2cos (problem 3.3.5)

Average Error: 39.5 → 0.5

Time: 21.8s

Precision: binary64

Cost: 39624

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0056:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\sin x \cdot \left(-\sin \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -2 + \left(t_0 + \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \end{array} \]

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 -0.0056)
     (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
     (if (<= eps 0.0045)
       (+
        (* (sin x) (- (sin eps)))
        (*
         (cos x)
         (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (pow eps 2.0)))))
       (+ (* t_0 -2.0) (+ t_0 (* (cos x) (+ (cos eps) -1.0))))))))

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 tmp;
	if (eps <= -0.0056) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else if (eps <= 0.0045) {
		tmp = (sin(x) * -sin(eps)) + (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * pow(eps, 2.0))));
	} else {
		tmp = (t_0 * -2.0) + (t_0 + (cos(x) * (cos(eps) + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(eps) * sin(x)
    if (eps <= (-0.0056d0)) then
        tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)))
    else if (eps <= 0.0045d0) then
        tmp = (sin(x) * -sin(eps)) + (cos(x) * ((0.041666666666666664d0 * (eps ** 4.0d0)) + ((-0.5d0) * (eps ** 2.0d0))))
    else
        tmp = (t_0 * (-2.0d0)) + (t_0 + (cos(x) * (cos(eps) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

↓

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) * Math.sin(x);
	double tmp;
	if (eps <= -0.0056) {
		tmp = (Math.cos(x) * Math.cos(eps)) - (Math.cos(x) + (Math.sin(x) * Math.sin(eps)));
	} else if (eps <= 0.0045) {
		tmp = (Math.sin(x) * -Math.sin(eps)) + (Math.cos(x) * ((0.041666666666666664 * Math.pow(eps, 4.0)) + (-0.5 * Math.pow(eps, 2.0))));
	} else {
		tmp = (t_0 * -2.0) + (t_0 + (Math.cos(x) * (Math.cos(eps) + -1.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

↓

def code(x, eps):
	t_0 = math.sin(eps) * math.sin(x)
	tmp = 0
	if eps <= -0.0056:
		tmp = (math.cos(x) * math.cos(eps)) - (math.cos(x) + (math.sin(x) * math.sin(eps)))
	elif eps <= 0.0045:
		tmp = (math.sin(x) * -math.sin(eps)) + (math.cos(x) * ((0.041666666666666664 * math.pow(eps, 4.0)) + (-0.5 * math.pow(eps, 2.0))))
	else:
		tmp = (t_0 * -2.0) + (t_0 + (math.cos(x) * (math.cos(eps) + -1.0)))
	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))
	tmp = 0.0
	if (eps <= -0.0056)
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	elseif (eps <= 0.0045)
		tmp = Float64(Float64(sin(x) * Float64(-sin(eps))) + Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * (eps ^ 2.0)))));
	else
		tmp = Float64(Float64(t_0 * -2.0) + Float64(t_0 + Float64(cos(x) * Float64(cos(eps) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end

↓

function tmp_2 = code(x, eps)
	t_0 = sin(eps) * sin(x);
	tmp = 0.0;
	if (eps <= -0.0056)
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	elseif (eps <= 0.0045)
		tmp = (sin(x) * -sin(eps)) + (cos(x) * ((0.041666666666666664 * (eps ^ 4.0)) + (-0.5 * (eps ^ 2.0))));
	else
		tmp = (t_0 * -2.0) + (t_0 + (cos(x) * (cos(eps) + -1.0)));
	end
	tmp_2 = 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, -0.0056], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0045], N[(N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * -2.0), $MachinePrecision] + N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00559999999999999994

   1. Initial program 30.9

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

   2. Applied egg-rr 0.8

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

   3. Taylor expanded in x around -inf 0.8

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

   if -0.00559999999999999994 < eps < 0.00449999999999999966

   1. Initial program 49.0

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

   2. Applied egg-rr 11.7

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

   3. Simplified 11.7

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

      [Start] 11.7	\[ \left(\left(-\sin x \cdot \sin \varepsilon\right) + \left(-\sin x \cdot \sin \varepsilon\right)\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-11 [=>] 11.7	\[ \left(\color{blue}{-1 \cdot \left(\sin x \cdot \sin \varepsilon\right)} + \left(-\sin x \cdot \sin \varepsilon\right)\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-2 [=>] 11.7	\[ \left(\color{blue}{\left(\sin x \cdot \sin \varepsilon\right) \cdot -1} + \left(-\sin x \cdot \sin \varepsilon\right)\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-11 [=>] 11.7	\[ \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot -1 + \color{blue}{-1 \cdot \left(\sin x \cdot \sin \varepsilon\right)}\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-2 [=>] 11.7	\[ \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot -1 + \color{blue}{\left(\sin x \cdot \sin \varepsilon\right) \cdot -1}\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-53 [=>] 11.7	\[ \color{blue}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(-1 + -1\right)} + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-2 [=>] 11.7	\[ \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)} \cdot \left(-1 + -1\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      metadata-eval [=>] 11.7	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot \color{blue}{-2} + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-13 [=>] 11.7	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \color{blue}{\left(0 - \sin x \cdot \sin \varepsilon\right)}\right) \]
      rational_best-simplify-47 [=>] 11.7	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \color{blue}{\left(\sin x \cdot \sin \varepsilon + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - 0\right)\right)} \]
      rational_best-simplify-2 [=>] 11.7	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \left(\color{blue}{\sin \varepsilon \cdot \sin x} + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - 0\right)\right) \]
      rational_best-simplify-6 [=>] 11.7	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \left(\sin \varepsilon \cdot \sin x + \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

   4. Taylor expanded in eps around inf 11.7

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

   5. Simplified 11.7

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

      [Start] 11.7	\[ -2 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \left(\cos x \cdot \left(\cos \varepsilon - 1\right) + \sin x \cdot \sin \varepsilon\right) \]
      rational_best-simplify-1 [=>] 11.7	\[ -2 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \color{blue}{\left(\sin x \cdot \sin \varepsilon + \cos x \cdot \left(\cos \varepsilon - 1\right)\right)} \]
      rational_best-simplify-2 [=>] 11.7	\[ -2 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \left(\color{blue}{\sin \varepsilon \cdot \sin x} + \cos x \cdot \left(\cos \varepsilon - 1\right)\right) \]
      rational_best-simplify-2 [=>] 11.7	\[ -2 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \left(\sin \varepsilon \cdot \sin x + \color{blue}{\left(\cos \varepsilon - 1\right) \cdot \cos x}\right) \]
      rational_best-simplify-18 [<=] 11.7	\[ -2 \cdot \left(\sin x \cdot \sin \varepsilon\right) + \left(\sin \varepsilon \cdot \sin x + \color{blue}{\left(\cos \varepsilon + -1\right)} \cdot \cos x\right) \]
      rational_best-simplify-43 [=>] 11.7	\[ \color{blue}{\left(\cos \varepsilon + -1\right) \cdot \cos x + \left(\sin \varepsilon \cdot \sin x + -2 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right)} \]
      rational_best-simplify-1 [=>] 11.7	\[ \color{blue}{\left(\sin \varepsilon \cdot \sin x + -2 \cdot \left(\sin x \cdot \sin \varepsilon\right)\right) + \left(\cos \varepsilon + -1\right) \cdot \cos x} \]

   6. Taylor expanded in eps around 0 0.1

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

   7. Simplified 0.1

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

      [Start] 0.1	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) \]
      rational_best-simplify-1 [=>] 0.1	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right)} \]
      rational_best-simplify-2 [=>] 0.1	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \left(-0.5 \cdot \color{blue}{\left(\cos x \cdot {\varepsilon}^{2}\right)} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational_best-simplify-46 [=>] 0.1	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \left(\color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) \]
      rational_best-simplify-2 [=>] 0.1	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + 0.041666666666666664 \cdot \color{blue}{\left(\cos x \cdot {\varepsilon}^{4}\right)}\right) \]
      rational_best-simplify-46 [=>] 0.1	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \left(\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) + \color{blue}{\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4}\right)}\right) \]
      rational_best-simplify-53 [=>] 0.1	\[ \sin x \cdot \left(-\sin \varepsilon\right) + \color{blue}{\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)} \]

   if 0.00449999999999999966 < eps

   1. Initial program 29.4

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

   2. Applied egg-rr 0.8

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

   3. Simplified 0.8

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

      [Start] 0.8	\[ \left(\left(-\sin x \cdot \sin \varepsilon\right) + \left(-\sin x \cdot \sin \varepsilon\right)\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-11 [=>] 0.8	\[ \left(\color{blue}{-1 \cdot \left(\sin x \cdot \sin \varepsilon\right)} + \left(-\sin x \cdot \sin \varepsilon\right)\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-2 [=>] 0.8	\[ \left(\color{blue}{\left(\sin x \cdot \sin \varepsilon\right) \cdot -1} + \left(-\sin x \cdot \sin \varepsilon\right)\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-11 [=>] 0.8	\[ \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot -1 + \color{blue}{-1 \cdot \left(\sin x \cdot \sin \varepsilon\right)}\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-2 [=>] 0.8	\[ \left(\left(\sin x \cdot \sin \varepsilon\right) \cdot -1 + \color{blue}{\left(\sin x \cdot \sin \varepsilon\right) \cdot -1}\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-53 [=>] 0.8	\[ \color{blue}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(-1 + -1\right)} + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-2 [=>] 0.8	\[ \color{blue}{\left(\sin \varepsilon \cdot \sin x\right)} \cdot \left(-1 + -1\right) + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      metadata-eval [=>] 0.8	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot \color{blue}{-2} + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \left(-\sin x \cdot \sin \varepsilon\right)\right) \]
      rational_best-simplify-13 [=>] 0.8	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - \color{blue}{\left(0 - \sin x \cdot \sin \varepsilon\right)}\right) \]
      rational_best-simplify-47 [=>] 0.8	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \color{blue}{\left(\sin x \cdot \sin \varepsilon + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - 0\right)\right)} \]
      rational_best-simplify-2 [=>] 0.8	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \left(\color{blue}{\sin \varepsilon \cdot \sin x} + \left(\cos x \cdot \left(\cos \varepsilon + -1\right) - 0\right)\right) \]
      rational_best-simplify-6 [=>] 0.8	\[ \left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \left(\sin \varepsilon \cdot \sin x + \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0056:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\sin x \cdot \left(-\sin \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \varepsilon \cdot \sin x\right) \cdot -2 + \left(\sin \varepsilon \cdot \sin x + \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	33288

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0052:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0045:\\ \;\;\;\;\sin x \cdot \left(-\sin \varepsilon\right) + \cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 2
Error	0.5
Cost	32708

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000112:\\ \;\;\;\;\sin x \cdot \left(-\sin \varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 3
Error	0.5
Cost	26568

\[\begin{array}{l} t_0 := \left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000112:\\ \;\;\;\;\sin x \cdot \left(-\sin \varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.6
Cost	26440

\[\begin{array}{l} t_0 := \left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.3
Cost	20168

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\sin \varepsilon\right) + \left(\cos \varepsilon - 1\right)\\ \mathbf{if}\;\varepsilon \leq -0.0048:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0018:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.5
Cost	19976

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\sin \varepsilon\right) + \left(\cos \varepsilon - 1\right)\\ \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000112:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.1
Cost	13640

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -205:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4400:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	20.9
Cost	13256

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.85 \cdot 10^{+14}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	32.5
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -3.9 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -6.3 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-97}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00085:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	20.9
Cost	6920

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -4.9 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00085:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	35.8
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	48.6
Cost	520

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 1.95 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 13
Error	51.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.8 \cdot 10^{-83}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-97}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 14
Error	57.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023097 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))