2tan (problem 3.3.2)

Average Error: 37.4 → 0.4

Time: 18.4s

Precision: binary64

Cost: 65544

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -0.0002832768356521709:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin x}{\cos \varepsilon \cdot \cos x} \cdot \sin \varepsilon} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.91964260788293 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(t_0 + {t_0}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (if (<= eps -0.0002832768356521709)
     (+
      (/ (tan eps) (- 1.0 (* (/ (sin x) (* (cos eps) (cos x))) (sin eps))))
      (- (/ (tan x) (- 1.0 (* (tan eps) (tan x)))) (tan x)))
     (if (<= eps 1.91964260788293e-9)
       (fma
        eps
        (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (* (* eps eps) (+ t_0 (pow t_0 3.0))))
       (-
        (* (/ 1.0 (fma (tan eps) (- (tan x)) 1.0)) (+ (tan eps) (tan x)))
        (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 tmp;
	if (eps <= -0.0002832768356521709) {
		tmp = (tan(eps) / (1.0 - ((sin(x) / (cos(eps) * cos(x))) * sin(eps)))) + ((tan(x) / (1.0 - (tan(eps) * tan(x)))) - tan(x));
	} else if (eps <= 1.91964260788293e-9) {
		tmp = fma(eps, (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))), ((eps * eps) * (t_0 + pow(t_0, 3.0))));
	} else {
		tmp = ((1.0 / fma(tan(eps), -tan(x), 1.0)) * (tan(eps) + tan(x))) - 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))
	tmp = 0.0
	if (eps <= -0.0002832768356521709)
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - Float64(Float64(sin(x) / Float64(cos(eps) * cos(x))) * sin(eps)))) + Float64(Float64(tan(x) / Float64(1.0 - Float64(tan(eps) * tan(x)))) - tan(x)));
	elseif (eps <= 1.91964260788293e-9)
		tmp = fma(eps, Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))), Float64(Float64(eps * eps) * Float64(t_0 + (t_0 ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / fma(tan(eps), Float64(-tan(x)), 1.0)) * Float64(tan(eps) + tan(x))) - 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]}, If[LessEqual[eps, -0.0002832768356521709], N[(N[(N[Tan[eps], $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.91964260788293e-9], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(t$95$0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Tan[eps], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Target

Original	37.4
Target	15.5
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.8327683565217089e-4

   1. Initial program 30.2

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

   2. Applied egg-rr 0.3

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

   3. Taylor expanded in x around inf 0.6

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

   4. Simplified 0.6

      \[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos \varepsilon \cdot \cos x} \cdot \sin \varepsilon} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos \varepsilon \cdot \cos x} \cdot \sin \varepsilon} - \frac{\sin x}{\cos x}\right)} \]
      Proof
      (+.f64 (/.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 eps) (cos.f64 x))) (sin.f64 eps)))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 eps) (cos.f64 x))) (sin.f64 eps)))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (cos.f64 eps)))) (sin.f64 eps)))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 eps) (cos.f64 x))) (sin.f64 eps)))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (-.f64 1 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 x) (cos.f64 eps)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 eps) (cos.f64 x))) (sin.f64 eps)))) (/.f64 (sin.f64 x) (cos.f64 x)))): 3 points increase in error, 3 points decrease in error
      (+.f64 (/.f64 (/.f64 (sin.f64 eps) (cos.f64 eps)) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 eps) (cos.f64 x))) (sin.f64 eps)))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x))))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 eps) (cos.f64 x))) (sin.f64 eps)))) (/.f64 (sin.f64 x) (cos.f64 x)))): 6 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (*.f64 (/.f64 (sin.f64 x) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 x) (cos.f64 eps)))) (sin.f64 eps)))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 x) (cos.f64 eps)))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 4 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (/.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 eps) (cos.f64 x)))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (-.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (sin.f64 x) (*.f64 (cos.f64 x) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x))))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 5 points increase in error, 3 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (sin.f64 eps) (*.f64 (cos.f64 eps) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x)))))) (/.f64 (sin.f64 x) (*.f64 (cos.f64 x) (-.f64 1 (/.f64 (*.f64 (sin.f64 x) (sin.f64 eps)) (*.f64 (cos.f64 eps) (cos.f64 x))))))) (/.f64 (sin.f64 x) (cos.f64 x)))): 102 points increase in error, 16 points decrease in error

   5. Applied egg-rr 0.5

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

   6. Applied egg-rr 0.4

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

   if -2.8327683565217089e-4 < eps < 1.91964260788293012e-9

   1. Initial program 44.8

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

   2. Applied egg-rr 44.3

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

   3. Applied egg-rr 51.5

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} - \tan x\right)}^{3}}} \]

   4. Taylor expanded in eps around 0 0.3

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

   5. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)} \]
      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 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)))): 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 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)))): 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 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)))): 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 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)))): 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 (sin.f64 x) (cos.f64 x)) (pow.f64 (/.f64 (sin.f64 x) (cos.f64 x)) 3)))): 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 (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)))))))): 0 points increase in error, 3 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 (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)))))))): 5 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 (pow.f64 eps 2) (+.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))))))): 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 (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))))))): 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 (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))))))): 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 (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)))))): 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 (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)))))): 1 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 (pow.f64 eps 2) (+.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)))))))): 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 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 3) (Rewrite<= cube-mult_binary64 (pow.f64 (cos.f64 x) 3)))))): 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) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 3) (pow.f64 (cos.f64 x) 3)) (/.f64 (sin.f64 x) (cos.f64 x)))))): 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) (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)))))))): 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) (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)))))))): 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) (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)))))))): 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-neg-in_binary64 (neg.f64 (*.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)))))))): 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<= mul-1-neg_binary64 (*.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)))))))): 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 -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)))))))): 0 points increase in error, 0 points decrease in error

   if 1.91964260788293012e-9 < eps

   1. Initial program 30.6

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

   2. Applied egg-rr 0.5

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

   3. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0002832768356521709:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin x}{\cos \varepsilon \cdot \cos x} \cdot \sin \varepsilon} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.91964260788293 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	59076

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0002832768356521709:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \frac{\sin x}{\cos \varepsilon \cdot \cos x} \cdot \sin \varepsilon} + \left(\frac{\tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.91964260788293 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \end{array} \]

Alternative 2
Error	0.4
Cost	39432

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0002832768356521709:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.91964260788293 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.4
Cost	39172

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0002832768356521709:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan \varepsilon, \tan x, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.91964260788293 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 4
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0002832768356521709:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.91964260788293 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.4
Cost	32968

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0002832768356521709:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.91964260788293 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 6
Error	14.7
Cost	19784

\[\begin{array}{l} t_0 := \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0006507855826718776:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.0071158077369693 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	14.8
Cost	13448

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -0.0006507855826718776:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.0071158077369693 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	14.8
Cost	13448

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -0.0006507855826718776:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.0071158077369693 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	27.4
Cost	12992

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

Alternative 10
Error	61.3
Cost	64

\[0 \]

Alternative 11
Error	44.2
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022294 
(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)))