2tan (problem 3.3.2)

Average Error: 36.9 → 0.4

Time: 24.2s

Precision: binary64

Cost: 163400

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := 1 + t_0\\ t_2 := \tan x \cdot \tan \varepsilon\\ t_3 := \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {t_2}^{3}}, \mathsf{fma}\left(t_2, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(t_1 \cdot -0.3333333333333333 + t_0 \cdot \left(-1 - t_0\right), -{\varepsilon}^{3}, \mathsf{fma}\left(\varepsilon, t_1, \left(t_1 \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (+ 1.0 t_0))
        (t_2 (* (tan x) (tan eps)))
        (t_3
         (fma
          (/ (+ (tan x) (tan eps)) (- 1.0 (pow t_2 3.0)))
          (fma t_2 (fma (tan x) (tan eps) 1.0) 1.0)
          (- (tan x)))))
   (if (<= eps -0.0081203423223153)
     t_3
     (if (<= eps 7.2260556432950395e-12)
       (fma
        (+ (* t_1 -0.3333333333333333) (* t_0 (- -1.0 t_0)))
        (- (pow eps 3.0))
        (fma eps t_1 (* (* t_1 (/ (sin x) (cos x))) (* eps eps))))
       t_3))))

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 = 1.0 + t_0;
	double t_2 = tan(x) * tan(eps);
	double t_3 = fma(((tan(x) + tan(eps)) / (1.0 - pow(t_2, 3.0))), fma(t_2, fma(tan(x), tan(eps), 1.0), 1.0), -tan(x));
	double tmp;
	if (eps <= -0.0081203423223153) {
		tmp = t_3;
	} else if (eps <= 7.2260556432950395e-12) {
		tmp = fma(((t_1 * -0.3333333333333333) + (t_0 * (-1.0 - t_0))), -pow(eps, 3.0), fma(eps, t_1, ((t_1 * (sin(x) / cos(x))) * (eps * eps))));
	} else {
		tmp = t_3;
	}
	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(1.0 + t_0)
	t_2 = Float64(tan(x) * tan(eps))
	t_3 = fma(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - (t_2 ^ 3.0))), fma(t_2, fma(tan(x), tan(eps), 1.0), 1.0), Float64(-tan(x)))
	tmp = 0.0
	if (eps <= -0.0081203423223153)
		tmp = t_3;
	elseif (eps <= 7.2260556432950395e-12)
		tmp = fma(Float64(Float64(t_1 * -0.3333333333333333) + Float64(t_0 * Float64(-1.0 - t_0))), Float64(-(eps ^ 3.0)), fma(eps, t_1, Float64(Float64(t_1 * Float64(sin(x) / cos(x))) * Float64(eps * eps))));
	else
		tmp = t_3;
	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[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[eps, -0.0081203423223153], t$95$3, If[LessEqual[eps, 7.2260556432950395e-12], N[(N[(N[(t$95$1 * -0.3333333333333333), $MachinePrecision] + N[(t$95$0 * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Power[eps, 3.0], $MachinePrecision]) + N[(eps * t$95$1 + N[(N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Target

Original	36.9
Target	15.0
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if eps < -0.00812034232231530033 or 7.22605564329503947e-12 < eps

   1. Initial program 29.4

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.5

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

   if -0.00812034232231530033 < eps < 7.22605564329503947e-12

   1. Initial program 44.8

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

   2. Applied egg-rr 45.3

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

   3. Taylor expanded in eps around 0 0.3

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

   4. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot -0.3333333333333333 - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), -{\varepsilon}^{3}, \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
      Proof
      (fma.f64 (-.f64 (*.f64 (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) -1/3) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (*.f64 (+.f64 1 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) -1/3) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (*.f64 (+.f64 1 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/3) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) -1/3) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (Rewrite<= metadata-eval (+.f64 1/6 -1/2))) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 26 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) 1/6) (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/2))) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (-.f64 (+.f64 (*.f64 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 1 (*.f64 (neg.f64 -1) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) 1/6) (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/2)) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (*.f64 (+.f64 1 (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) 1/6) (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/2)) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (*.f64 (+.f64 1 (Rewrite=> *-lft-identity_binary64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) 1/6) (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/2)) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1)) 1/6) (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/2)) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) 1/6) 1/6)) (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/2)) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 2 points decrease in error
      (fma.f64 (-.f64 (+.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) 1/6) (*.f64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) -1/2)) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (+.f64 1 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (*.f64 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 (sin.f64 x) 2) (/.f64 (pow.f64 (cos.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 2 points increase in error, 4 points decrease in error
      (fma.f64 (-.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2)))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 4 points increase in error, 3 points decrease in error
      (fma.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (neg.f64 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (*.f64 -1/2 (-.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 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (+.f64 (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2)))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 1 points increase in error, 4 points decrease in error
      (fma.f64 (+.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) 1/6) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))))) (neg.f64 (pow.f64 eps 3)) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 4 points increase in error, 4 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 eps 3))) (fma.f64 eps (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (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 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (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 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (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 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (+.f64 1 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (*.f64 (/.f64 (sin.f64 x) (cos.f64 x)) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (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 (sin.f64 x) (/.f64 (cos.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (cos.f64 x))) (*.f64 eps eps)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (*.f64 (/.f64 (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (cos.f64 x)) (Rewrite<= unpow2_binary64 (pow.f64 eps 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (/.f64 (cos.f64 x) (pow.f64 eps 2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 eps 2)) (cos.f64 x))))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (fma.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 eps 2) (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3)) (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 (pow.f64 eps 2) (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (cos.f64 x))))): 2 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))) (*.f64 -1 (pow.f64 eps 3))) (+.f64 (*.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (/.f64 (*.f64 (pow.f64 eps 2) (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (cos.f64 x))))): 2 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1 (pow.f64 eps 3)) (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))))) (+.f64 (*.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (/.f64 (*.f64 (pow.f64 eps 2) (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (pow.f64 eps 3) (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))))))) (+.f64 (*.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (/.f64 (*.f64 (pow.f64 eps 2) (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (cos.f64 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (/.f64 (*.f64 (pow.f64 eps 2) (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (cos.f64 x))) (*.f64 -1 (*.f64 (pow.f64 eps 3) (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 eps (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (+.f64 (/.f64 (*.f64 (pow.f64 eps 2) (*.f64 (sin.f64 x) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2)))))) (cos.f64 x)) (*.f64 -1 (*.f64 (pow.f64 eps 3) (+.f64 (*.f64 1/6 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))) (+.f64 1/6 (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 (sin.f64 x) 2) (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))) (pow.f64 (cos.f64 x) 2))) (*.f64 -1/2 (-.f64 1 (*.f64 -1 (/.f64 (pow.f64 (sin.f64 x) 2) (pow.f64 (cos.f64 x) 2))))))))))))): 0 points increase in error, 1 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot -0.3333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), -{\varepsilon}^{3}, \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	117896

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_2 := \tan x \cdot \tan \varepsilon\\ t_3 := \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {t_2}^{3}}, \mathsf{fma}\left(t_2, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + t_1, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(t_0 + {t_0}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + t_1 \cdot 1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	0.4
Cost	84680

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {t_1}^{3}}, \mathsf{fma}\left(t_1, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, t_0 + {t_0}^{3}, \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	0.4
Cost	65544

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, t_0 + {t_0}^{3}, \varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 4
Error	0.4
Cost	59144

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \varepsilon \cdot \left(t_1 + {t_1}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 5
Error	0.4
Cost	39172

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.266119606404827 \cdot 10^{-8}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot \left(0.5 + \cos \left(x + x\right) \cdot -0.5\right)}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 6
Error	0.5
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -5.266119606404827 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot \left(0.5 + \cos \left(x + x\right) \cdot -0.5\right)}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.5
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.266119606404827 \cdot 10^{-8}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot \left(0.5 + \cos \left(x + x\right) \cdot -0.5\right)}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 8
Error	14.4
Cost	20360

\[\begin{array}{l} t_0 := \left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot \left(0.5 + \cos \left(x + x\right) \cdot -0.5\right)}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	14.4
Cost	19976

\[\begin{array}{l} t_0 := \left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	14.5
Cost	19784

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := \left(\tan x + \tan \varepsilon\right) - \tan x\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot \left(0.5 + t_0 \cdot -0.5\right)}{0.5 + 0.5 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	14.6
Cost	14280

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;\varepsilon \leq -0.0081203423223153:\\ \;\;\;\;\tan \varepsilon - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot \left(0.5 + t_0 \cdot -0.5\right)}{0.5 + 0.5 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 12
Error	26.7
Cost	13256

\[\begin{array}{l} t_0 := \tan \varepsilon - \tan x\\ \mathbf{if}\;\varepsilon \leq -5.266119606404827 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.2260556432950395 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	26.6
Cost	12992

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

Alternative 14
Error	44.0
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022301 
(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)))