2tan (problem 3.3.2)

Average Error: 37.1 → 0.3

Time: 24.0s

Precision: binary64

Cost: 117896

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := -\tan x\\ t_3 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan \varepsilon, t_2, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + t_0, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(t_3 + {t_3}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + t_0 \cdot 1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_2\right)\\ \end{array} \]

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 (+ (tan x) (tan eps)))
        (t_2 (- (tan x)))
        (t_3 (/ (sin x) (cos x))))
   (if (<= eps -0.0005964228779477913)
     (- (/ t_1 (fma (tan eps) t_2 1.0)) (tan x))
     (if (<= eps 3.714164716202234e-8)
       (fma
        eps
        (+ 1.0 t_0)
        (*
         (* eps eps)
         (+
          (+ t_3 (pow t_3 3.0))
          (*
           eps
           (+
            0.3333333333333333
            (+
             (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
             (* t_0 1.3333333333333333)))))))
       (fma t_1 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_2)))))

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 = tan(x) + tan(eps);
	double t_2 = -tan(x);
	double t_3 = sin(x) / cos(x);
	double tmp;
	if (eps <= -0.0005964228779477913) {
		tmp = (t_1 / fma(tan(eps), t_2, 1.0)) - tan(x);
	} else if (eps <= 3.714164716202234e-8) {
		tmp = fma(eps, (1.0 + t_0), ((eps * eps) * ((t_3 + pow(t_3, 3.0)) + (eps * (0.3333333333333333 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) + (t_0 * 1.3333333333333333)))))));
	} else {
		tmp = fma(t_1, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_2);
	}
	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(tan(x) + tan(eps))
	t_2 = Float64(-tan(x))
	t_3 = Float64(sin(x) / cos(x))
	tmp = 0.0
	if (eps <= -0.0005964228779477913)
		tmp = Float64(Float64(t_1 / fma(tan(eps), t_2, 1.0)) - tan(x));
	elseif (eps <= 3.714164716202234e-8)
		tmp = fma(eps, Float64(1.0 + t_0), 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_0 * 1.3333333333333333)))))));
	else
		tmp = fma(t_1, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_2);
	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[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0005964228779477913], N[(N[(t$95$1 / N[(N[Tan[eps], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.714164716202234e-8], N[(eps * N[(1.0 + t$95$0), $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$0 * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]

Target

Original	37.1
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 < -5.96422877947791279e-4

   1. Initial program 30.1

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.3

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

   if -5.96422877947791279e-4 < eps < 3.71416471620223408e-8

   1. Initial program 44.5

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

   2. Applied egg-rr 43.9

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

   3. Applied egg-rr 43.9

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

   4. Taylor expanded in eps around 0 0.3

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

   5. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left({\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}\right) + \left(0.3333333333333333 - \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -1.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) \cdot \varepsilon\right)\right)} \]
      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 (pow.f64 (/.f64 (sin.f64 x) (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 (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 (pow.f64 (/.f64 (sin.f64 x) (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 (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 (pow.f64 (/.f64 (sin.f64 x) (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 (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 (pow.f64 (/.f64 (sin.f64 x) (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 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (+.f64 (+.f64 (pow.f64 (/.f64 (sin.f64 x) (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 (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 (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 (/.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 (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 (/.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 (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 (/.f64 (sin.f64 x) (cos.f64 x)) (/.f64 (pow.f64 (sin.f64 x) 2) (Rewrite<= unpow2_binary64 (pow.f64 (cos.f64 x) 2)))) (/.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 (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 (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 (*.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 (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 (Rewrite<= cube-mult_binary64 (pow.f64 (sin.f64 x) 3)) (*.f64 (cos.f64 x) (pow.f64 (cos.f64 x) 2))) (/.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) (*.f64 (cos.f64 x) (Rewrite=> unpow2_binary64 (*.f64 (cos.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 (pow.f64 (sin.f64 x) 3) (Rewrite<= cube-mult_binary64 (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)))): 3 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))))): 0 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 (*.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)))))): 9 points increase in error, 10 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))))): 3 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 (*.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 3.71416471620223408e-8 < eps

   1. Initial program 29.7

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

   2. Applied egg-rr 30.4

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

   3. Applied egg-rr 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	104968

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\tan \varepsilon, t_0, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{3}, 0.3333333333333333 + t_2 \cdot \left(0.3333333333333333 - \left(-1 - t_2\right)\right), \left(1 + t_2\right) \cdot \left(\varepsilon + \frac{\varepsilon \cdot \sin x}{\frac{\cos x}{\varepsilon}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_0\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	78984

\[\begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \frac{{\sin x}^{2}}{\cos x}\\ t_2 := \tan x + \tan \varepsilon\\ t_3 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(\tan \varepsilon, t_3, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + t_1\right) + {\varepsilon}^{3} \cdot \left(\cos x \cdot 0.3333333333333333 + 0.3333333333333333 \cdot t_1\right)}{\cos x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{t_0}, t_3\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	46088

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ t_2 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(\tan \varepsilon, t_2, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, t_2\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	39752

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(\tan \varepsilon, t_1, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \left(\varepsilon + \frac{\varepsilon \cdot \sin x}{\frac{\cos x}{\varepsilon}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1\right)\\ \end{array} \]

Alternative 5
Error	0.8
Cost	39304

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} - \tan x\\ \mathbf{if}\;\varepsilon \leq -2.5627444998452892 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.0141950659280984 \cdot 10^{-30}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.8
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -2.5627444998452892 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.0141950659280984 \cdot 10^{-30}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	14.4
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \tan x\\ \end{array} \]

Alternative 8
Error	14.4
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0005964228779477913:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.714164716202234 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 9
Error	26.9
Cost	6464

\[\tan \varepsilon \]

Alternative 10
Error	43.9
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022295 
(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)))