2tan (problem 3.3.2)

Average Error: 37.0 → 0.3

Time: 21.6s

Precision: binary64

Cost: 150280

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := {t_1}^{3}\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_1 + t_2, \varepsilon \cdot \varepsilon, \mathsf{fma}\left({\varepsilon}^{3}, \frac{t_2 + t_1 \cdot 0.3333333333333333}{\frac{\cos x}{\sin x}} + \left(0.3333333333333333 + t_0\right), \varepsilon + \varepsilon \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + t_1, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_1 (/ (sin x) (cos x)))
        (t_2 (pow t_1 3.0)))
   (if (<= eps -3.4e-5)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (* (tan x) (sin eps)) (cos eps))))
      (tan x))
     (if (<= eps 4.5e-5)
       (fma
        (+ t_1 t_2)
        (* eps eps)
        (fma
         (pow eps 3.0)
         (+
          (/ (+ t_2 (* t_1 0.3333333333333333)) (/ (cos x) (sin x)))
          (+ 0.3333333333333333 t_0))
         (+ eps (* eps t_0))))
       (fma
        (+ (tan eps) t_1)
        (/ 1.0 (- 1.0 (* (tan x) (tan eps))))
        (- (tan x)))))))

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) / pow(cos(x), 2.0);
	double t_1 = sin(x) / cos(x);
	double t_2 = pow(t_1, 3.0);
	double tmp;
	if (eps <= -3.4e-5) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 4.5e-5) {
		tmp = fma((t_1 + t_2), (eps * eps), fma(pow(eps, 3.0), (((t_2 + (t_1 * 0.3333333333333333)) / (cos(x) / sin(x))) + (0.3333333333333333 + t_0)), (eps + (eps * t_0))));
	} else {
		tmp = fma((tan(eps) + t_1), (1.0 / (1.0 - (tan(x) * tan(eps)))), -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) ^ 2.0) / (cos(x) ^ 2.0))
	t_1 = Float64(sin(x) / cos(x))
	t_2 = t_1 ^ 3.0
	tmp = 0.0
	if (eps <= -3.4e-5)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	elseif (eps <= 4.5e-5)
		tmp = fma(Float64(t_1 + t_2), Float64(eps * eps), fma((eps ^ 3.0), Float64(Float64(Float64(t_2 + Float64(t_1 * 0.3333333333333333)) / Float64(cos(x) / sin(x))) + Float64(0.3333333333333333 + t_0)), Float64(eps + Float64(eps * t_0))));
	else
		tmp = fma(Float64(tan(eps) + t_1), Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), Float64(-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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 3.0], $MachinePrecision]}, If[LessEqual[eps, -3.4e-5], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.5e-5], N[(N[(t$95$1 + t$95$2), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[(t$95$2 + N[(t$95$1 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(eps + N[(eps * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[eps], $MachinePrecision] + t$95$1), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]]

Target

Original	37.0
Target	15.1
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -3.4e-5

   1. Initial program 29.9

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

   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 0.3

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      Proof
      (/.f64 (+.f64 (tan.f64 x) (tan.f64 eps)) (-.f64 1 (*.f64 (tan.f64 x) (tan.f64 eps)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (tan.f64 x) (tan.f64 eps)) 1)) (-.f64 1 (*.f64 (tan.f64 x) (tan.f64 eps)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (tan.f64 x) (tan.f64 eps)) (/.f64 1 (-.f64 1 (*.f64 (tan.f64 x) (tan.f64 eps)))))): 5 points increase in error, 9 points decrease in error

   4. Applied egg-rr 0.4

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

   if -3.4e-5 < eps < 4.50000000000000028e-5

   1. Initial program 44.1

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

   2. Applied egg-rr 43.6

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

   3. Taylor expanded in eps around 0 0.3

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

   4. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}, \varepsilon \cdot \varepsilon, \mathsf{fma}\left({\varepsilon}^{3}, \frac{{\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x} \cdot 0.3333333333333333}{\frac{\cos x}{\sin x}} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 0.3333333333333333\right), \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
      Proof
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)) (*.f64 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.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 eps eps) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (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 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.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 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 1 points increase in error, 3 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.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 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.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 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (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 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 2 points increase in error, 2 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (/.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 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (/.f64 (Rewrite<= cube-mult_binary64 (pow.f64 (sin.f64 x) 3)) (*.f64 (cos.f64 x) (pow.f64 (cos.f64 x) 2))) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 2 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (*.f64 (cos.f64 x) (Rewrite=> unpow2_binary64 (*.f64 (cos.f64 x) (cos.f64 x))))) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (Rewrite<= cube-mult_binary64 (pow.f64 (cos.f64 x) 3))) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 1 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) 1)) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (Rewrite<= metadata-eval (*.f64 -1 -1))) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) -1) -1)) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)))) -1) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 1/3)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) -1) (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (Rewrite<= metadata-eval (*.f64 -1/3 -1)))) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) -1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) -1/3) -1))) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (+.f64 (*.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) -1) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) -1)) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 -1 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))))) (/.f64 (cos.f64 x) (sin.f64 x))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.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 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (/.f64 (cos.f64 x) (sin.f64 x))))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 (*.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x)))) (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 2 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x))) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/3)) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 3 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (Rewrite<= +-commutative_binary64 (+.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x))) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (+.f64 eps (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x))) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 eps 1)) (*.f64 eps (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (fma.f64 (pow.f64 eps 3) (+.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x))) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (Rewrite=> distribute-lft-out_binary64 (*.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))): 17 points increase in error, 6 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 eps 3) (+.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x))) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (*.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))): 1 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 3) (+.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x))) (/.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 (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (pow.f64 eps 2)) (+.f64 (*.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (pow.f64 eps 3) (+.f64 1/3 (+.f64 (*.f64 -1 (/.f64 (*.f64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3))) (*.f64 -1/3 (/.f64 (sin.f64 x) (cos.f64 x)))) (sin.f64 x)) (cos.f64 x))) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))): 0 points increase in error, 0 points decrease in error

   if 4.50000000000000028e-5 < eps

   1. Initial program 30.1

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.4

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

   4. Simplified 0.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin x}{\cos x} + \tan \varepsilon}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      Proof
      (+.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (tan.f64 eps)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (sin.f64 x) 1)) (cos.f64 x)) (tan.f64 eps)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (sin.f64 x) (/.f64 1 (cos.f64 x)))) (tan.f64 eps)): 20 points increase in error, 12 points decrease in error
      (+.f64 (*.f64 (sin.f64 x) (Rewrite<= unpow-1_binary64 (pow.f64 (cos.f64 x) -1))) (tan.f64 eps)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-udef_binary64 (fma.f64 (sin.f64 x) (pow.f64 (cos.f64 x) -1) (tan.f64 eps))): 6 points increase in error, 7 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}, \varepsilon \cdot \varepsilon, \mathsf{fma}\left({\varepsilon}^{3}, \frac{{\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x} \cdot 0.3333333333333333}{\frac{\cos x}{\sin x}} + \left(0.3333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \frac{\sin x}{\cos x}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	137992

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -1.15:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 + \frac{-1}{\frac{1}{\tan x} \cdot \frac{1}{\tan \varepsilon}}} + \left({\sin x}^{2} \cdot \frac{\varepsilon}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot \varepsilon}{{\cos x}^{3}} \cdot {\sin x}^{3} + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + t_0 \cdot \frac{\sin x \cdot 0.3333333333333333}{\cos x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	85512

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.15:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 + \frac{-1}{\frac{1}{\tan x} \cdot \frac{1}{\tan \varepsilon}}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{3} \cdot {\varepsilon}^{2}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \frac{\sin x}{\cos x}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	79176

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.15:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 + \frac{-1}{\frac{1}{\tan x} \cdot \frac{1}{\tan \varepsilon}}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \frac{\sin x}{\cos x}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	59144

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(t_0 + {t_0}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	45960

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \]

Alternative 6
Error	0.4
Cost	45960

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \frac{\sin x}{\cos x}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 7
Error	0.4
Cost	39432

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.4
Cost	39432

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 9
Error	0.4
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

Alternative 10
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	14.5
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.32 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 12
Error	14.4
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \end{array} \]

Alternative 13
Error	27.4
Cost	13256

\[\begin{array}{l} t_0 := \tan \varepsilon - \tan x\\ \mathbf{if}\;\varepsilon \leq -1.52 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.5 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	27.2
Cost	12992

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

Alternative 15
Error	61.3
Cost	64

\[0 \]

Alternative 16
Error	44.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022335 
(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)))