2cos (problem 3.3.5)

Average Error: 39.8 → 0.6

Time: 19.4s

Precision: binary64

Cost: 39432

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \sin x + \left(\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \frac{\sin \varepsilon \cdot {\sin x}^{2}}{\sin x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.000165)
   (- (- (* (cos x) (cos eps)) (* (sin eps) (sin x))) (cos x))
   (if (<= eps 0.00016)
     (/
      (*
       (+
        (* eps (sin x))
        (+
         (* (cos x) (* eps (* eps 0.5)))
         (* (sin x) (* -0.16666666666666666 (pow eps 3.0)))))
       -2.0)
      2.0)
     (-
      (* (cos x) (+ (cos eps) -1.0))
      (/ (* (sin eps) (pow (sin x) 2.0)) (sin x))))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.000165) {
		tmp = ((cos(x) * cos(eps)) - (sin(eps) * sin(x))) - cos(x);
	} else if (eps <= 0.00016) {
		tmp = (((eps * sin(x)) + ((cos(x) * (eps * (eps * 0.5))) + (sin(x) * (-0.16666666666666666 * pow(eps, 3.0))))) * -2.0) / 2.0;
	} else {
		tmp = (cos(x) * (cos(eps) + -1.0)) - ((sin(eps) * pow(sin(x), 2.0)) / sin(x));
	}
	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) :: tmp
    if (eps <= (-0.000165d0)) then
        tmp = ((cos(x) * cos(eps)) - (sin(eps) * sin(x))) - cos(x)
    else if (eps <= 0.00016d0) then
        tmp = (((eps * sin(x)) + ((cos(x) * (eps * (eps * 0.5d0))) + (sin(x) * ((-0.16666666666666666d0) * (eps ** 3.0d0))))) * (-2.0d0)) / 2.0d0
    else
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - ((sin(eps) * (sin(x) ** 2.0d0)) / sin(x))
    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 tmp;
	if (eps <= -0.000165) {
		tmp = ((Math.cos(x) * Math.cos(eps)) - (Math.sin(eps) * Math.sin(x))) - Math.cos(x);
	} else if (eps <= 0.00016) {
		tmp = (((eps * Math.sin(x)) + ((Math.cos(x) * (eps * (eps * 0.5))) + (Math.sin(x) * (-0.16666666666666666 * Math.pow(eps, 3.0))))) * -2.0) / 2.0;
	} else {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - ((Math.sin(eps) * Math.pow(Math.sin(x), 2.0)) / Math.sin(x));
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	tmp = 0
	if eps <= -0.000165:
		tmp = ((math.cos(x) * math.cos(eps)) - (math.sin(eps) * math.sin(x))) - math.cos(x)
	elif eps <= 0.00016:
		tmp = (((eps * math.sin(x)) + ((math.cos(x) * (eps * (eps * 0.5))) + (math.sin(x) * (-0.16666666666666666 * math.pow(eps, 3.0))))) * -2.0) / 2.0
	else:
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - ((math.sin(eps) * math.pow(math.sin(x), 2.0)) / math.sin(x))
	return tmp

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.000165)
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - Float64(sin(eps) * sin(x))) - cos(x));
	elseif (eps <= 0.00016)
		tmp = Float64(Float64(Float64(Float64(eps * sin(x)) + Float64(Float64(cos(x) * Float64(eps * Float64(eps * 0.5))) + Float64(sin(x) * Float64(-0.16666666666666666 * (eps ^ 3.0))))) * -2.0) / 2.0);
	else
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(Float64(sin(eps) * (sin(x) ^ 2.0)) / sin(x)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.000165)
		tmp = ((cos(x) * cos(eps)) - (sin(eps) * sin(x))) - cos(x);
	elseif (eps <= 0.00016)
		tmp = (((eps * sin(x)) + ((cos(x) * (eps * (eps * 0.5))) + (sin(x) * (-0.16666666666666666 * (eps ^ 3.0))))) * -2.0) / 2.0;
	else
		tmp = (cos(x) * (cos(eps) + -1.0)) - ((sin(eps) * (sin(x) ^ 2.0)) / sin(x));
	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_] := If[LessEqual[eps, -0.000165], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00016], N[(N[(N[(N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(eps * N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(-0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[eps], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.65e-4

   1. Initial program 30.7

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

   2. Applied egg-rr 0.9

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

   if -1.65e-4 < eps < 1.60000000000000013e-4

   1. Initial program 49.5

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

   2. Applied egg-rr 49.5

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

   3. Taylor expanded in eps around 0 48.8

      \[\leadsto \frac{\color{blue}{\left(\left(\varepsilon \cdot \sin x + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \cos \left(-x\right)\right)\right)\right) - \cos x\right)} \cdot -2}{2} \]

   4. Simplified 0.2

      \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \sin x + \left(\left(\cos x \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \varepsilon\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + 0\right)\right)} \cdot -2}{2} \]
      Proof

      [Start] 48.8	\[ \frac{\left(\left(\varepsilon \cdot \sin x + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \cos \left(-x\right)\right)\right)\right) - \cos x\right) \cdot -2}{2} \]
      associate--l+ [=>] 12.1	\[ \frac{\color{blue}{\left(\varepsilon \cdot \sin x + \left(\left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \cos \left(-x\right)\right)\right) - \cos x\right)\right)} \cdot -2}{2} \]
      associate-+r+ [=>] 12.1	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\color{blue}{\left(\left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \cos \left(-x\right)\right)} - \cos x\right)\right) \cdot -2}{2} \]
      cos-neg [=>] 12.1	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \color{blue}{\cos x}\right) - \cos x\right)\right) \cdot -2}{2} \]
      associate--l+ [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \color{blue}{\left(\left(0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(\cos x - \cos x\right)\right)}\right) \cdot -2}{2} \]
      associate-*r* [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\color{blue}{\left(0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(\cos x - \cos x\right)\right)\right) \cdot -2}{2} \]
      *-commutative [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\color{blue}{\cos x \cdot \left(0.5 \cdot {\varepsilon}^{2}\right)} + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(\cos x - \cos x\right)\right)\right) \cdot -2}{2} \]
      unpow2 [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\cos x \cdot \left(0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(\cos x - \cos x\right)\right)\right) \cdot -2}{2} \]
      associate-*r* [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\cos x \cdot \color{blue}{\left(\left(0.5 \cdot \varepsilon\right) \cdot \varepsilon\right)} + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(\cos x - \cos x\right)\right)\right) \cdot -2}{2} \]
      *-commutative [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\cos x \cdot \left(\color{blue}{\left(\varepsilon \cdot 0.5\right)} \cdot \varepsilon\right) + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) + \left(\cos x - \cos x\right)\right)\right) \cdot -2}{2} \]
      associate-*r* [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\cos x \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \varepsilon\right) + \color{blue}{\left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x}\right) + \left(\cos x - \cos x\right)\right)\right) \cdot -2}{2} \]
      *-commutative [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\cos x \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \varepsilon\right) + \color{blue}{\sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) + \left(\cos x - \cos x\right)\right)\right) \cdot -2}{2} \]
      +-inverses [=>] 0.2	\[ \frac{\left(\varepsilon \cdot \sin x + \left(\left(\cos x \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \varepsilon\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + \color{blue}{0}\right)\right) \cdot -2}{2} \]

   if 1.60000000000000013e-4 < eps

   1. Initial program 29.7

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.9

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

   4. Applied egg-rr 0.9

      \[\leadsto \cos x \cdot \left(\cos \varepsilon + -1\right) + \color{blue}{\frac{\left(0 - {\sin x}^{2}\right) \cdot \sin \varepsilon}{\sin x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \sin x + \left(\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \frac{\sin \varepsilon \cdot {\sin x}^{2}}{\sin x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	39296

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

Alternative 2
Error	0.6
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.000165:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \sin x + \left(\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \sin x + \left(\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \end{array} \]

Alternative 4
Error	0.6
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - t_0\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \sin x + \left(\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \end{array} \]

Alternative 5
Error	0.6
Cost	27273

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \sin x + \left(\cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right) + \sin x \cdot \left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right) \cdot -2}{2}\\ \end{array} \]

Alternative 6
Error	0.6
Cost	26441

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00016\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - t_0\\ \end{array} \]

Alternative 7
Error	15.0
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 8
Error	14.7
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0275 \lor \neg \left(\varepsilon \leq 0.0285\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 9
Error	21.3
Cost	13520

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.25 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -2.9 \cdot 10^{-126}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0024:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	21.8
Cost	7500

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -8 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -5.1 \cdot 10^{-126}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) - x \cdot \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.8:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	22.2
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(-\sin x\right)\\ \mathbf{if}\;\varepsilon \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.45 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -5.2 \cdot 10^{-126}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;\varepsilon \leq 4.8:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	32.9
Cost	7120

\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.9 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-106}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	49.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-157} \lor \neg \left(\varepsilon \leq 1.5 \cdot 10^{-105}\right):\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14
Error	52.9
Cost	256

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

Reproduce

herbie shell --seed 2023237 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))