2tan (problem 3.3.2)

Average Accuracy: 42.3% → 99.5%

Time: 24.9s

Precision: binary64

Cost: 294408

Math

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

↓

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t_0}{t_1}\\ t_3 := 1 + t_2\\ t_4 := \tan x + \tan \varepsilon\\ t_5 := \sin x \cdot t_3\\ t_6 := 0.3333333333333333 + \left(\frac{t_0 \cdot t_3}{t_1} + 0.3333333333333333 \cdot t_2\right)\\ \mathbf{if}\;\varepsilon \leq -0.000215:\\ \;\;\;\;\frac{t_4}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\frac{\sin x \cdot t_6}{\cos x} + 0.3333333333333333 \cdot \frac{t_5}{\cos x}\right) + \left(\varepsilon \cdot t_3 + \left(\frac{t_5 \cdot {\varepsilon}^{2}}{\cos x} + t_6 \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_4}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ 1.0 t_2))
        (t_4 (+ (tan x) (tan eps)))
        (t_5 (* (sin x) t_3))
        (t_6
         (+
          0.3333333333333333
          (+ (/ (* t_0 t_3) t_1) (* 0.3333333333333333 t_2)))))
   (if (<= eps -0.000215)
     (- (/ t_4 (- 1.0 (/ (tan eps) (/ 1.0 (tan x))))) (tan x))
     (if (<= eps 0.00016)
       (+
        (*
         (pow eps 4.0)
         (+
          (/ (* (sin x) t_6) (cos x))
          (* 0.3333333333333333 (/ t_5 (cos x)))))
        (+
         (* eps t_3)
         (+ (/ (* t_5 (pow eps 2.0)) (cos x)) (* t_6 (pow eps 3.0)))))
       (- (- (tan x)) (/ t_4 (fma (tan x) (tan eps) -1.0)))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = 1.0 + t_2;
	double t_4 = tan(x) + tan(eps);
	double t_5 = sin(x) * t_3;
	double t_6 = 0.3333333333333333 + (((t_0 * t_3) / t_1) + (0.3333333333333333 * t_2));
	double tmp;
	if (eps <= -0.000215) {
		tmp = (t_4 / (1.0 - (tan(eps) / (1.0 / tan(x))))) - tan(x);
	} else if (eps <= 0.00016) {
		tmp = (pow(eps, 4.0) * (((sin(x) * t_6) / cos(x)) + (0.3333333333333333 * (t_5 / cos(x))))) + ((eps * t_3) + (((t_5 * pow(eps, 2.0)) / cos(x)) + (t_6 * pow(eps, 3.0))));
	} else {
		tmp = -tan(x) - (t_4 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(1.0 + t_2)
	t_4 = Float64(tan(x) + tan(eps))
	t_5 = Float64(sin(x) * t_3)
	t_6 = Float64(0.3333333333333333 + Float64(Float64(Float64(t_0 * t_3) / t_1) + Float64(0.3333333333333333 * t_2)))
	tmp = 0.0
	if (eps <= -0.000215)
		tmp = Float64(Float64(t_4 / Float64(1.0 - Float64(tan(eps) / Float64(1.0 / tan(x))))) - tan(x));
	elseif (eps <= 0.00016)
		tmp = Float64(Float64((eps ^ 4.0) * Float64(Float64(Float64(sin(x) * t_6) / cos(x)) + Float64(0.3333333333333333 * Float64(t_5 / cos(x))))) + Float64(Float64(eps * t_3) + Float64(Float64(Float64(t_5 * (eps ^ 2.0)) / cos(x)) + Float64(t_6 * (eps ^ 3.0)))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_4 / fma(tan(x), tan(eps), -1.0)));
	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[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(0.3333333333333333 + N[(N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(0.3333333333333333 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.000215], N[(N[(t$95$4 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00016], N[(N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$6), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(t$95$5 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * t$95$3), $MachinePrecision] + N[(N[(N[(t$95$5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$4 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Target

Original	42.3%
Target	77.2%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.14999999999999995e-4

   1. Initial program 53.9%

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

   2. Applied egg-rr 99.5%

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

   3. Simplified 99.5%

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

      [Start] 99.5	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 99.5	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 99.5	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 99.5%

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

   5. Simplified 99.5%

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

      [Start] 99.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x \]
      associate-/r/ [=>] 99.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan x}{1} \cdot \tan \varepsilon}} - \tan x \]
      associate-*l/ [=>] 99.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan x \cdot \tan \varepsilon}{1}}} - \tan x \]
      *-commutative [=>] 99.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\tan \varepsilon \cdot \tan x}}{1}} - \tan x \]
      associate-*l/ [<=] 99.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan \varepsilon}{1} \cdot \tan x}} - \tan x \]
      associate-/r/ [<=] 99.5	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan \varepsilon}{\frac{1}{\tan x}}}} - \tan x \]

   if -2.14999999999999995e-4 < eps < 1.60000000000000013e-4

   1. Initial program 30.4%

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

   2. Applied egg-rr 31.7%

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

   3. Simplified 31.7%

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

      [Start] 31.7	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 31.7	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 31.7	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 31.7%

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

   5. Simplified 31.7%

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

      [Start] 31.7	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      *-commutative [<=] 31.7	\[ \color{blue}{\frac{-1}{-1 + \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      associate-*l/ [=>] 31.7	\[ \color{blue}{\frac{-1 \cdot \left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      associate-*r/ [<=] 31.7	\[ \color{blue}{-1 \cdot \frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      neg-mul-1 [<=] 31.7	\[ \color{blue}{\left(-\frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}\right)} - \tan x \]
      distribute-neg-frac [=>] 31.7	\[ \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      +-commutative [=>] 31.7	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon + -1}} - \tan x \]
      metadata-eval [<=] 31.7	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\tan x \cdot \tan \varepsilon + \color{blue}{\left(-1\right)}} - \tan x \]
      sub-neg [<=] 31.7	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon - 1}} - \tan x \]
      fma-neg [=>] 31.7	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]
      metadata-eval [=>] 31.7	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, \color{blue}{-1}\right)} - \tan x \]

   6. Taylor expanded in eps around 0 99.6%

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

   if 1.60000000000000013e-4 < eps

   1. Initial program 54.9%

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

   2. Applied egg-rr 99.5%

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

   3. Simplified 99.5%

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

      [Start] 99.5	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 99.5	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 99.5	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 99.5%

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

   5. Simplified 99.5%

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

      [Start] 99.5	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      *-commutative [<=] 99.5	\[ \color{blue}{\frac{-1}{-1 + \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{-1 \cdot \left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      associate-*r/ [<=] 99.5	\[ \color{blue}{-1 \cdot \frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      neg-mul-1 [<=] 99.5	\[ \color{blue}{\left(-\frac{\tan x + \tan \varepsilon}{-1 + \tan x \cdot \tan \varepsilon}\right)} - \tan x \]
      distribute-neg-frac [=>] 99.5	\[ \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      +-commutative [=>] 99.5	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon + -1}} - \tan x \]
      metadata-eval [<=] 99.5	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\tan x \cdot \tan \varepsilon + \color{blue}{\left(-1\right)}} - \tan x \]
      sub-neg [<=] 99.5	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\tan x \cdot \tan \varepsilon - 1}} - \tan x \]
      fma-neg [=>] 99.5	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]
      metadata-eval [=>] 99.5	\[ \frac{-\left(\tan x + \tan \varepsilon\right)}{\mathsf{fma}\left(\tan x, \tan \varepsilon, \color{blue}{-1}\right)} - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000215:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\frac{\sin x \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + 0.3333333333333333 \cdot \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot {\varepsilon}^{2}}{\cos x} + \left(0.3333333333333333 + \left(\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot {\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	163080

\[\begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + t_0, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(t_0 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333 \cdot t_0\right)\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	157128

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := {\cos x}^{2}\\ t_3 := \frac{t_0}{t_2}\\ t_4 := 1 + t_3\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot t_4 + \left(\frac{\left(\sin x \cdot t_4\right) \cdot {\varepsilon}^{2}}{\cos x} + \left(0.3333333333333333 + \left(\frac{t_0 \cdot t_4}{t_2} + 0.3333333333333333 \cdot t_3\right)\right) \cdot {\varepsilon}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	39304

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sqrt[3]{{\tan x}^{6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 4
Accuracy	98.9%
Cost	33097

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.5 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sqrt[3]{{\tan x}^{6}}\right)\\ \end{array} \]

Alternative 5
Accuracy	99.0%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.5 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sqrt[3]{{\tan x}^{6}}\right)\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sqrt[3]{{\tan x}^{6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 7
Accuracy	78.5%
Cost	26952

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.018:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sqrt[3]{{\tan x}^{6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{x} + x \cdot -0.3333333333333333}} - \tan x\\ \end{array} \]

Alternative 8
Accuracy	78.4%
Cost	26697

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.018 \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{x}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sqrt[3]{{\tan x}^{6}}\right)\\ \end{array} \]

Alternative 9
Accuracy	77.4%
Cost	19848

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.018:\\ \;\;\;\;\frac{1}{\frac{1}{\tan \left(\varepsilon + x\right)}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sqrt[3]{{\tan x}^{6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 10
Accuracy	77.5%
Cost	19720

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.02:\\ \;\;\;\;\frac{1}{\frac{1}{\tan \left(\varepsilon + x\right)}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11
Accuracy	77.5%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.018:\\ \;\;\;\;\frac{1}{\frac{1}{\tan \left(\varepsilon + x\right)}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12
Accuracy	77.5%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.018:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 13
Accuracy	58.5%
Cost	6464

\[\tan \varepsilon \]

Alternative 14
Accuracy	31.7%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023129 
(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)))