2cos (problem 3.3.5)

Average Error: 39.8 → 0.5

Time: 28.0s

Precision: binary64

Cost: 46664

Math

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

↓

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0039:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0055:\\ \;\;\;\;\left(\varepsilon \cdot \left(-\sin x\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\right) + \left(\cos x \cdot \left({\varepsilon}^{4} \cdot 0.041666666666666664\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) \cdot \cos x - t_0\\ \end{array} \]

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 -0.0039)
     (- (- (* (cos eps) (cos x)) (cos x)) t_0)
     (if (<= eps 0.0055)
       (+
        (+ (* eps (- (sin x))) (* (cos x) (* -0.5 (pow eps 2.0))))
        (+
         (* (cos x) (* (pow eps 4.0) 0.041666666666666664))
         (* (sin x) (* (pow eps 3.0) 0.16666666666666666))))
       (- (* (+ (cos eps) -1.0) (cos x)) 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 tmp;
	if (eps <= -0.0039) {
		tmp = ((cos(eps) * cos(x)) - cos(x)) - t_0;
	} else if (eps <= 0.0055) {
		tmp = ((eps * -sin(x)) + (cos(x) * (-0.5 * pow(eps, 2.0)))) + ((cos(x) * (pow(eps, 4.0) * 0.041666666666666664)) + (sin(x) * (pow(eps, 3.0) * 0.16666666666666666)));
	} else {
		tmp = ((cos(eps) + -1.0) * cos(x)) - t_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 <= (-0.0039d0)) then
        tmp = ((cos(eps) * cos(x)) - cos(x)) - t_0
    else if (eps <= 0.0055d0) then
        tmp = ((eps * -sin(x)) + (cos(x) * ((-0.5d0) * (eps ** 2.0d0)))) + ((cos(x) * ((eps ** 4.0d0) * 0.041666666666666664d0)) + (sin(x) * ((eps ** 3.0d0) * 0.16666666666666666d0)))
    else
        tmp = ((cos(eps) + (-1.0d0)) * cos(x)) - t_0
    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 <= -0.0039) {
		tmp = ((Math.cos(eps) * Math.cos(x)) - Math.cos(x)) - t_0;
	} else if (eps <= 0.0055) {
		tmp = ((eps * -Math.sin(x)) + (Math.cos(x) * (-0.5 * Math.pow(eps, 2.0)))) + ((Math.cos(x) * (Math.pow(eps, 4.0) * 0.041666666666666664)) + (Math.sin(x) * (Math.pow(eps, 3.0) * 0.16666666666666666)));
	} else {
		tmp = ((Math.cos(eps) + -1.0) * Math.cos(x)) - t_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 <= -0.0039:
		tmp = ((math.cos(eps) * math.cos(x)) - math.cos(x)) - t_0
	elif eps <= 0.0055:
		tmp = ((eps * -math.sin(x)) + (math.cos(x) * (-0.5 * math.pow(eps, 2.0)))) + ((math.cos(x) * (math.pow(eps, 4.0) * 0.041666666666666664)) + (math.sin(x) * (math.pow(eps, 3.0) * 0.16666666666666666)))
	else:
		tmp = ((math.cos(eps) + -1.0) * math.cos(x)) - 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))
	tmp = 0.0
	if (eps <= -0.0039)
		tmp = Float64(Float64(Float64(cos(eps) * cos(x)) - cos(x)) - t_0);
	elseif (eps <= 0.0055)
		tmp = Float64(Float64(Float64(eps * Float64(-sin(x))) + Float64(cos(x) * Float64(-0.5 * (eps ^ 2.0)))) + Float64(Float64(cos(x) * Float64((eps ^ 4.0) * 0.041666666666666664)) + Float64(sin(x) * Float64((eps ^ 3.0) * 0.16666666666666666))));
	else
		tmp = Float64(Float64(Float64(cos(eps) + -1.0) * cos(x)) - t_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 <= -0.0039)
		tmp = ((cos(eps) * cos(x)) - cos(x)) - t_0;
	elseif (eps <= 0.0055)
		tmp = ((eps * -sin(x)) + (cos(x) * (-0.5 * (eps ^ 2.0)))) + ((cos(x) * ((eps ^ 4.0) * 0.041666666666666664)) + (sin(x) * ((eps ^ 3.0) * 0.16666666666666666)));
	else
		tmp = ((cos(eps) + -1.0) * cos(x)) - t_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, -0.0039], N[(N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[eps, 0.0055], N[(N[(N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Power[eps, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Power[eps, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0038999999999999998

   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

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

   4. Simplified 0.8

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

      [Start] 0.8	\[ \cos \varepsilon \cdot \cos x - \left(\cos x + \sin x \cdot \sin \varepsilon\right) \]
      rational.json-simplify-2 [<=] 0.8	\[ \color{blue}{\cos x \cdot \cos \varepsilon} - \left(\cos x + \sin x \cdot \sin \varepsilon\right) \]
      rational.json-simplify-46 [=>] 0.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon} \]
      rational.json-simplify-2 [=>] 0.8	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} - \cos x\right) - \sin x \cdot \sin \varepsilon \]
      rational.json-simplify-2 [=>] 0.8	\[ \left(\cos \varepsilon \cdot \cos x - \cos x\right) - \color{blue}{\sin \varepsilon \cdot \sin x} \]

   if -0.0038999999999999998 < eps < 0.0054999999999999997

   1. Initial program 49.1

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

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

      [Start] 0.2	\[ 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)\right) \]
      rational.json-simplify-41 [<=] 0.2	\[ \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} \]
      rational.json-simplify-1 [=>] 0.2	\[ \color{blue}{\left(-1 \cdot \left(\varepsilon \cdot \sin x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)} + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      rational.json-simplify-43 [=>] 0.2	\[ \left(\color{blue}{\varepsilon \cdot \left(\sin x \cdot -1\right)} + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      rational.json-simplify-9 [=>] 0.2	\[ \left(\varepsilon \cdot \color{blue}{\left(-\sin x\right)} + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      rational.json-simplify-43 [=>] 0.2	\[ \left(\varepsilon \cdot \left(-\sin x\right) + \color{blue}{{\varepsilon}^{2} \cdot \left(\cos x \cdot -0.5\right)}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      rational.json-simplify-43 [=>] 0.2	\[ \left(\varepsilon \cdot \left(-\sin x\right) + \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(\varepsilon \cdot \left(-\sin x\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\right) + \left(0.041666666666666664 \cdot \color{blue}{\left(\cos x \cdot {\varepsilon}^{4}\right)} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      rational.json-simplify-43 [=>] 0.2	\[ \left(\varepsilon \cdot \left(-\sin x\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\right) + \left(\color{blue}{\cos x \cdot \left({\varepsilon}^{4} \cdot 0.041666666666666664\right)} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(\varepsilon \cdot \left(-\sin x\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\right) + \left(\cos x \cdot \left({\varepsilon}^{4} \cdot 0.041666666666666664\right) + 0.16666666666666666 \cdot \color{blue}{\left(\sin x \cdot {\varepsilon}^{3}\right)}\right) \]
      rational.json-simplify-43 [=>] 0.2	\[ \left(\varepsilon \cdot \left(-\sin x\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\right) + \left(\cos x \cdot \left({\varepsilon}^{4} \cdot 0.041666666666666664\right) + \color{blue}{\sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666\right)}\right) \]

   if 0.0054999999999999997 < eps

   1. Initial program 29.7

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

   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

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

   4. Simplified 0.8

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

      [Start] 0.8	\[ \cos \varepsilon \cdot \cos x - \left(\cos x + \sin x \cdot \sin \varepsilon\right) \]
      rational.json-simplify-2 [<=] 0.8	\[ \color{blue}{\cos x \cdot \cos \varepsilon} - \left(\cos x + \sin x \cdot \sin \varepsilon\right) \]
      rational.json-simplify-46 [=>] 0.8	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon} \]
      rational.json-simplify-2 [=>] 0.8	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} - \cos x\right) - \sin x \cdot \sin \varepsilon \]
      rational.json-simplify-2 [=>] 0.8	\[ \left(\cos \varepsilon \cdot \cos x - \cos x\right) - \color{blue}{\sin \varepsilon \cdot \sin x} \]

   5. Applied egg-rr 0.8

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0039:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.0055:\\ \;\;\;\;\left(\varepsilon \cdot \left(-\sin x\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\right) + \left(\cos x \cdot \left({\varepsilon}^{4} \cdot 0.041666666666666664\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	32708

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \cos x\right) - t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00014:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon + -1\right) \cdot \cos x - t_0\\ \end{array} \]

Alternative 2
Error	0.5
Cost	26504

\[\begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ t_1 := \left(\cos \varepsilon + -1\right) \cdot \cos x - t_0\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 0.00014:\\ \;\;\;\;\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.6
Cost	26440

\[\begin{array}{l} t_0 := \left(\cos \varepsilon + -1\right) \cdot \cos x - \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.000108:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.9
Cost	26312

\[\begin{array}{l} t_0 := \left(\cos \varepsilon - \cos x\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0035:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0015:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.4
Cost	20168

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0028:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00042:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right) + \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \varepsilon - 1\right) - \sin \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 6
Error	20.9
Cost	19912

\[\begin{array}{l} t_0 := \left(\cos \varepsilon - 1\right) - \sin \varepsilon \cdot \sin x\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-282}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	21.6
Cost	13520

\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ t_1 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.9 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -6.8 \cdot 10^{-113}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.04 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	32.6
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := -0.5 \cdot {\varepsilon}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.46 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-101}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00014:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	22.0
Cost	7184

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ t_1 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.9 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq -6.4 \cdot 10^{-113}:\\ \;\;\;\;-0.5 \cdot {\varepsilon}^{2}\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	36.7
Cost	6856

\[\begin{array}{l} t_0 := \cos \varepsilon - 1\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-28}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	49.0
Cost	520

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-90}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{-73}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12
Error	51.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-70}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-75}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 13
Error	58.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))