2cos (problem 3.3.5)

Average Accuracy: 38.8% → 99.2%

Time: 21.2s

Precision: binary64

Cost: 39108

Math

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

↓

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0046:\\ \;\;\;\;\cos x \cdot \log \left(e^{t_0}\right) - t_1\\ \mathbf{elif}\;\varepsilon \leq 0.0056:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0 - t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* (sin eps) (sin x))))
   (if (<= eps -0.0046)
     (- (* (cos x) (log (exp t_0))) t_1)
     (if (<= eps 0.0056)
       (-
        (*
         (cos x)
         (fma 0.041666666666666664 (pow eps 4.0) (* -0.5 (* eps eps))))
        t_1)
       (- (* (cos x) t_0) t_1)))))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

↓

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double t_1 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0046) {
		tmp = (cos(x) * log(exp(t_0))) - t_1;
	} else if (eps <= 0.0056) {
		tmp = (cos(x) * fma(0.041666666666666664, pow(eps, 4.0), (-0.5 * (eps * eps)))) - t_1;
	} else {
		tmp = (cos(x) * t_0) - t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

↓

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0046)
		tmp = Float64(Float64(cos(x) * log(exp(t_0))) - t_1);
	elseif (eps <= 0.0056)
		tmp = Float64(Float64(cos(x) * fma(0.041666666666666664, (eps ^ 4.0), Float64(-0.5 * Float64(eps * eps)))) - t_1);
	else
		tmp = Float64(Float64(cos(x) * t_0) - t_1);
	end
	return tmp
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0046], N[(N[(N[Cos[x], $MachinePrecision] * N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[eps, 0.0056], N[(N[(N[Cos[x], $MachinePrecision] * N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0045999999999999999

   1. Initial program 52.3%

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

   2. Applied egg-rr 98.7%

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

      [Start] 52.3	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 52.3	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 52.3	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 98.7	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 98.7	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 98.7	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      +-commutative [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 98.7	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      neg-mul-1 [=>] 98.7	\[ \left(\cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-out [=>] 98.7	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   4. Applied egg-rr 98.6%

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

      [Start] 98.7	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      add-log-exp [=>] 98.6	\[ \cos x \cdot \color{blue}{\log \left(e^{\cos \varepsilon + -1}\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   if -0.0045999999999999999 < eps < 0.00559999999999999994

   1. Initial program 23.7%

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

   2. Applied egg-rr 82.0%

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

      [Start] 23.7	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 23.7	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 23.7	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 24.8	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 24.8	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 82.0	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 82.0	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Applied egg-rr 82.0%

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

      [Start] 82.0	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      +-commutative [=>] 82.0	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 82.0	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      neg-mul-1 [=>] 82.0	\[ \left(\cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-out [=>] 82.0	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   4. Taylor expanded in eps around 0 99.8%

      \[\leadsto \cos x \cdot \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   5. Simplified 99.8%

      \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      Proof

      [Start] 99.8	\[ \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      fma-def [=>] 99.8	\[ \cos x \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot {\varepsilon}^{2}\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      unpow2 [=>] 99.8	\[ \cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]

   if 0.00559999999999999994 < eps

   1. Initial program 53.8%

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

   2. Applied egg-rr 98.7%

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

      [Start] 53.8	\[ \cos \left(x + \varepsilon\right) - \cos x \]
      sub-neg [=>] 53.8	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
      +-commutative [=>] 53.8	\[ \color{blue}{\left(-\cos x\right) + \cos \left(x + \varepsilon\right)} \]
      cos-sum [=>] 98.7	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \]
      cancel-sign-sub-inv [=>] 98.7	\[ \left(-\cos x\right) + \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\sin x\right) \cdot \sin \varepsilon\right)} \]
      associate-+r+ [=>] 98.7	\[ \color{blue}{\left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
      *-commutative [=>] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \color{blue}{\sin \varepsilon \cdot \left(-\sin x\right)} \]

   3. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \left(\left(-\cos x\right) + \cos x \cdot \cos \varepsilon\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      +-commutative [=>] 98.7	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon + \left(-\cos x\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 98.7	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} + \left(-\cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      neg-mul-1 [=>] 98.7	\[ \left(\cos \varepsilon \cdot \cos x + \color{blue}{-1 \cdot \cos x}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-out [=>] 98.7	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   4. Applied egg-rr 98.7%

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

      [Start] 98.7	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-neg-out [=>] 98.7	\[ \cos x \cdot \left(\cos \varepsilon + -1\right) + \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)} \]
      unsub-neg [=>] 98.7	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x} \]
      *-commutative [=>] 98.7	\[ \color{blue}{\left(\cos \varepsilon + -1\right) \cdot \cos x} - \sin \varepsilon \cdot \sin x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0046:\\ \;\;\;\;\cos x \cdot \log \left(e^{-1 + \cos \varepsilon}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0056:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	39168

\[\frac{{\sin \varepsilon}^{2}}{\frac{-1 - \cos \varepsilon}{\cos x}} - \sin \varepsilon \cdot \sin x \]

Alternative 2
Accuracy	99.2%
Cost	39108

\[\begin{array}{l} t_0 := \cos x \cdot \left(-1 + \cos \varepsilon\right)\\ t_1 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0049:\\ \;\;\;\;t_0 - \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0056:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_1\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	33161

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0049 \lor \neg \left(\varepsilon \leq 0.0056\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-25} \lor \neg \left(x \leq 3.4 \cdot 10^{-33}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	76.1%
Cost	14152

\[\begin{array}{l} t_0 := x + \left(\varepsilon + x\right)\\ \mathbf{if}\;\varepsilon \leq -1.95:\\ \;\;\;\;\frac{\cos x - \cos t_0}{-1}\\ \mathbf{elif}\;\varepsilon \leq 0.00285:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(x + \left(\varepsilon - x\right)\right)\right) \cdot \sin \left(0.5 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	76.1%
Cost	13888

\[-2 \cdot \left(\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right)\right) \]

Alternative 7
Accuracy	74.0%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.95:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]

Alternative 8
Accuracy	74.1%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.95:\\ \;\;\;\;\frac{\cos x - \cos \left(x + \left(\varepsilon + x\right)\right)}{-1}\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]

Alternative 9
Accuracy	65.1%
Cost	13576

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.004:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -4 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]

Alternative 10
Accuracy	65.1%
Cost	13516

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000195:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -6.5 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \end{array} \]

Alternative 11
Accuracy	66.7%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -6.2 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 12
Accuracy	66.3%
Cost	7052

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000175:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -6.5 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Accuracy	47.1%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000175 \lor \neg \left(\varepsilon \leq 0.000205\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 14
Accuracy	20.8%
Cost	320

\[-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))