2tan (problem 3.3.2)

Average Accuracy: 42.5% → 99.6%

Time: 27.5s

Precision: binary64

Cost: 111304

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 - {t_1}^{2}\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{1 - t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{\frac{t_2}{t_1}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_2} \cdot \left(1 + t_1\right) - \tan x\\ \end{array} \]

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)))
        (t_2 (- 1.0 (pow t_1 2.0))))
   (if (<= eps -3.7e-5)
     (- (/ t_0 (- 1.0 t_1)) (tan x))
     (if (<= eps 5e-5)
       (+
        (/ t_0 (/ t_2 t_1))
        (+
         (* (/ (* eps eps) (pow (cos x) 3.0)) (pow (sin x) 3.0))
         (fma
          (pow eps 3.0)
          (+ 0.3333333333333333 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
          eps)))
       (- (* (/ t_0 t_2) (+ 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 = tan(x) + tan(eps);
	double t_1 = tan(x) * tan(eps);
	double t_2 = 1.0 - pow(t_1, 2.0);
	double tmp;
	if (eps <= -3.7e-5) {
		tmp = (t_0 / (1.0 - t_1)) - tan(x);
	} else if (eps <= 5e-5) {
		tmp = (t_0 / (t_2 / t_1)) + ((((eps * eps) / pow(cos(x), 3.0)) * pow(sin(x), 3.0)) + fma(pow(eps, 3.0), (0.3333333333333333 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))), eps));
	} else {
		tmp = ((t_0 / t_2) * (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 = Float64(tan(x) + tan(eps))
	t_1 = Float64(tan(x) * tan(eps))
	t_2 = Float64(1.0 - (t_1 ^ 2.0))
	tmp = 0.0
	if (eps <= -3.7e-5)
		tmp = Float64(Float64(t_0 / Float64(1.0 - t_1)) - tan(x));
	elseif (eps <= 5e-5)
		tmp = Float64(Float64(t_0 / Float64(t_2 / t_1)) + Float64(Float64(Float64(Float64(eps * eps) / (cos(x) ^ 3.0)) * (sin(x) ^ 3.0)) + fma((eps ^ 3.0), Float64(0.3333333333333333 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))), eps)));
	else
		tmp = Float64(Float64(Float64(t_0 / t_2) * 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[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.7e-5], N[(N[(t$95$0 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5e-5], N[(N[(t$95$0 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(eps * eps), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / t$95$2), $MachinePrecision] * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	42.5%
Target	76.8%
Herbie	99.6%

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

Derivation

1. Split input into 3 regimes

2. if eps < -3.69999999999999981e-5

   1. Initial program 53.1%

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

   2. Applied egg-rr 99.4%

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

      [Start] 53.1	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.4	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.4	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   3. Simplified 99.4%

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

      [Start] 99.4	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 99.4	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 99.4	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   if -3.69999999999999981e-5 < eps < 5.00000000000000024e-5

   1. Initial program 30.7%

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

   2. Applied egg-rr 31.6%

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

      [Start] 30.7	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 31.6	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      flip-- [=>] 31.6	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x \]
      associate-/r/ [=>] 31.6	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x \]
      metadata-eval [=>] 31.6	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{1} - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   3. Applied egg-rr 38.7%

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

      [Start] 31.6	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      distribute-rgt-in [=>] 31.6	\[ \color{blue}{\left(1 \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} + \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right)} - \tan x \]
      *-un-lft-identity [<=] 31.6	\[ \left(\color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}} + \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) - \tan x \]
      +-commutative [=>] 31.6	\[ \color{blue}{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} + \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right)} - \tan x \]
      *-commutative [<=] 31.6	\[ \left(\color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(\tan x \cdot \tan \varepsilon\right)} + \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) - \tan x \]

   4. Taylor expanded in eps around 0 99.7%

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

   5. Simplified 99.7%

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

      [Start] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right)\right) \]
      associate-+r+ [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \color{blue}{\left(\left({\varepsilon}^{3} \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon\right) + \frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}}\right)} \]
      +-commutative [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \color{blue}{\left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon\right)\right)} \]
      associate-/l* [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\color{blue}{\frac{{\varepsilon}^{2}}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}} + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon\right)\right) \]
      associate-/r/ [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\color{blue}{\frac{{\varepsilon}^{2}}{{\cos x}^{3}} \cdot {\sin x}^{3}} + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon\right)\right) \]
      unpow2 [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\frac{\color{blue}{\varepsilon \cdot \varepsilon}}{{\cos x}^{3}} \cdot {\sin x}^{3} + \left({\varepsilon}^{3} \cdot \left(0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon\right)\right) \]
      fma-def [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + \color{blue}{\mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)}\right) \]
      sub-neg [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + \mathsf{fma}\left({\varepsilon}^{3}, \color{blue}{0.3333333333333333 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}, \varepsilon\right)\right) \]
      mul-1-neg [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right), \varepsilon\right)\right) \]
      remove-double-neg [=>] 99.7	\[ \frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}, \varepsilon\right)\right) \]

   if 5.00000000000000024e-5 < eps

   1. Initial program 55.2%

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

   2. Applied egg-rr 99.4%

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

      [Start] 55.2	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.4	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      flip-- [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x \]
      associate-/r/ [=>] 99.4	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x \]
      metadata-eval [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{1} - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   3. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      expm1-log1p-u [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      expm1-udef [=>] 99.3	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)} - 1\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      log1p-udef [=>] 99.3	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(e^{\color{blue}{\log \left(1 + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}} - 1\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      add-exp-log [<=] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\color{blue}{\left(1 + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)} - 1\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      associate--l+ [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) - 1\right)\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      pow2 [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(1 + \left(\color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}} - 1\right)\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   4. Simplified 99.4%

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

      [Start] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(1 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} - 1\right)\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      associate-+r- [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\left(1 + {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) - 1\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      +-commutative [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\color{blue}{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} + 1\right)} - 1\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      associate--l+ [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} + \left(1 - 1\right)\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      metadata-eval [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left({\left(\tan x \cdot \tan \varepsilon\right)}^{2} + \color{blue}{0}\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      +-rgt-identity [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\left(\tan x \cdot \tan \varepsilon\right)}^{2}}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}{\tan x \cdot \tan \varepsilon}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	52616

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - {t_1}^{2}} \cdot \left(1 + t_1\right) - \tan x\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	46280

\[\begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{t_0} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon \cdot \frac{1}{t_0} + \left(\frac{\tan x}{t_0} - \tan x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	46152

\[\begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{t_0} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\tan x}{t_0} - \tan x\right) + \frac{\tan \varepsilon}{t_0}\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	33480

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t_0 \cdot \frac{1}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}}}} - \tan x\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

Alternative 6
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7
Accuracy	77.9%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8
Accuracy	77.9%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 9
Accuracy	58.5%
Cost	6464

\[\tan \varepsilon \]

Alternative 10
Accuracy	31.6%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023137 
(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)))