2cos (problem 3.3.5)

Average Error: 62.05% → 0.75%

Time: 20.9s

Precision: binary64

Cost: 151432

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ t_1 := \mathsf{fma}\left(-\sin \varepsilon, \sin x, t_0\right)\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \mathsf{fma}\left(\sin \varepsilon, -\sin x, t_1\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00175:\\ \;\;\;\;\mathsf{log1p}\left({\varepsilon}^{3} \cdot \left(-0.16666666666666666 \cdot {\sin x}^{3} + \left(0.5 \cdot \left(\cos x \cdot \sin x\right) + \sin x \cdot 0.16666666666666666\right)\right) + \left({\varepsilon}^{4} \cdot \left(\cos x \cdot 0.041666666666666664 + \left(-0.25 \cdot \left(\cos x \cdot t_2\right) + \left(0.125 \cdot {\cos x}^{2} + \left(0.041666666666666664 \cdot {\sin x}^{4} + -0.16666666666666666 \cdot t_2\right)\right)\right)\right) + \left({\varepsilon}^{2} \cdot \left(0.5 \cdot t_2 + \cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x)))
        (t_1 (fma (- (sin eps)) (sin x) t_0))
        (t_2 (pow (sin x) 2.0)))
   (if (<= eps -0.00155)
     (- (fma (cos x) (cos eps) (fma (sin eps) (- (sin x)) t_1)) (cos x))
     (if (<= eps 0.00175)
       (log1p
        (+
         (*
          (pow eps 3.0)
          (+
           (* -0.16666666666666666 (pow (sin x) 3.0))
           (+ (* 0.5 (* (cos x) (sin x))) (* (sin x) 0.16666666666666666))))
         (+
          (*
           (pow eps 4.0)
           (+
            (* (cos x) 0.041666666666666664)
            (+
             (* -0.25 (* (cos x) t_2))
             (+
              (* 0.125 (pow (cos x) 2.0))
              (+
               (* 0.041666666666666664 (pow (sin x) 4.0))
               (* -0.16666666666666666 t_2))))))
          (-
           (* (pow eps 2.0) (+ (* 0.5 t_2) (* (cos x) -0.5)))
           (* eps (sin x))))))
       (+ t_1 (- (* (cos x) (+ (cos eps) -1.0)) t_0))))))

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 t_1 = fma(-sin(eps), sin(x), t_0);
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (eps <= -0.00155) {
		tmp = fma(cos(x), cos(eps), fma(sin(eps), -sin(x), t_1)) - cos(x);
	} else if (eps <= 0.00175) {
		tmp = log1p(((pow(eps, 3.0) * ((-0.16666666666666666 * pow(sin(x), 3.0)) + ((0.5 * (cos(x) * sin(x))) + (sin(x) * 0.16666666666666666)))) + ((pow(eps, 4.0) * ((cos(x) * 0.041666666666666664) + ((-0.25 * (cos(x) * t_2)) + ((0.125 * pow(cos(x), 2.0)) + ((0.041666666666666664 * pow(sin(x), 4.0)) + (-0.16666666666666666 * t_2)))))) + ((pow(eps, 2.0) * ((0.5 * t_2) + (cos(x) * -0.5))) - (eps * sin(x))))));
	} else {
		tmp = t_1 + ((cos(x) * (cos(eps) + -1.0)) - t_0);
	}
	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))
	t_1 = fma(Float64(-sin(eps)), sin(x), t_0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (eps <= -0.00155)
		tmp = Float64(fma(cos(x), cos(eps), fma(sin(eps), Float64(-sin(x)), t_1)) - cos(x));
	elseif (eps <= 0.00175)
		tmp = log1p(Float64(Float64((eps ^ 3.0) * Float64(Float64(-0.16666666666666666 * (sin(x) ^ 3.0)) + Float64(Float64(0.5 * Float64(cos(x) * sin(x))) + Float64(sin(x) * 0.16666666666666666)))) + Float64(Float64((eps ^ 4.0) * Float64(Float64(cos(x) * 0.041666666666666664) + Float64(Float64(-0.25 * Float64(cos(x) * t_2)) + Float64(Float64(0.125 * (cos(x) ^ 2.0)) + Float64(Float64(0.041666666666666664 * (sin(x) ^ 4.0)) + Float64(-0.16666666666666666 * t_2)))))) + Float64(Float64((eps ^ 2.0) * Float64(Float64(0.5 * t_2) + Float64(cos(x) * -0.5))) - Float64(eps * sin(x))))));
	else
		tmp = Float64(t_1 + Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - t_0));
	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]}, Block[{t$95$1 = N[((-N[Sin[eps], $MachinePrecision]) * N[Sin[x], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[eps, -0.00155], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + t$95$1), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00175], N[Log[1 + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] * 0.041666666666666664), $MachinePrecision] + N[(N[(-0.25 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(0.125 * N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(0.5 * t$95$2), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00154999999999999995

   1. Initial program 49.28

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

   2. Applied egg-rr 1.34

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

   3. Simplified 1.35

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

      [Start] 1.34	\[ \left(\cos x \cdot \cos \varepsilon + \left(\sin \varepsilon \cdot \left(-\sin x\right) + \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin x \cdot \sin \varepsilon\right)\right)\right) - \cos x \]
      fma-def [=>] 1.31	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin \varepsilon \cdot \left(-\sin x\right) + \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin x \cdot \sin \varepsilon\right)\right)} - \cos x \]
      fma-def [=>] 1.35	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, \color{blue}{\mathsf{fma}\left(\sin \varepsilon, -\sin x, \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin x \cdot \sin \varepsilon\right)\right)}\right) - \cos x \]
      *-commutative [=>] 1.35	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, \mathsf{fma}\left(\sin \varepsilon, -\sin x, \mathsf{fma}\left(-\sin \varepsilon, \sin x, \color{blue}{\sin \varepsilon \cdot \sin x}\right)\right)\right) - \cos x \]

   if -0.00154999999999999995 < eps < 0.00175000000000000004

   1. Initial program 76.91

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

   2. Applied egg-rr 76.91

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

   3. Taylor expanded in eps around 0 0.22

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{\varepsilon}^{3} \cdot \left(-0.16666666666666666 \cdot {\sin x}^{3} + \left(0.5 \cdot \left(\cos x \cdot \sin x\right) + 0.16666666666666666 \cdot \sin x\right)\right) + \left({\varepsilon}^{4} \cdot \left(0.041666666666666664 \cdot \cos x + \left(-0.25 \cdot \left(\cos x \cdot {\sin x}^{2}\right) + \left(0.125 \cdot {\cos x}^{2} + \left(0.041666666666666664 \cdot {\sin x}^{4} + -0.16666666666666666 \cdot {\sin x}^{2}\right)\right)\right)\right) + \left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + {\varepsilon}^{2} \cdot \left(0.5 \cdot {\sin x}^{2} + -0.5 \cdot \cos x\right)\right)\right)}\right) \]

   if 0.00175000000000000004 < eps

   1. Initial program 45.71

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

   2. Applied egg-rr 1.17

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

   3. Applied egg-rr 1.18

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

   4. Applied egg-rr 1.18

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

   5. Simplified 1.17

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

      [Start] 1.18	\[ \left(\cos \varepsilon + -1\right) \cdot \cos x - \left(\sin \varepsilon \cdot \sin x - \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin \varepsilon \cdot \sin x\right)\right) \]
      associate--r- [=>] 1.17	\[ \color{blue}{\left(\left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\right) + \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin \varepsilon \cdot \sin x\right)} \]
      *-commutative [=>] 1.17	\[ \left(\color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)} - \sin \varepsilon \cdot \sin x\right) + \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin \varepsilon \cdot \sin x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.75

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \mathsf{fma}\left(\sin \varepsilon, -\sin x, \mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin \varepsilon \cdot \sin x\right)\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00175:\\ \;\;\;\;\mathsf{log1p}\left({\varepsilon}^{3} \cdot \left(-0.16666666666666666 \cdot {\sin x}^{3} + \left(0.5 \cdot \left(\cos x \cdot \sin x\right) + \sin x \cdot 0.16666666666666666\right)\right) + \left({\varepsilon}^{4} \cdot \left(\cos x \cdot 0.041666666666666664 + \left(-0.25 \cdot \left(\cos x \cdot {\sin x}^{2}\right) + \left(0.125 \cdot {\cos x}^{2} + \left(0.041666666666666664 \cdot {\sin x}^{4} + -0.16666666666666666 \cdot {\sin x}^{2}\right)\right)\right)\right) + \left({\varepsilon}^{2} \cdot \left(0.5 \cdot {\sin x}^{2} + \cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sin \varepsilon, \sin x, \sin \varepsilon \cdot \sin x\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.84%
Cost	77764

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ t_1 := \mathsf{fma}\left(-\sin \varepsilon, \sin x, t_0\right)\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \mathsf{fma}\left(\sin \varepsilon, -\sin x, t_1\right)\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\right)\\ \end{array} \]

Alternative 2
Error	0.85%
Cost	58888

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \varepsilon}{\frac{\sin x}{-{\sin x}^{2}}} + \left(\cos x \cdot \cos \varepsilon - \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 6.4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sin \varepsilon, \sin x, t_0\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\right)\\ \end{array} \]

Alternative 3
Error	0.89%
Cost	45764

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin \varepsilon}{\frac{\sin x}{-{\sin x}^{2}}} + \left(\cos x \cdot \cos \varepsilon - \cos x\right)\\ \mathbf{elif}\;\varepsilon \leq 6.4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \log \left(e^{\cos \varepsilon + -1}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4
Error	0.88%
Cost	45380

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \mathsf{fma}\left(-\sin x, \sin \varepsilon, -\cos x\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 6.4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \log \left(e^{\cos \varepsilon + -1}\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 5
Error	0.91%
Cost	39241

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \log \left(e^{\cos \varepsilon + -1}\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \end{array} \]

Alternative 6
Error	0.84%
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \end{array} \]

Alternative 7
Error	0.85%
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \end{array} \]

Alternative 8
Error	0.84%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 0.5, \sin x, \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.25\right)\right)\right) \cdot -2\\ \end{array} \]

Alternative 9
Error	23.31%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072 \lor \neg \left(\varepsilon \leq 3450\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 10
Error	23.88%
Cost	13632

\[-2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \]

Alternative 11
Error	23.66%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.01 \lor \neg \left(\varepsilon \leq 3450\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	24.4%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0026 \lor \neg \left(\varepsilon \leq 1850000\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	32.91%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.6 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 1.2 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14
Error	53.33%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.5 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.000156\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 15
Error	77.93%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.89:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos x\\ \end{array} \]

Alternative 16
Error	79.13%
Cost	320

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

Reproduce

herbie shell --seed 2023088 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))