2cos (problem 3.3.5)

Average Error: 39.7 → 0.5

Time: 15.2s

Precision: binary64

Cost: 66312

Math

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

↓

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.105:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.1:\\ \;\;\;\;\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(2.48015873015873 \cdot 10^{-5} \cdot \left({\varepsilon}^{8} \cdot \cos x\right) + \left(-0.001388888888888889 \cdot \left({\varepsilon}^{6} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)\right) + \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{{\cos x}^{2} \cdot t_0}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0\\ \end{array} - t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* (sin x) (sin eps))))
   (if (<= eps -0.105)
     (- (- (* (cos x) (cos eps)) t_1) (cos x))
     (if (<= eps 0.1)
       (+
        (+
         (* 0.041666666666666664 (* (pow eps 4.0) (cos x)))
         (+
          (* 2.48015873015873e-5 (* (pow eps 8.0) (cos x)))
          (+
           (* -0.001388888888888889 (* (pow eps 6.0) (cos x)))
           (* -0.5 (* (pow eps 2.0) (cos x))))))
        (* (sin x) (- (sin eps))))
       (-
        (if (!= (- (cos x)) 0.0)
          (/ (* (pow (cos x) 2.0) t_0) (cos x))
          (* (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 = cos(eps) + -1.0;
	double t_1 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.105) {
		tmp = ((cos(x) * cos(eps)) - t_1) - cos(x);
	} else if (eps <= 0.1) {
		tmp = ((0.041666666666666664 * (pow(eps, 4.0) * cos(x))) + ((2.48015873015873e-5 * (pow(eps, 8.0) * cos(x))) + ((-0.001388888888888889 * (pow(eps, 6.0) * cos(x))) + (-0.5 * (pow(eps, 2.0) * cos(x)))))) + (sin(x) * -sin(eps));
	} else {
		double tmp_1;
		if (-cos(x) != 0.0) {
			tmp_1 = (pow(cos(x), 2.0) * t_0) / cos(x);
		} else {
			tmp_1 = cos(x) * t_0;
		}
		tmp = tmp_1 - t_1;
	}
	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) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    t_0 = cos(eps) + (-1.0d0)
    t_1 = sin(x) * sin(eps)
    if (eps <= (-0.105d0)) then
        tmp = ((cos(x) * cos(eps)) - t_1) - cos(x)
    else if (eps <= 0.1d0) then
        tmp = ((0.041666666666666664d0 * ((eps ** 4.0d0) * cos(x))) + ((2.48015873015873d-5 * ((eps ** 8.0d0) * cos(x))) + (((-0.001388888888888889d0) * ((eps ** 6.0d0) * cos(x))) + ((-0.5d0) * ((eps ** 2.0d0) * cos(x)))))) + (sin(x) * -sin(eps))
    else
        if (-cos(x) /= 0.0d0) then
            tmp_1 = ((cos(x) ** 2.0d0) * t_0) / cos(x)
        else
            tmp_1 = cos(x) * t_0
        end if
        tmp = tmp_1 - t_1
    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.cos(eps) + -1.0;
	double t_1 = Math.sin(x) * Math.sin(eps);
	double tmp;
	if (eps <= -0.105) {
		tmp = ((Math.cos(x) * Math.cos(eps)) - t_1) - Math.cos(x);
	} else if (eps <= 0.1) {
		tmp = ((0.041666666666666664 * (Math.pow(eps, 4.0) * Math.cos(x))) + ((2.48015873015873e-5 * (Math.pow(eps, 8.0) * Math.cos(x))) + ((-0.001388888888888889 * (Math.pow(eps, 6.0) * Math.cos(x))) + (-0.5 * (Math.pow(eps, 2.0) * Math.cos(x)))))) + (Math.sin(x) * -Math.sin(eps));
	} else {
		double tmp_1;
		if (-Math.cos(x) != 0.0) {
			tmp_1 = (Math.pow(Math.cos(x), 2.0) * t_0) / Math.cos(x);
		} else {
			tmp_1 = Math.cos(x) * t_0;
		}
		tmp = tmp_1 - t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	t_1 = math.sin(x) * math.sin(eps)
	tmp = 0
	if eps <= -0.105:
		tmp = ((math.cos(x) * math.cos(eps)) - t_1) - math.cos(x)
	elif eps <= 0.1:
		tmp = ((0.041666666666666664 * (math.pow(eps, 4.0) * math.cos(x))) + ((2.48015873015873e-5 * (math.pow(eps, 8.0) * math.cos(x))) + ((-0.001388888888888889 * (math.pow(eps, 6.0) * math.cos(x))) + (-0.5 * (math.pow(eps, 2.0) * math.cos(x)))))) + (math.sin(x) * -math.sin(eps))
	else:
		tmp_1 = 0
		if -math.cos(x) != 0.0:
			tmp_1 = (math.pow(math.cos(x), 2.0) * t_0) / math.cos(x)
		else:
			tmp_1 = math.cos(x) * t_0
		tmp = tmp_1 - 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(cos(eps) + -1.0)
	t_1 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.105)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - t_1) - cos(x));
	elseif (eps <= 0.1)
		tmp = Float64(Float64(Float64(0.041666666666666664 * Float64((eps ^ 4.0) * cos(x))) + Float64(Float64(2.48015873015873e-5 * Float64((eps ^ 8.0) * cos(x))) + Float64(Float64(-0.001388888888888889 * Float64((eps ^ 6.0) * cos(x))) + Float64(-0.5 * Float64((eps ^ 2.0) * cos(x)))))) + Float64(sin(x) * Float64(-sin(eps))));
	else
		tmp_1 = 0.0
		if (Float64(-cos(x)) != 0.0)
			tmp_1 = Float64(Float64((cos(x) ^ 2.0) * t_0) / cos(x));
		else
			tmp_1 = Float64(cos(x) * t_0);
		end
		tmp = Float64(tmp_1 - t_1);
	end
	return tmp
end

MATLAB

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

↓

function tmp_3 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	t_1 = sin(x) * sin(eps);
	tmp = 0.0;
	if (eps <= -0.105)
		tmp = ((cos(x) * cos(eps)) - t_1) - cos(x);
	elseif (eps <= 0.1)
		tmp = ((0.041666666666666664 * ((eps ^ 4.0) * cos(x))) + ((2.48015873015873e-5 * ((eps ^ 8.0) * cos(x))) + ((-0.001388888888888889 * ((eps ^ 6.0) * cos(x))) + (-0.5 * ((eps ^ 2.0) * cos(x)))))) + (sin(x) * -sin(eps));
	else
		tmp_2 = 0.0;
		if (-cos(x) ~= 0.0)
			tmp_2 = ((cos(x) ^ 2.0) * t_0) / cos(x);
		else
			tmp_2 = cos(x) * t_0;
		end
		tmp = tmp_2 - t_1;
	end
	tmp_3 = 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[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.105], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.1], N[(N[(N[(0.041666666666666664 * N[(N[Power[eps, 4.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.48015873015873e-5 * N[(N[Power[eps, 8.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.001388888888888889 * N[(N[Power[eps, 6.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[eps, 2.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(If[Unequal[(-N[Cos[x], $MachinePrecision]), 0.0], N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.104999999999999996

   1. Initial program 29.6

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

   2. Applied egg-rr 0.8

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

   if -0.104999999999999996 < eps < 0.10000000000000001

   1. Initial program 49.9

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

   2. Applied egg-rr 11.8

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

   3. Taylor expanded in eps around 0 0.1

      \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(2.48015873015873 \cdot 10^{-5} \cdot \left({\varepsilon}^{8} \cdot \cos x\right) + \left(-0.001388888888888889 \cdot \left({\varepsilon}^{6} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)\right)} + \sin x \cdot \left(-\sin \varepsilon\right) \]

   if 0.10000000000000001 < eps

   1. Initial program 30.0

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

   2. Applied egg-rr 0.8

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

   3. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\cos x \cdot \left(\cos x \cdot \left(-1 - \left(-\cos \varepsilon\right)\right)\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 - \left(-\cos \varepsilon\right)\right)\\ } \end{array}} + \sin x \cdot \left(-\sin \varepsilon\right) \]

   4. Simplified 0.8

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\left(\cos \varepsilon + -1\right) \cdot \left(\cos x \cdot \cos x\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ } \end{array}} + \sin x \cdot \left(-\sin \varepsilon\right) \]
      Proof

      [Start] 0.8	\[ \begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\cos x \cdot \left(\cos x \cdot \left(-1 - \left(-\cos \varepsilon\right)\right)\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 - \left(-\cos \varepsilon\right)\right)\\ \end{array} + \sin x \cdot \left(-\sin \varepsilon\right) \]
      rational_best-simplify-3 [=>] 0.8	\[ \begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\cos x \cdot \color{blue}{\left(\left(-1 - \left(-\cos \varepsilon\right)\right) \cdot \cos x\right)}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 - \left(-\cos \varepsilon\right)\right)\\ \end{array} + \sin x \cdot \left(-\sin \varepsilon\right) \]
      rational_best-simplify-113 [=>] 0.8	\[ \begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\color{blue}{\left(-1 - \left(-\cos \varepsilon\right)\right) \cdot \left(\cos x \cdot \cos x\right)}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 - \left(-\cos \varepsilon\right)\right)\\ \end{array} + \sin x \cdot \left(-\sin \varepsilon\right) \]
      rational_best-simplify-63 [=>] 0.8	\[ \begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\color{blue}{\left(\cos \varepsilon + -1\right)} \cdot \left(\cos x \cdot \cos x\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(-1 - \left(-\cos \varepsilon\right)\right)\\ \end{array} + \sin x \cdot \left(-\sin \varepsilon\right) \]
      rational_best-simplify-63 [=>] 0.8	\[ \begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\left(\cos \varepsilon + -1\right) \cdot \left(\cos x \cdot \cos x\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right)}\\ \end{array} + \sin x \cdot \left(-\sin \varepsilon\right) \]

   5. Taylor expanded in x around inf 0.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{\left(\cos \varepsilon + -1\right) \cdot \color{blue}{{\cos x}^{2}}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} + \sin x \cdot \left(-\sin \varepsilon\right) \]

   6. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{{\cos x}^{2} \cdot \left(\cos \varepsilon + -1\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} - \sin x \cdot \sin \varepsilon} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.105:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.1:\\ \;\;\;\;\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(2.48015873015873 \cdot 10^{-5} \cdot \left({\varepsilon}^{8} \cdot \cos x\right) + \left(-0.001388888888888889 \cdot \left({\varepsilon}^{6} \cdot \cos x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)\right) + \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{{\cos x}^{2} \cdot \left(\cos \varepsilon + -1\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} - \sin x \cdot \sin \varepsilon\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	46028

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;-\cos x \ne 0:\\ \;\;\;\;\frac{{\cos x}^{2} \cdot t_0}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot t_0\\ \end{array} - t_1\\ \end{array} \]

Alternative 2
Error	0.6
Cost	32708

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) \cdot \cos x - t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	32708

\[\begin{array}{l} t_0 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) \cdot \cos x - t_0\\ \end{array} \]

Alternative 4
Error	0.6
Cost	26440

\[\begin{array}{l} t_0 := \left(\cos \varepsilon + -1\right) \cdot \cos x - \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.7 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.0
Cost	26376

\[\begin{array}{l} t_0 := \left(\cos \varepsilon - \cos x\right) + \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.032:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.0
Cost	26376

\[\begin{array}{l} t_0 := \sin x \cdot \left(-\sin \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -0.012:\\ \;\;\;\;\left(t_0 - \cos x\right) + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.059:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon - \cos x\right) + t_0\\ \end{array} \]

Alternative 7
Error	21.2
Cost	20308

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-\sin x\right)\\ t_1 := \varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.00012:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq -1.3 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 4.9 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3.45 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \left(1 + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 8
Error	21.2
Cost	20176

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ t_1 := \cos \varepsilon + -1\\ t_2 := t_1 - \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000166:\\ \;\;\;\;\cos x \cdot t_1\\ \mathbf{elif}\;\varepsilon \leq -2.3 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 4.1 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	14.4
Cost	20168

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right) + -0.5 \cdot \left(\cos x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \left(1 + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 10
Error	21.4
Cost	13780

\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right)\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ t_2 := \varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.0001:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -4.8 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 5.3 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7.6 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	21.7
Cost	13652

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ t_2 := \varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.000135:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.6 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 5.3 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	22.1
Cost	7504

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ t_2 := \varepsilon \cdot \left(-x\right) + -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.000104:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -9.6 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 4.9 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{-57}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	23.0
Cost	7316

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(-\sin x\right)\\ t_1 := \cos \varepsilon - 1\\ t_2 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -1.55 \cdot 10^{-76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 5.3 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{-35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	32.6
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.5 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-149}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.03:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	35.9
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	53.0
Cost	256

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

Reproduce

herbie shell --seed 2023104 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))