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

↓

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_3 := \frac{\sin x}{\cos x}\\ t_4 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{t_1 \cdot \cos x + \sin x \cdot \left(t_0 + -1\right)}{\cos x \cdot \left(1 - t_0\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.017864840351748 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + t_2, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(t_3 + {t_3}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + t_2 \cdot 1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, t_4, 1\right)}, t_1, t_4\right)\\ \end{array} \]

(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 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_3 (/ (sin x) (cos x)))
        (t_4 (- (tan x))))
   (if (<= eps -273179843922033.63)
     (/ (+ (* t_1 (cos x)) (* (sin x) (+ t_0 -1.0))) (* (cos x) (- 1.0 t_0)))
     (if (<= eps 1.017864840351748e-7)
       (fma
        eps
        (+ 1.0 t_2)
        (*
         (* eps eps)
         (+
          (+ t_3 (pow t_3 3.0))
          (*
           eps
           (+
            0.3333333333333333
            (+
             (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
             (* t_2 1.3333333333333333)))))))
       (fma (/ 1.0 (fma (tan eps) t_4 1.0)) t_1 t_4)))))

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 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_3 = sin(x) / cos(x);
	double t_4 = -tan(x);
	double tmp;
	if (eps <= -273179843922033.63) {
		tmp = ((t_1 * cos(x)) + (sin(x) * (t_0 + -1.0))) / (cos(x) * (1.0 - t_0));
	} else if (eps <= 1.017864840351748e-7) {
		tmp = fma(eps, (1.0 + t_2), ((eps * eps) * ((t_3 + pow(t_3, 3.0)) + (eps * (0.3333333333333333 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) + (t_2 * 1.3333333333333333)))))));
	} else {
		tmp = fma((1.0 / fma(tan(eps), t_4, 1.0)), t_1, t_4);
	}
	return tmp;
}

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((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_3 = Float64(sin(x) / cos(x))
	t_4 = Float64(-tan(x))
	tmp = 0.0
	if (eps <= -273179843922033.63)
		tmp = Float64(Float64(Float64(t_1 * cos(x)) + Float64(sin(x) * Float64(t_0 + -1.0))) / Float64(cos(x) * Float64(1.0 - t_0)));
	elseif (eps <= 1.017864840351748e-7)
		tmp = fma(eps, Float64(1.0 + t_2), Float64(Float64(eps * eps) * Float64(Float64(t_3 + (t_3 ^ 3.0)) + Float64(eps * Float64(0.3333333333333333 + Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) + Float64(t_2 * 1.3333333333333333)))))));
	else
		tmp = fma(Float64(1.0 / fma(tan(eps), t_4, 1.0)), t_1, t_4);
	end
	return tmp
end

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[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = (-N[Tan[x], $MachinePrecision])}, If[LessEqual[eps, -273179843922033.63], N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.017864840351748e-7], N[(eps * N[(1.0 + t$95$2), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(t$95$3 + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(0.3333333333333333 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Tan[eps], $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$4), $MachinePrecision]]]]]]]]

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

↓

\begin{array}{l}
t_0 := \tan x \cdot \tan \varepsilon\\
t_1 := \tan x + \tan \varepsilon\\
t_2 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\
t_3 := \frac{\sin x}{\cos x}\\
t_4 := -\tan x\\
\mathbf{if}\;\varepsilon \leq -273179843922033.63:\\
\;\;\;\;\frac{t_1 \cdot \cos x + \sin x \cdot \left(t_0 + -1\right)}{\cos x \cdot \left(1 - t_0\right)}\\

\mathbf{elif}\;\varepsilon \leq 1.017864840351748 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + t_2, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(t_3 + {t_3}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + t_2 \cdot 1.3333333333333333\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, t_4, 1\right)}, t_1, t_4\right)\\


\end{array}

Original	36.8
Target	14.6
Herbie	1.2

Derivation

Split input into 3 regimes
if eps < -273179843922033.62
1. Initial program 28.0
  \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Applied egg-rr45.5
  \[\leadsto \color{blue}{{\left(\sqrt{\tan \left(x + \varepsilon\right)}\right)}^{2}} - \tan x \]
3. Applied egg-rr0.4
  \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}} \]
if -273179843922033.62 < eps < 1.01786484035174795e-7
1. Initial program 44.5
  \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Applied egg-rr43.0
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
3. Applied egg-rr43.0
  \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right) - 1\right)}} - \tan x \]
4. Applied egg-rr43.0
  \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}} - \tan x \]
5. Taylor expanded in eps around 0 1.9
  \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(-1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]
6. Simplified1.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) + \left(0.3333333333333333 - \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -1.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) \cdot \varepsilon\right)\right)} \]
  Proof
  (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 eps eps) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (+.f64 1 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (*.f64 (*.f64 eps eps) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (+.f64 1 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (*.f64 eps eps) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (*.f64 (*.f64 eps eps) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (Rewrite<= cube-unmult_binary64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (sin.f64 x) (cos.f64 x)))))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sin.f64 x) (sin.f64 x)) (*.f64 (cos.f64 x) (cos.f64 x)))))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 x) 2)) (*.f64 (cos.f64 x) (cos.f64 x))))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 2) (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 x) 2))))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sin.f64 x) (pow.f64 (sin.f64 x) 2)) (*.f64 (cos.f64 x) (pow.f64 (cos.f64 x) 2))))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (*.f64 (sin.f64 x) (Rewrite=> unpow2_binary64 (*.f64 (sin.f64 x) (sin.f64 x)))) (*.f64 (cos.f64 x) (pow.f64 (cos.f64 x) 2)))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (Rewrite<= cube-mult_binary64 (pow.f64 (sin.f64 x) 3)) (*.f64 (cos.f64 x) (pow.f64 (cos.f64 x) 2)))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (*.f64 (cos.f64 x) (Rewrite=> unpow2_binary64 (*.f64 (cos.f64 x) (cos.f64 x)))))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (Rewrite<= cube-mult_binary64 (pow.f64 (cos.f64 x) 3)))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x)))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) -4/3) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x))) (*.f64 (-.f64 1/3 (-.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (Rewrite<= metadata-eval (+.f64 -1 -1/3))) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x))) (*.f64 (-.f64 1/3 (-.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x))) (*.f64 (-.f64 1/3 (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (neg.f64 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x))) (*.f64 (-.f64 1/3 (+.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4)))))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x))) (*.f64 (-.f64 1/3 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))))))) eps)))): 2 points increase in error, 1 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 2) (+.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x))) (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))) eps)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x))) (pow.f64 eps 2)) (*.f64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) eps) (pow.f64 eps 2))))): 3 points increase in error, 2 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x)))))) (pow.f64 eps 2)) (*.f64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) eps) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x)))))) (pow.f64 eps 2)) (*.f64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) eps) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 (neg.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))))) (pow.f64 eps 2)) (*.f64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) eps) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))))) (pow.f64 eps 2)) (*.f64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) eps) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))) (pow.f64 eps 2)))) (*.f64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) eps) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 -1 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x))))))) (*.f64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) eps) (pow.f64 eps 2)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 -1 (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))))) (Rewrite<= associate-*r*_binary64 (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (*.f64 eps (pow.f64 eps 2)))))): 6 points increase in error, 7 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 -1 (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))))) (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (*.f64 eps (Rewrite=> unpow2_binary64 (*.f64 eps eps)))))): 0 points increase in error, 0 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 -1 (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))))) (*.f64 (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (Rewrite<= cube-mult_binary64 (pow.f64 eps 3))))): 5 points increase in error, 5 points decrease in error
  (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (*.f64 -1 (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))))) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 3) (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= fma-def_binary64 (+.f64 (*.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (+.f64 (*.f64 -1 (*.f64 (pow.f64 eps 2) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1 (/.f64 (sin.f64 x) (cos.f64 x)))))) (*.f64 (pow.f64 eps 3) (-.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 4) (pow.f64 (cos.f64 x) 4))) (*.f64 -1/3 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))))): 0 points increase in error, 0 points decrease in error
if 1.01786484035174795e-7 < eps
1. Initial program 29.6
  \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Applied egg-rr0.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
3. Applied egg-rr0.4
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)}, \tan x + \tan \varepsilon, -\tan x\right)} \]
Recombined 3 regimes into one program.
Final simplification1.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x + \sin x \cdot \left(\tan x \cdot \tan \varepsilon + -1\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.017864840351748 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)}, \tan x + \tan \varepsilon, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	65544

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := -\tan x\\ t_3 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{t_3 \cdot \cos x + \sin x \cdot \left(t_0 + -1\right)}{\cos x \cdot \left(1 - t_0\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.017864840351748 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(t_1 + {t_1}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, t_2, 1\right)}, t_3, t_2\right)\\ \end{array} \]

Alternative 2
Error	1.3
Cost	59076

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{t_0 \cdot \cos x + \sin x \cdot \left(t_1 + -1\right)}{\cos x \cdot \left(1 - t_1\right)}\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, t_2, 1\right)}, t_0, t_2\right)\\ \end{array} \]

Alternative 3
Error	1.3
Cost	45768

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\tan \varepsilon, t_1, 1\right)}, t_0, t_1\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	39432

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{t_1}{1 + 0.3333333333333333 \cdot \left(t_0 \cdot -3\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - t_0}, -\tan x\right)\\ \end{array} \]

Alternative 5
Error	1.3
Cost	39432

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 6
Error	1.3
Cost	33096

\[\begin{array}{l} t_0 := \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.3
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_0}} - \tan x\\ \end{array} \]

Alternative 8
Error	1.4
Cost	33096

\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;\frac{t_1}{1 + 0.3333333333333333 \cdot \left(t_0 \cdot -3\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - t_0}{t_1}} - \tan x\\ \end{array} \]

Alternative 9
Error	1.3
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9.142722786589536 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	14.3
Cost	32712

\[\begin{array}{l} t_0 := \left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.017864840351748 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	14.3
Cost	26440

\[\begin{array}{l} t_0 := \left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{if}\;\varepsilon \leq -273179843922033.63:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.017864840351748 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	26.5
Cost	12992

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

Alternative 13
Error	44.0
Cost	64

\[\varepsilon \]

Error

Target

Derivation

`if eps < -273179843922033.62`

`if -273179843922033.62 < eps < 1.01786484035174795e-7`

`if 1.01786484035174795e-7 < eps`

Alternatives

Error

Reproduce