[(t$95$0 / t$95$4), $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$4 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 + N[(N[(N[Tan[x], $MachinePrecision] / N[(1.0 + N[(1.0 - N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	36.9
Target	15.1
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -42

   1. Initial program 29.7

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

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

   if -42 < eps < 6.00000000000000015e-5

   1. Initial program 44.4

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

   2. Applied egg-rr 43.5

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

   3. Applied egg-rr 25.0

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

   4. Taylor expanded in eps around 0 0.3

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

   5. Simplified 0.3

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

      [Start] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      associate-/l* [=>] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\color{blue}{\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      associate-/l* [=>] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\color{blue}{\frac{{\varepsilon}^{2}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      unpow2 [=>] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\color{blue}{\varepsilon \cdot \varepsilon}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + -1 \cdot \left({\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
      mul-1-neg [=>] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \color{blue}{\left(-{\varepsilon}^{3} \cdot \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right)\right) \]
      fma-def [=>] 0.3	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} + \left(-{\varepsilon}^{3} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{{\sin x}^{4}}{{\cos x}^{4}}, -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right)\right) \]

   if 6.00000000000000015e-5 < eps

   1. Initial program 29.5

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -42:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \left(\frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}} - {\varepsilon}^{3} \cdot \mathsf{fma}\left(-1, \frac{{\sin x}^{4}}{{\cos x}^{4}}, -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 + \left(1 - \left(1 + \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	72264

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := 1 - t_0\\ t_2 := \frac{\tan \varepsilon}{t_1}\\ \mathbf{if}\;\varepsilon \leq -42:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_1}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;t_2 + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{\tan x}{1 + \left(1 - \left(1 + t_0\right)\right)} - \tan x\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	46408

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := 1 - t_0\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_1}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_1} + \left(\frac{\tan x}{1 + \left(1 - \left(1 + t_0\right)\right)} - \tan x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	46152

\[\begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_0}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_0} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	39880

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := 1 - t_0\\ \mathbf{if}\;\varepsilon \leq -3.15 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t_1}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{t_1} + \tan x \cdot \left(-1 + \frac{-1}{t_0 + -1}\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	39433

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}, \tan x + \tan \varepsilon, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	39236

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 + \left(1 - \left(1 + \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \end{array} \]

Alternative 7
Error	0.4
Cost	33352

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 + \left(1 - \left(1 + t_1\right)\right)} - \tan x\\ \end{array} \]

Alternative 8
Error	0.4
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-14}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1} \cdot t_0 - \tan x\\ \end{array} \]

Alternative 9
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 10
Error	14.5
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0108:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \end{array} \]

Alternative 11
Error	14.5
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0108:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \end{array} \]

Alternative 12
Error	26.7
Cost	6464

\[\tan \varepsilon \]

Alternative 13
Error	44.2
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023045 
(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)))