mixedcos

Average Error: 28.4 → 2.8

Time: 16.8s

Precision: binary64

Cost: 13572

\[ \begin{array}{c}[c, s] = \mathsf{sort}([c, s])\\ \end{array} \]

Math

\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

↓

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-130}:\\ \;\;\;\;t_1 \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;{t_0}^{-2}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot x\right)} \cdot \frac{t_1}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{c \cdot \left(\left(x \cdot s\right) \cdot t_0\right)}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

↓

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (+ x x))))
   (if (<= x -2e-130)
     (* t_1 (pow (* x (* c s)) -2.0))
     (if (<= x 4.2e-43)
       (pow t_0 -2.0)
       (if (<= x 4.5e+115)
         (* (/ 1.0 (* (* c s) (* x x))) (/ t_1 (* c s)))
         (/ t_1 (* c (* (* x s) t_0))))))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

↓

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((x + x));
	double tmp;
	if (x <= -2e-130) {
		tmp = t_1 * pow((x * (c * s)), -2.0);
	} else if (x <= 4.2e-43) {
		tmp = pow(t_0, -2.0);
	} else if (x <= 4.5e+115) {
		tmp = (1.0 / ((c * s) * (x * x))) * (t_1 / (c * s));
	} else {
		tmp = t_1 / (c * ((x * s) * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

↓

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = cos((x + x))
    if (x <= (-2d-130)) then
        tmp = t_1 * ((x * (c * s)) ** (-2.0d0))
    else if (x <= 4.2d-43) then
        tmp = t_0 ** (-2.0d0)
    else if (x <= 4.5d+115) then
        tmp = (1.0d0 / ((c * s) * (x * x))) * (t_1 / (c * s))
    else
        tmp = t_1 / (c * ((x * s) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

↓

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((x + x));
	double tmp;
	if (x <= -2e-130) {
		tmp = t_1 * Math.pow((x * (c * s)), -2.0);
	} else if (x <= 4.2e-43) {
		tmp = Math.pow(t_0, -2.0);
	} else if (x <= 4.5e+115) {
		tmp = (1.0 / ((c * s) * (x * x))) * (t_1 / (c * s));
	} else {
		tmp = t_1 / (c * ((x * s) * t_0));
	}
	return tmp;
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

↓

def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((x + x))
	tmp = 0
	if x <= -2e-130:
		tmp = t_1 * math.pow((x * (c * s)), -2.0)
	elif x <= 4.2e-43:
		tmp = math.pow(t_0, -2.0)
	elif x <= 4.5e+115:
		tmp = (1.0 / ((c * s) * (x * x))) * (t_1 / (c * s))
	else:
		tmp = t_1 / (c * ((x * s) * t_0))
	return tmp

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

↓

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -2e-130)
		tmp = Float64(t_1 * (Float64(x * Float64(c * s)) ^ -2.0));
	elseif (x <= 4.2e-43)
		tmp = t_0 ^ -2.0;
	elseif (x <= 4.5e+115)
		tmp = Float64(Float64(1.0 / Float64(Float64(c * s) * Float64(x * x))) * Float64(t_1 / Float64(c * s)));
	else
		tmp = Float64(t_1 / Float64(c * Float64(Float64(x * s) * t_0)));
	end
	return tmp
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

↓

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((x + x));
	tmp = 0.0;
	if (x <= -2e-130)
		tmp = t_1 * ((x * (c * s)) ^ -2.0);
	elseif (x <= 4.2e-43)
		tmp = t_0 ^ -2.0;
	elseif (x <= 4.5e+115)
		tmp = (1.0 / ((c * s) * (x * x))) * (t_1 / (c * s));
	else
		tmp = t_1 / (c * ((x * s) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e-130], N[(t$95$1 * N[Power[N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-43], N[Power[t$95$0, -2.0], $MachinePrecision], If[LessEqual[x, 4.5e+115], N[(N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(c * N[(N[(x * s), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0000000000000002e-130

   1. Initial program 25.3

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 1.9

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]
      Proof

      [Start] 25.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 25.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 26.4	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 26.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 26.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 26.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
      unpow2 [=>] 26.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      unswap-sqr [=>] 13.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
      unswap-sqr [=>] 1.9	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 1.6

      \[\leadsto \color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{-2} \cdot \cos \left(x + x\right)} \]

   if -2.0000000000000002e-130 < x < 4.2000000000000001e-43

   1. Initial program 38.0

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 34.9

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x}} \]
      Proof

      [Start] 38.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 34.9	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      unpow2 [=>] 34.9	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x} \]
      unpow2 [=>] 34.9	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right) \cdot x} \]

   3. Taylor expanded in x around 0 49.3

      \[\leadsto \color{blue}{\frac{1}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)}} \]

   4. Simplified 28.9

      \[\leadsto \color{blue}{\frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)}} \]
      Proof

      [Start] 49.3	\[ \frac{1}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      associate-*r* [=>] 49.3	\[ \frac{1}{\color{blue}{\left({s}^{2} \cdot {c}^{2}\right) \cdot {x}^{2}}} \]
      *-commutative [<=] 49.3	\[ \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right)} \cdot {x}^{2}} \]
      associate-*r* [<=] 49.0	\[ \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
      unpow2 [=>] 49.0	\[ \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]
      sqr-pow [=>] 49.0	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left({s}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)} \cdot {x}^{2}\right)} \]
      unpow2 [=>] 49.0	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left({s}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]
      unswap-sqr [=>] 25.5	\[ \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left({s}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)}} \]
      metadata-eval [=>] 25.5	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left({s}^{\color{blue}{1}} \cdot x\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)} \]
      unpow1 [=>] 25.5	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(\color{blue}{s} \cdot x\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)} \]
      rem-square-sqrt [<=] 39.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)} \]
      metadata-eval [=>] 39.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right) \cdot \left({s}^{\color{blue}{1}} \cdot x\right)\right)} \]
      unpow1 [=>] 39.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)} \]
      rem-square-sqrt [<=] 39.3	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right) \cdot \left(s \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)} \]
      associate-*l* [<=] 39.3	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(\left(s \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)} \cdot \left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right)\right)} \]
      associate-*l* [<=] 39.3	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(\left(s \cdot \sqrt{x}\right) \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(s \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)}\right)} \]

   5. Applied egg-rr 34.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)}{\frac{1}{x}}}} \]

   6. Applied egg-rr 24.4

      \[\leadsto \color{blue}{{\left(\left(c \cdot \left(s \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right)}^{-1} \cdot {\left(\left(c \cdot \left(s \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right)}^{-1}} \]

   7. Simplified 4.5

      \[\leadsto \color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{-2}} \]
      Proof

      [Start] 24.4	\[ {\left(\left(c \cdot \left(s \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right)}^{-1} \cdot {\left(\left(c \cdot \left(s \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right)}^{-1} \]
      pow-sqr [=>] 24.4	\[ \color{blue}{{\left(\left(c \cdot \left(s \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right)}^{\left(2 \cdot -1\right)}} \]
      associate-*l* [=>] 26.3	\[ {\color{blue}{\left(c \cdot \left(\left(s \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)\right)}}^{\left(2 \cdot -1\right)} \]
      associate-*l* [=>] 26.3	\[ {\left(c \cdot \color{blue}{\left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \]
      rem-square-sqrt [=>] 4.5	\[ {\left(c \cdot \left(s \cdot \color{blue}{x}\right)\right)}^{\left(2 \cdot -1\right)} \]
      metadata-eval [=>] 4.5	\[ {\left(c \cdot \left(s \cdot x\right)\right)}^{\color{blue}{-2}} \]

   if 4.2000000000000001e-43 < x < 4.49999999999999963e115

   1. Initial program 22.3

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 0.6

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]
      Proof

      [Start] 22.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 22.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 22.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 22.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 22.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 22.6	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
      unpow2 [=>] 22.6	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      unswap-sqr [=>] 4.8	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
      unswap-sqr [=>] 0.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot x\right)} \cdot \frac{\cos \left(x + x\right)}{c \cdot s}} \]

   if 4.49999999999999963e115 < x

   1. Initial program 27.8

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. Simplified 14.9

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}} \]
      Proof

      [Start] 27.8	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 26.0	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      *-commutative [=>] 26.0	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
      *-commutative [=>] 26.0	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot x\right)}\right)} \]
      associate-*r* [=>] 28.6	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]
      *-commutative [=>] 28.6	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]
      unpow2 [=>] 28.6	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)\right)} \]
      unpow2 [=>] 28.6	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)} \]
      unswap-sqr [=>] 14.9	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right)} \]

   3. Taylor expanded in x around inf 30.7

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)}} \]

   4. Simplified 2.4

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
      Proof

      [Start] 30.7	\[ \frac{\cos \left(2 \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      count-2 [<=] 30.7	\[ \frac{\cos \color{blue}{\left(x + x\right)}}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      associate-*r* [=>] 30.9	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left({s}^{2} \cdot {c}^{2}\right) \cdot {x}^{2}}} \]
      associate-/r* [=>] 30.9	\[ \color{blue}{\frac{\frac{\cos \left(x + x\right)}{{s}^{2} \cdot {c}^{2}}}{{x}^{2}}} \]
      *-commutative [<=] 30.9	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{{c}^{2} \cdot {s}^{2}}}}{{x}^{2}} \]
      unpow2 [=>] 30.9	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}}}{{x}^{2}} \]
      unpow2 [=>] 30.9	\[ \frac{\frac{\cos \left(x + x\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}}}{{x}^{2}} \]
      swap-sqr [<=] 21.3	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}}}{{x}^{2}} \]
      unpow2 [<=] 21.3	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}}}}{{x}^{2}} \]
      associate-/l/ [=>] 21.2	\[ \color{blue}{\frac{\cos \left(x + x\right)}{{x}^{2} \cdot {\left(c \cdot s\right)}^{2}}} \]
      unpow2 [=>] 21.2	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot x\right)} \cdot {\left(c \cdot s\right)}^{2}} \]
      unpow2 [=>] 21.2	\[ \frac{\cos \left(x + x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
      swap-sqr [<=] 2.9	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]
      unpow2 [<=] 2.9	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
      *-commutative [=>] 2.9	\[ \frac{\cos \left(x + x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]
      associate-*l* [=>] 2.4	\[ \frac{\cos \left(x + x\right)}{{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}}^{2}} \]

   5. Applied egg-rr 4.9

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)\right) \cdot c}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-130}:\\ \;\;\;\;\cos \left(x + x\right) \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-43}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot x\right)} \cdot \frac{\cos \left(x + x\right)}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.7
Cost	13705

\[\begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;c \leq -3.8 \cdot 10^{-141} \lor \neg \left(c \leq 1.2 \cdot 10^{-288}\right):\\ \;\;\;\;\frac{t_1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} \cdot \frac{1}{t_0}\\ \end{array} \]

Alternative 2
Error	2.8
Cost	7884

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x + x\right)\\ t_2 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{t_1}{t_2} \cdot \frac{1}{t_2}\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-43}:\\ \;\;\;\;{t_0}^{-2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot x\right)} \cdot \frac{t_1}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{c \cdot \left(\left(x \cdot s\right) \cdot t_0\right)}\\ \end{array} \]

Alternative 3
Error	6.9
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-42} \lor \neg \left(x \leq 5.1 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 4
Error	4.1
Cost	7625

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-46} \lor \neg \left(x \leq 1.35 \cdot 10^{-252}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{-2}\\ \end{array} \]

Alternative 5
Error	2.5
Cost	7625

\[\begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-130} \lor \neg \left(x \leq 4.6 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 6
Error	13.0
Cost	7624

\[\begin{array}{l} \mathbf{if}\;s \leq 4.2 \cdot 10^{-135}:\\ \;\;\;\;{\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\\ \mathbf{elif}\;s \leq 8 \cdot 10^{+65}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 7
Error	7.6
Cost	7624

\[\begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	17.0
Cost	6784

\[{\left(c \cdot \left(x \cdot s\right)\right)}^{-2} \]

Alternative 9
Error	24.7
Cost	1228

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \mathbf{if}\;s \leq -2.3 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;s \leq 4.8 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{elif}\;s \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	25.5
Cost	1097

\[\begin{array}{l} \mathbf{if}\;s \leq -1.5 \cdot 10^{-111} \lor \neg \left(s \leq 4.8 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	19.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-226} \lor \neg \left(x \leq 2.55 \cdot 10^{-239}\right):\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	17.0
Cost	960

\[\begin{array}{l} t_0 := \frac{1}{s \cdot \left(c \cdot x\right)}\\ t_0 \cdot t_0 \end{array} \]

Alternative 13
Error	17.0
Cost	960

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ t_0 \cdot t_0 \end{array} \]

Alternative 14
Error	27.0
Cost	832

\[\frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)} \]

Alternative 15
Error	17.1
Cost	832

\[\begin{array}{l} t_0 := s \cdot \left(c \cdot x\right)\\ \frac{1}{t_0 \cdot t_0} \end{array} \]

Reproduce

herbie shell --seed 2023046 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))