2cos (problem 3.3.5)

Average Error: 40.0 → 0.5

Time: 17.1s

Precision: binary64

Cost: 39112

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))))
   (if (<= eps -0.0055)
     (- (* (cos x) (+ -1.0 (cos eps))) t_0)
     (if (<= eps 0.0054)
       (-
        (*
         (cos x)
         (+ (* -0.5 (* eps eps)) (* (pow eps 4.0) 0.041666666666666664)))
        t_0)
       (- (* (cos eps) (cos x)) (fma (sin eps) (sin x) (cos x)))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double tmp;
	if (eps <= -0.0055) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - t_0;
	} else if (eps <= 0.0054) {
		tmp = (cos(x) * ((-0.5 * (eps * eps)) + (pow(eps, 4.0) * 0.041666666666666664))) - t_0;
	} else {
		tmp = (cos(eps) * cos(x)) - fma(sin(eps), sin(x), cos(x));
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	tmp = 0.0
	if (eps <= -0.0055)
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - t_0);
	elseif (eps <= 0.0054)
		tmp = Float64(Float64(cos(x) * Float64(Float64(-0.5 * Float64(eps * eps)) + Float64((eps ^ 4.0) * 0.041666666666666664))) - t_0);
	else
		tmp = Float64(Float64(cos(eps) * cos(x)) - fma(sin(eps), sin(x), cos(x)));
	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[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0055], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[eps, 0.0054], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0054999999999999997

   1. Initial program 30.2

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.8

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

   4. Applied egg-rr 0.8

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

   if -0.0054999999999999997 < eps < 0.0054000000000000003

   1. Initial program 49.5

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

   2. Applied egg-rr 12.3

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

   3. Applied egg-rr 12.3

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

   4. Applied egg-rr 0.1

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

   5. Simplified 0.1

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

      [Start] 0.1	\[ \frac{-{\sin \varepsilon}^{2}}{\frac{\cos \varepsilon + 1}{\cos x}} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-/l* [<=] 0.1	\[ \color{blue}{\frac{\left(-{\sin \varepsilon}^{2}\right) \cdot \cos x}{\cos \varepsilon + 1}} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      /-rgt-identity [<=] 0.1	\[ \frac{\left(-{\sin \varepsilon}^{2}\right) \cdot \cos x}{\color{blue}{\frac{\cos \varepsilon + 1}{1}}} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-*l/ [<=] 0.1	\[ \color{blue}{\frac{-{\sin \varepsilon}^{2}}{\frac{\cos \varepsilon + 1}{1}} \cdot \cos x} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-/l* [<=] 0.1	\[ \color{blue}{\frac{\left(-{\sin \varepsilon}^{2}\right) \cdot 1}{\cos \varepsilon + 1}} \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-*r/ [<=] 0.1	\[ \color{blue}{\left(\left(-{\sin \varepsilon}^{2}\right) \cdot \frac{1}{\cos \varepsilon + 1}\right)} \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-lft-neg-out [=>] 0.1	\[ \color{blue}{\left(-{\sin \varepsilon}^{2} \cdot \frac{1}{\cos \varepsilon + 1}\right)} \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      neg-mul-1 [=>] 0.1	\[ \color{blue}{\left(-1 \cdot \left({\sin \varepsilon}^{2} \cdot \frac{1}{\cos \varepsilon + 1}\right)\right)} \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      metadata-eval [<=] 0.1	\[ \left(\color{blue}{\frac{1}{-1}} \cdot \left({\sin \varepsilon}^{2} \cdot \frac{1}{\cos \varepsilon + 1}\right)\right) \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-*r/ [=>] 0.1	\[ \left(\frac{1}{-1} \cdot \color{blue}{\frac{{\sin \varepsilon}^{2} \cdot 1}{\cos \varepsilon + 1}}\right) \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-rgt-identity [=>] 0.1	\[ \left(\frac{1}{-1} \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{\cos \varepsilon + 1}\right) \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      times-frac [<=] 0.1	\[ \color{blue}{\frac{1 \cdot {\sin \varepsilon}^{2}}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-lft-identity [=>] 0.1	\[ \frac{\color{blue}{{\sin \varepsilon}^{2}}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      neg-mul-1 [<=] 0.1	\[ \frac{{\sin \varepsilon}^{2}}{\color{blue}{-\left(\cos \varepsilon + 1\right)}} \cdot \cos x + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-/r/ [<=] 0.1	\[ \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{-\left(\cos \varepsilon + 1\right)}{\cos x}}} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   6. Taylor expanded in eps around 0 0.1

      \[\leadsto \color{blue}{\left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]

   7. Simplified 0.1

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

      [Start] 0.1	\[ \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      mul-1-neg [=>] 0.1	\[ \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \color{blue}{\left(-{\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      unsub-neg [=>] 0.1	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-*r* [=>] 0.1	\[ \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 0.1	\[ \left(\color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} - {\varepsilon}^{4} \cdot \left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right)\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      *-commutative [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\left(-0.16666666666666666 \cdot \cos x - -0.125 \cdot \cos x\right) \cdot {\varepsilon}^{4}}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-rgt-out-- [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\left(\cos x \cdot \left(-0.16666666666666666 - -0.125\right)\right)} \cdot {\varepsilon}^{4}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      metadata-eval [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \left(\cos x \cdot \color{blue}{-0.041666666666666664}\right) \cdot {\varepsilon}^{4}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      associate-*l* [=>] 0.1	\[ \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - \color{blue}{\cos x \cdot \left(-0.041666666666666664 \cdot {\varepsilon}^{4}\right)}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]
      distribute-lft-out-- [=>] 0.1	\[ \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} - -0.041666666666666664 \cdot {\varepsilon}^{4}\right)} + \sin \varepsilon \cdot \left(-\sin x\right) \]
      unpow2 [=>] 0.1	\[ \cos x \cdot \left(-0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - -0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin \varepsilon \cdot \left(-\sin x\right) \]

   if 0.0054000000000000003 < eps

   1. Initial program 30.7

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

   2. Applied egg-rr 0.8

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

   3. Simplified 0.8

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

      [Start] 0.8	\[ \cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \left(-\cos x\right)\right) \]
      *-commutative [=>] 0.8	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\left(-\sin x\right) \cdot \sin \varepsilon} + \left(-\cos x\right)\right) \]
      distribute-lft-neg-in [<=] 0.8	\[ \cos x \cdot \cos \varepsilon + \left(\color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right)\right) \]
      distribute-neg-out [=>] 0.8	\[ \cos x \cdot \cos \varepsilon + \color{blue}{\left(-\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)} \]
      unsub-neg [=>] 0.8	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
      *-commutative [=>] 0.8	\[ \color{blue}{\cos \varepsilon \cdot \cos x} - \left(\sin x \cdot \sin \varepsilon + \cos x\right) \]
      *-commutative [=>] 0.8	\[ \cos \varepsilon \cdot \cos x - \left(\color{blue}{\sin \varepsilon \cdot \sin x} + \cos x\right) \]
      fma-def [=>] 0.8	\[ \cos \varepsilon \cdot \cos x - \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	39168

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

Alternative 2
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0055:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - t_0\right) - \cos x\\ \end{array} \]

Alternative 3
Error	0.5
Cost	32840

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058:\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - t_0\\ \end{array} \]

Alternative 4
Error	0.4
Cost	26889

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0058 \lor \neg \left(\varepsilon \leq 0.0044\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + {\varepsilon}^{4} \cdot 0.041666666666666664\right) - t_0\\ \end{array} \]

Alternative 5
Error	0.5
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.000165 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\cos x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - t_0\\ \end{array} \]

Alternative 6
Error	15.2
Cost	13888

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

Alternative 7
Error	15.1
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.032:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4800000:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 8
Error	14.9
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00045 \lor \neg \left(\varepsilon \leq 0.0062\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9
Error	21.6
Cost	13388

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{-138}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	22.0
Cost	7372

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-134}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) - \sin \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	32.7
Cost	7120

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -7.4 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{-182}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	22.2
Cost	6988

\[\begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000175:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{-124}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	49.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14
Error	52.7
Cost	256

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

Alternative 15
Error	55.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023012 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))