2tan (problem 3.3.2)

Average Accuracy: 42.7% → 99.5%

Time: 26.6s

Precision: binary64

Cost: 325832

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_3 := 1 + t_2\\ t_4 := {t_3}^{2}\\ \mathbf{if}\;\varepsilon \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.5, t_4, t_3 \cdot t_0\right), \mathsf{fma}\left({\varepsilon}^{3}, \mathsf{fma}\left(0.16666666666666666, {t_3}^{3}, t_4 \cdot t_0\right) + \left(\left(t_2 \cdot t_3 - \mathsf{fma}\left(0.16666666666666666, t_2, 0.16666666666666666\right)\right) - \left(-0.5 + t_2 \cdot -0.5\right)\right), \varepsilon + \varepsilon \cdot t_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_3 (+ 1.0 t_2))
        (t_4 (pow t_3 2.0)))
   (if (<= eps -1.65e-6)
     (- (/ t_1 (- 1.0 (/ (sin eps) (/ (cos eps) (tan x))))) (tan x))
     (if (<= eps 2e-5)
       (log1p
        (fma
         (* eps eps)
         (fma 0.5 t_4 (* t_3 t_0))
         (fma
          (pow eps 3.0)
          (+
           (fma 0.16666666666666666 (pow t_3 3.0) (* t_4 t_0))
           (-
            (- (* t_2 t_3) (fma 0.16666666666666666 t_2 0.16666666666666666))
            (+ -0.5 (* t_2 -0.5))))
          (+ eps (* eps t_2)))))
       (- (/ t_1 (fma (tan x) (- (tan eps)) 1.0)) (tan x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = tan(x) + tan(eps);
	double t_2 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_3 = 1.0 + t_2;
	double t_4 = pow(t_3, 2.0);
	double tmp;
	if (eps <= -1.65e-6) {
		tmp = (t_1 / (1.0 - (sin(eps) / (cos(eps) / tan(x))))) - tan(x);
	} else if (eps <= 2e-5) {
		tmp = log1p(fma((eps * eps), fma(0.5, t_4, (t_3 * t_0)), fma(pow(eps, 3.0), (fma(0.16666666666666666, pow(t_3, 3.0), (t_4 * t_0)) + (((t_2 * t_3) - fma(0.16666666666666666, t_2, 0.16666666666666666)) - (-0.5 + (t_2 * -0.5)))), (eps + (eps * t_2)))));
	} else {
		tmp = (t_1 / fma(tan(x), -tan(eps), 1.0)) - 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(sin(x) / cos(x))
	t_1 = Float64(tan(x) + tan(eps))
	t_2 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_3 = Float64(1.0 + t_2)
	t_4 = t_3 ^ 2.0
	tmp = 0.0
	if (eps <= -1.65e-6)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(sin(eps) / Float64(cos(eps) / tan(x))))) - tan(x));
	elseif (eps <= 2e-5)
		tmp = log1p(fma(Float64(eps * eps), fma(0.5, t_4, Float64(t_3 * t_0)), fma((eps ^ 3.0), Float64(fma(0.16666666666666666, (t_3 ^ 3.0), Float64(t_4 * t_0)) + Float64(Float64(Float64(t_2 * t_3) - fma(0.16666666666666666, t_2, 0.16666666666666666)) - Float64(-0.5 + Float64(t_2 * -0.5)))), Float64(eps + Float64(eps * t_2)))));
	else
		tmp = Float64(Float64(t_1 / fma(tan(x), Float64(-tan(eps)), 1.0)) - 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[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, If[LessEqual[eps, -1.65e-6], N[(N[(t$95$1 / N[(1.0 - N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2e-5], N[Log[1 + N[(N[(eps * eps), $MachinePrecision] * N[(0.5 * t$95$4 + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[t$95$3, 3.0], $MachinePrecision] + N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 * t$95$3), $MachinePrecision] - N[(0.16666666666666666 * t$95$2 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(-0.5 + N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps + N[(eps * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$1 / N[(N[Tan[x], $MachinePrecision] * (-N[Tan[eps], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	42.7%
Target	76.6%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.65000000000000008e-6

   1. Initial program 54.6%

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

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

   4. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      tan-quot [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x \]
      associate-*r/ [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}} - \tan x \]

   5. Simplified 99.4%

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

      [Start] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x \]
      *-commutative [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\sin \varepsilon \cdot \tan x}}{\cos \varepsilon}} - \tan x \]
      associate-/l* [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}}} - \tan x \]

   if -1.65000000000000008e-6 < eps < 2.00000000000000016e-5

   1. Initial program 31.1%

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

   2. Applied egg-rr 31.1%

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

      [Start] 31.1	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      log1p-expm1-u [=>] 31.1	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(x + \varepsilon\right) - \tan x\right)\right)} \]

   3. Taylor expanded in eps around 0 99.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + 0.5 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}\right) + \left(\left(0.16666666666666666 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{3} + \left(\frac{\sin x \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}}{\cos x} + -1 \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]

   4. Simplified 99.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.5, {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}, \frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right), \mathsf{fma}\left({\varepsilon}^{3}, \mathsf{fma}\left(0.16666666666666666, {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{3}, \frac{\sin x}{\cos x} \cdot {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}\right) - \left(\left(\mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(-0.5 + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.5\right)\right), \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]
      Proof

      [Start] 99.6

      \[ \mathsf{log1p}\left({\varepsilon}^{2} \cdot \left(\frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + 0.5 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}\right) + \left(\left(0.16666666666666666 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{3} + \left(\frac{\sin x \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}}{\cos x} + -1 \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

      fma-def [=>] 99.6

      \[ \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + 0.5 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}, \left(0.16666666666666666 \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{3} + \left(\frac{\sin x \cdot {\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}}{\cos x} + -1 \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right) \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]

   if 2.00000000000000016e-5 < eps

   1. Initial program 53.3%

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

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

   4. Applied egg-rr 97.9%

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

      [Start] 99.4	\[ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      log1p-expm1-u [=>] 97.9	\[ \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan x \cdot \tan \varepsilon\right)\right)}} - \tan x \]

   5. Applied egg-rr 99.4%

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

      [Start] 97.9	\[ \frac{\tan x + \tan \varepsilon}{1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan x \cdot \tan \varepsilon\right)\right)} - \tan x \]
      sub-neg [=>] 97.9	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan x \cdot \tan \varepsilon\right)\right)} + \left(-\tan x\right)} \]
      div-inv [=>] 97.8	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan x \cdot \tan \varepsilon\right)\right)}} + \left(-\tan x\right) \]
      div-inv [<=] 97.9	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan x \cdot \tan \varepsilon\right)\right)}} + \left(-\tan x\right) \]
      sub-neg [=>] 97.9	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{1 + \left(-\mathsf{log1p}\left(\mathsf{expm1}\left(\tan x \cdot \tan \varepsilon\right)\right)\right)}} + \left(-\tan x\right) \]
      +-commutative [=>] 97.9	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{\left(-\mathsf{log1p}\left(\mathsf{expm1}\left(\tan x \cdot \tan \varepsilon\right)\right)\right) + 1}} + \left(-\tan x\right) \]
      log1p-expm1-u [<=] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\left(-\color{blue}{\tan x \cdot \tan \varepsilon}\right) + 1} + \left(-\tan x\right) \]
      distribute-rgt-neg-in [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{\tan x \cdot \left(-\tan \varepsilon\right)} + 1} + \left(-\tan x\right) \]
      fma-def [=>] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\color{blue}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}} + \left(-\tan x\right) \]

   6. Simplified 99.4%

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

      [Start] 99.4	\[ \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} + \left(-\tan x\right) \]
      sub-neg [<=] 99.4	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.5, {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2}, \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \mathsf{fma}\left({\varepsilon}^{3}, \mathsf{fma}\left(0.16666666666666666, {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{3}, {\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}^{2} \cdot \frac{\sin x}{\cos x}\right) + \left(\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666\right)\right) - \left(-0.5 + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.5\right)\right), \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	150664

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -3.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{1 - t_2 \cdot t_0} + \left({\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - t_2 \cdot \left(t_2 \cdot -0.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	91848

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x}{\cos x} \cdot t_0} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	65544

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(t_0 + {t_0}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	39364

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon}{\tan x}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	39304

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)} - \tan x\\ \end{array} \]

Alternative 6
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.15 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7
Accuracy	99.4%
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 8
Accuracy	77.5%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 9
Accuracy	77.5%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 10
Accuracy	31.4%
Cost	6464

\[\mathsf{log1p}\left(\varepsilon\right) \]

Alternative 11
Accuracy	58.3%
Cost	6464

\[\tan \varepsilon \]

Alternative 12
Accuracy	4.2%
Cost	64

\[0 \]

Alternative 13
Accuracy	31.3%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023147 
(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)))