2tan (problem 3.3.2)

Average Error: 37.3 → 0.3

Time: 20.7s

Precision: binary64

Cost: 169352

Math

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

↓

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.95 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{t_2}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 3.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(t_1 + \mathsf{fma}\left(0.3333333333333333, t_1, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon + \varepsilon \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x)))
        (t_1 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_2 (+ (tan x) (tan eps))))
   (if (<= eps -2.95e-5)
     (fma
      (fma (tan x) (tan eps) 1.0)
      (/ t_2 (- 1.0 (pow (* (tan x) (tan eps)) 2.0)))
      t_0)
     (if (<= eps 3.05e-5)
       (fma
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (* eps eps)
        (fma
         (pow eps 3.0)
         (+
          0.3333333333333333
          (+
           t_1
           (fma
            0.3333333333333333
            t_1
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))
         (+ eps (* eps t_1))))
       (fma t_2 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_0)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_2 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2.95e-5) {
		tmp = fma(fma(tan(x), tan(eps), 1.0), (t_2 / (1.0 - pow((tan(x) * tan(eps)), 2.0))), t_0);
	} else if (eps <= 3.05e-5) {
		tmp = fma(((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))), (eps * eps), fma(pow(eps, 3.0), (0.3333333333333333 + (t_1 + fma(0.3333333333333333, t_1, (pow(sin(x), 4.0) / pow(cos(x), 4.0))))), (eps + (eps * t_1))));
	} else {
		tmp = fma(t_2, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_0);
	}
	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))
	t_1 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_2 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2.95e-5)
		tmp = fma(fma(tan(x), tan(eps), 1.0), Float64(t_2 / Float64(1.0 - (Float64(tan(x) * tan(eps)) ^ 2.0))), t_0);
	elseif (eps <= 3.05e-5)
		tmp = fma(Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))), Float64(eps * eps), fma((eps ^ 3.0), Float64(0.3333333333333333 + Float64(t_1 + fma(0.3333333333333333, t_1, Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))))), Float64(eps + Float64(eps * t_1))));
	else
		tmp = fma(t_2, Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), t_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[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.95e-5], N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$2 / N[(1.0 - N[Power[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[eps, 3.05e-5], N[(N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(t$95$1 + N[(0.3333333333333333 * t$95$1 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps + N[(eps * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]

Target

Original	37.3
Target	15.0
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.9499999999999999e-5

   1. Initial program 29.2

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

   2. Applied egg-rr 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      swap-sqr [=>] 0.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      unpow2 [<=] 0.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\tan x}^{2}} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   4. Applied egg-rr 0.4

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

   if -2.9499999999999999e-5 < eps < 3.04999999999999994e-5

   1. Initial program 45.2

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

   2. Applied egg-rr 44.7

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

   3. Simplified 44.7

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

      [Start] 44.7	\[ \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      swap-sqr [=>] 44.7	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]
      unpow2 [<=] 44.7	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{{\tan x}^{2}} \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x \]

   4. Applied egg-rr 44.6

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

   5. Taylor expanded in eps around 0 0.3

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

   6. Simplified 0.2

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

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

   if 3.04999999999999994e-5 < eps

   1. Initial program 30.0

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

   2. Applied egg-rr 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.4	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.4	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 2.4

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan x + \tan \varepsilon\right)\right)}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   5. Applied egg-rr 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{-1 + \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      +-commutative [=>] 0.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\color{blue}{\tan x \cdot \tan \varepsilon + -1}}, -\tan x\right) \]
      fma-def [=>] 0.4	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}}, -\tan x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.95 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}, \varepsilon \cdot \varepsilon, \mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \mathsf{fma}\left(0.3333333333333333, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right), \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	78984

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{{\sin x}^{2}}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -0.00034:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00039:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\cos x + t_1\right) + {\varepsilon}^{3} \cdot \left(\cos x \cdot 0.3333333333333333 + 0.3333333333333333 \cdot t_1\right)}{1 - \tan x \cdot \tan \varepsilon}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	65736

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.35 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{t_1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{2}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_0\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	65092

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \frac{t_2}{1 - {t_0}^{2}}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{1 - t_0}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_1\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	46088

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -7 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{1 - \tan x \cdot \tan \varepsilon}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	45704

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 6
Error	0.4
Cost	39432

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.95 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{t_1 + -1}, -\tan x\right)\\ \end{array} \]

Alternative 7
Error	0.4
Cost	39432

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.95 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\tan x \cdot \tan \varepsilon + -1}, -\tan x\right)\\ \end{array} \]

Alternative 8
Error	0.4
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.95 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.95 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 9
Error	14.4
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{elif}\;\varepsilon \leq 0.000135:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 10
Error	26.8
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00048 \lor \neg \left(\varepsilon \leq 1.25 \cdot 10^{-6}\right):\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right)\\ \end{array} \]

Alternative 11
Error	27.3
Cost	12992

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

Alternative 12
Error	43.8
Cost	7104

\[\varepsilon + \varepsilon \cdot \left(0.5 - \frac{\cos \left(x + x\right)}{2}\right) \]

Alternative 13
Error	47.0
Cost	704

\[\varepsilon + \left(-1 + \left(1 + x \cdot \left(\varepsilon \cdot x\right)\right)\right) \]

Alternative 14
Error	47.6
Cost	448

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

Alternative 15
Error	47.5
Cost	448

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

Reproduce

herbie shell --seed 2023016 
(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)))