mixedcos

Average Accuracy: 54.8% → 97.5%

Time: 18.8s

Precision: binary64

Cost: 7949

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

Math

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

↓

\[\begin{array}{l} t_0 := s \cdot \left(c \cdot x\right)\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;c \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot s} \cdot \frac{t_1}{x \cdot \left(c \cdot s\right)}\\ \mathbf{elif}\;c \leq -6.6 \cdot 10^{-200} \lor \neg \left(c \leq 9 \cdot 10^{-124}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{-1}{x}}{s}}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \end{array} \]

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 (* s (* c x))) (t_1 (cos (+ x x))))
   (if (<= c -6e+220)
     (* (/ (/ 1.0 x) (* c s)) (/ t_1 (* x (* c s))))
     (if (or (<= c -6.6e-200) (not (<= c 9e-124)))
       (* (/ (cos (* x -2.0)) (* c (* x s))) (/ (/ (/ -1.0 x) s) (- c)))
       (/ (/ t_1 t_0) 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 = s * (c * x);
	double t_1 = cos((x + x));
	double tmp;
	if (c <= -6e+220) {
		tmp = ((1.0 / x) / (c * s)) * (t_1 / (x * (c * s)));
	} else if ((c <= -6.6e-200) || !(c <= 9e-124)) {
		tmp = (cos((x * -2.0)) / (c * (x * s))) * (((-1.0 / x) / s) / -c);
	} else {
		tmp = (t_1 / t_0) / 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 = s * (c * x)
    t_1 = cos((x + x))
    if (c <= (-6d+220)) then
        tmp = ((1.0d0 / x) / (c * s)) * (t_1 / (x * (c * s)))
    else if ((c <= (-6.6d-200)) .or. (.not. (c <= 9d-124))) then
        tmp = (cos((x * (-2.0d0))) / (c * (x * s))) * ((((-1.0d0) / x) / s) / -c)
    else
        tmp = (t_1 / t_0) / 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 = s * (c * x);
	double t_1 = Math.cos((x + x));
	double tmp;
	if (c <= -6e+220) {
		tmp = ((1.0 / x) / (c * s)) * (t_1 / (x * (c * s)));
	} else if ((c <= -6.6e-200) || !(c <= 9e-124)) {
		tmp = (Math.cos((x * -2.0)) / (c * (x * s))) * (((-1.0 / x) / s) / -c);
	} else {
		tmp = (t_1 / t_0) / 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 = s * (c * x)
	t_1 = math.cos((x + x))
	tmp = 0
	if c <= -6e+220:
		tmp = ((1.0 / x) / (c * s)) * (t_1 / (x * (c * s)))
	elif (c <= -6.6e-200) or not (c <= 9e-124):
		tmp = (math.cos((x * -2.0)) / (c * (x * s))) * (((-1.0 / x) / s) / -c)
	else:
		tmp = (t_1 / t_0) / 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(s * Float64(c * x))
	t_1 = cos(Float64(x + x))
	tmp = 0.0
	if (c <= -6e+220)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(c * s)) * Float64(t_1 / Float64(x * Float64(c * s))));
	elseif ((c <= -6.6e-200) || !(c <= 9e-124))
		tmp = Float64(Float64(cos(Float64(x * -2.0)) / Float64(c * Float64(x * s))) * Float64(Float64(Float64(-1.0 / x) / s) / Float64(-c)));
	else
		tmp = Float64(Float64(t_1 / t_0) / 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 = s * (c * x);
	t_1 = cos((x + x));
	tmp = 0.0;
	if (c <= -6e+220)
		tmp = ((1.0 / x) / (c * s)) * (t_1 / (x * (c * s)));
	elseif ((c <= -6.6e-200) || ~((c <= 9e-124)))
		tmp = (cos((x * -2.0)) / (c * (x * s))) * (((-1.0 / x) / s) / -c);
	else
		tmp = (t_1 / t_0) / 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[(s * N[(c * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, -6e+220], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c, -6.6e-200], N[Not[LessEqual[c, 9e-124]], $MachinePrecision]], N[(N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-1.0 / x), $MachinePrecision] / s), $MachinePrecision] / (-c)), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.00000000000000048e220

   1. Initial program 60.1%

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

   2. Simplified 61.3%

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

      [Start] 60.1	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 60.1	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 53.4	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 53.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 53.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 53.6	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      associate-*r* [=>] 61.3	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot s\right)}} \]
      associate-*r* [=>] 61.3	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left({c}^{2} \cdot s\right)\right) \cdot s}} \]
      *-commutative [=>] 61.3	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{s \cdot \left(\left(x \cdot x\right) \cdot \left({c}^{2} \cdot s\right)\right)}} \]
      unpow2 [=>] 61.3	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)\right)} \]

   3. Applied egg-rr 49.9%

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

      [Start] 61.3	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]
      *-un-lft-identity [=>] 61.3	\[ \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]
      add-sqr-sqrt [=>] 61.3	\[ \frac{1 \cdot \cos \left(2 \cdot x\right)}{\color{blue}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \cdot \sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}}} \]
      times-frac [=>] 61.3	\[ \color{blue}{\frac{1}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}}} \]
      *-commutative [=>] 61.3	\[ \frac{1}{\sqrt{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right) \cdot s}}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 32.2	\[ \frac{1}{\color{blue}{\sqrt{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)} \cdot \sqrt{s}}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 32.2	\[ \frac{1}{\color{blue}{\left(\sqrt{x \cdot x} \cdot \sqrt{\left(c \cdot c\right) \cdot s}\right)} \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 17.1	\[ \frac{1}{\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\left(c \cdot c\right) \cdot s}\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      add-sqr-sqrt [<=] 32.2	\[ \frac{1}{\left(\color{blue}{x} \cdot \sqrt{\left(c \cdot c\right) \cdot s}\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 32.2	\[ \frac{1}{\left(x \cdot \color{blue}{\left(\sqrt{c \cdot c} \cdot \sqrt{s}\right)}\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 0.0	\[ \frac{1}{\left(x \cdot \left(\color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)} \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      add-sqr-sqrt [<=] 32.2	\[ \frac{1}{\left(x \cdot \left(\color{blue}{c} \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]

   4. Simplified 96.5%

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

      [Start] 49.9	\[ \frac{1}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l/ [=>] 49.9	\[ \color{blue}{\frac{1 \cdot \frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}} \]
      *-lft-identity [=>] 49.9	\[ \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 49.1	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{x \cdot \left(\left(c \cdot \sqrt{s}\right) \cdot \sqrt{s}\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 49.1	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \color{blue}{\left(c \cdot \left(\sqrt{s} \cdot \sqrt{s}\right)\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      rem-square-sqrt [=>] 49.2	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot \color{blue}{s}\right)}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      *-commutative [=>] 49.2	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \color{blue}{\left(s \cdot c\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 49.2	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{\color{blue}{x \cdot \left(\left(c \cdot \sqrt{s}\right) \cdot \sqrt{s}\right)}} \]
      associate-*l* [=>] 49.2	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \color{blue}{\left(c \cdot \left(\sqrt{s} \cdot \sqrt{s}\right)\right)}} \]
      rem-square-sqrt [=>] 96.5	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(c \cdot \color{blue}{s}\right)} \]
      *-commutative [=>] 96.5	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \color{blue}{\left(s \cdot c\right)}} \]

   5. Taylor expanded in x around 0 88.7%

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

   6. Applied egg-rr 96.5%

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

      [Start] 88.7	\[ \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{x \cdot \left(s \cdot c\right)} \]
      div-inv [=>] 88.7	\[ \color{blue}{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)} \cdot \frac{1}{x \cdot \left(s \cdot c\right)}} \]
      *-commutative [=>] 88.7	\[ \color{blue}{\frac{1}{x \cdot \left(s \cdot c\right)} \cdot \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}} \]
      associate-/r* [=>] 88.7	\[ \color{blue}{\frac{\frac{1}{x}}{s \cdot c}} \cdot \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)} \]
      *-commutative [=>] 88.7	\[ \frac{\frac{1}{x}}{\color{blue}{c \cdot s}} \cdot \frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)} \]
      associate-*r* [=>] 96.5	\[ \frac{\frac{1}{x}}{c \cdot s} \cdot \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]
      *-commutative [<=] 96.5	\[ \frac{\frac{1}{x}}{c \cdot s} \cdot \frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot c\right)} \cdot x} \]
      *-commutative [<=] 96.5	\[ \frac{\frac{1}{x}}{c \cdot s} \cdot \frac{\cos \left(x + x\right)}{\color{blue}{x \cdot \left(s \cdot c\right)}} \]
      *-commutative [=>] 96.5	\[ \frac{\frac{1}{x}}{c \cdot s} \cdot \frac{\cos \left(x + x\right)}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

   if -6.00000000000000048e220 < c < -6.5999999999999995e-200 or 8.9999999999999992e-124 < c

   1. Initial program 62.8%

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

   2. Simplified 66.0%

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

      [Start] 62.8	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 62.8	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 56.8	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 57.1	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 57.1	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 57.1	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      associate-*r* [=>] 62.8	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot s\right)}} \]
      associate-*r* [=>] 66.0	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left({c}^{2} \cdot s\right)\right) \cdot s}} \]
      *-commutative [=>] 66.0	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{s \cdot \left(\left(x \cdot x\right) \cdot \left({c}^{2} \cdot s\right)\right)}} \]
      unpow2 [=>] 66.0	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)\right)} \]

   3. Applied egg-rr 75.1%

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

      [Start] 66.0	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]
      *-un-lft-identity [=>] 66.0	\[ \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]
      add-sqr-sqrt [=>] 66.0	\[ \frac{1 \cdot \cos \left(2 \cdot x\right)}{\color{blue}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \cdot \sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}}} \]
      times-frac [=>] 66.0	\[ \color{blue}{\frac{1}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}}} \]
      *-commutative [=>] 66.0	\[ \frac{1}{\sqrt{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right) \cdot s}}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 50.9	\[ \frac{1}{\color{blue}{\sqrt{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)} \cdot \sqrt{s}}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 50.9	\[ \frac{1}{\color{blue}{\left(\sqrt{x \cdot x} \cdot \sqrt{\left(c \cdot c\right) \cdot s}\right)} \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 24.9	\[ \frac{1}{\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\left(c \cdot c\right) \cdot s}\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      add-sqr-sqrt [<=] 42.7	\[ \frac{1}{\left(\color{blue}{x} \cdot \sqrt{\left(c \cdot c\right) \cdot s}\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 42.7	\[ \frac{1}{\left(x \cdot \color{blue}{\left(\sqrt{c \cdot c} \cdot \sqrt{s}\right)}\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      sqrt-prod [=>] 16.0	\[ \frac{1}{\left(x \cdot \left(\color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)} \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]
      add-sqr-sqrt [<=] 43.2	\[ \frac{1}{\left(x \cdot \left(\color{blue}{c} \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)}} \]

   4. Simplified 96.0%

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

      [Start] 75.1	\[ \frac{1}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l/ [=>] 75.1	\[ \color{blue}{\frac{1 \cdot \frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}} \]
      *-lft-identity [=>] 75.1	\[ \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 72.6	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{x \cdot \left(\left(c \cdot \sqrt{s}\right) \cdot \sqrt{s}\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 72.7	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \color{blue}{\left(c \cdot \left(\sqrt{s} \cdot \sqrt{s}\right)\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      rem-square-sqrt [=>] 72.8	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot \color{blue}{s}\right)}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      *-commutative [=>] 72.8	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \color{blue}{\left(s \cdot c\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 72.8	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{\color{blue}{x \cdot \left(\left(c \cdot \sqrt{s}\right) \cdot \sqrt{s}\right)}} \]
      associate-*l* [=>] 72.8	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \color{blue}{\left(c \cdot \left(\sqrt{s} \cdot \sqrt{s}\right)\right)}} \]
      rem-square-sqrt [=>] 96.0	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(c \cdot \color{blue}{s}\right)} \]
      *-commutative [=>] 96.0	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \color{blue}{\left(s \cdot c\right)}} \]

   5. Applied egg-rr 95.4%

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

      [Start] 96.0	\[ \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(s \cdot c\right)} \]
      frac-2neg [=>] 96.0	\[ \color{blue}{\frac{-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{-x \cdot \left(s \cdot c\right)}} \]
      div-inv [=>] 96.0	\[ \color{blue}{\left(-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}\right) \cdot \frac{1}{-x \cdot \left(s \cdot c\right)}} \]
      associate-*r* [=>] 95.4	\[ \left(-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}\right) \cdot \frac{1}{-\color{blue}{\left(x \cdot s\right) \cdot c}} \]
      distribute-rgt-neg-in [=>] 95.4	\[ \left(-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}\right) \cdot \frac{1}{\color{blue}{\left(x \cdot s\right) \cdot \left(-c\right)}} \]

   6. Simplified 98.6%

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

      [Start] 95.4	\[ \left(-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}\right) \cdot \frac{1}{\left(x \cdot s\right) \cdot \left(-c\right)} \]
      *-commutative [=>] 95.4	\[ \color{blue}{\frac{1}{\left(x \cdot s\right) \cdot \left(-c\right)} \cdot \left(-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}\right)} \]
      associate-/r* [=>] 95.3	\[ \color{blue}{\frac{\frac{1}{x \cdot s}}{-c}} \cdot \left(-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}\right) \]
      associate-/r* [=>] 95.4	\[ \frac{\color{blue}{\frac{\frac{1}{x}}{s}}}{-c} \cdot \left(-\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}\right) \]
      distribute-neg-frac [=>] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \color{blue}{\frac{-\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}} \]
      count-2 [=>] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\cos \color{blue}{\left(2 \cdot x\right)}}{x \cdot \left(s \cdot c\right)} \]
      metadata-eval [<=] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\cos \left(\color{blue}{\left(--2\right)} \cdot x\right)}{x \cdot \left(s \cdot c\right)} \]
      distribute-lft-neg-in [<=] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\cos \color{blue}{\left(--2 \cdot x\right)}}{x \cdot \left(s \cdot c\right)} \]
      cos-neg [=>] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\color{blue}{\cos \left(-2 \cdot x\right)}}{x \cdot \left(s \cdot c\right)} \]
      *-commutative [=>] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\cos \color{blue}{\left(x \cdot -2\right)}}{x \cdot \left(s \cdot c\right)} \]
      *-commutative [=>] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\cos \left(x \cdot -2\right)}{\color{blue}{\left(s \cdot c\right) \cdot x}} \]
      *-commutative [=>] 95.4	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\cos \left(x \cdot -2\right)}{\color{blue}{\left(c \cdot s\right)} \cdot x} \]
      associate-*r* [<=] 98.6	\[ \frac{\frac{\frac{1}{x}}{s}}{-c} \cdot \frac{-\cos \left(x \cdot -2\right)}{\color{blue}{c \cdot \left(s \cdot x\right)}} \]

   if -6.5999999999999995e-200 < c < 8.9999999999999992e-124

   1. Initial program 7.9%

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

   2. Simplified 10.2%

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

      [Start] 7.9	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 7.9	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 6.8	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 7.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 7.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 7.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      associate-*r* [=>] 10.4	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left({c}^{2} \cdot s\right) \cdot s\right)}} \]
      associate-*r* [=>] 10.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left({c}^{2} \cdot s\right)\right) \cdot s}} \]
      *-commutative [=>] 10.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{s \cdot \left(\left(x \cdot x\right) \cdot \left({c}^{2} \cdot s\right)\right)}} \]
      unpow2 [=>] 10.2	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)\right)} \]

   3. Applied egg-rr 37.8%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(0 + c \cdot \left(c \cdot s\right)\right)}\right)} \]
      Proof

      [Start] 10.2	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]
      add-log-exp [=>] 1.0	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\log \left(e^{\left(c \cdot c\right) \cdot s}\right)}\right)} \]
      *-un-lft-identity [=>] 1.0	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \log \color{blue}{\left(1 \cdot e^{\left(c \cdot c\right) \cdot s}\right)}\right)} \]
      log-prod [=>] 1.0	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\log 1 + \log \left(e^{\left(c \cdot c\right) \cdot s}\right)\right)}\right)} \]
      metadata-eval [=>] 1.0	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{0} + \log \left(e^{\left(c \cdot c\right) \cdot s}\right)\right)\right)} \]
      add-log-exp [<=] 10.2	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \color{blue}{\left(c \cdot c\right) \cdot s}\right)\right)} \]
      associate-*l* [=>] 37.8	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \color{blue}{c \cdot \left(c \cdot s\right)}\right)\right)} \]

   4. Applied egg-rr 96.5%

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

      [Start] 37.8	\[ \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + c \cdot \left(c \cdot s\right)\right)\right)} \]
      *-un-lft-identity [=>] 37.8	\[ \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + c \cdot \left(c \cdot s\right)\right)\right)} \]
      add-sqr-sqrt [=>] 37.7	\[ \frac{1 \cdot \cos \left(2 \cdot x\right)}{\color{blue}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + c \cdot \left(c \cdot s\right)\right)\right)} \cdot \sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + c \cdot \left(c \cdot s\right)\right)\right)}}} \]
      times-frac [=>] 37.7	\[ \color{blue}{\frac{1}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + c \cdot \left(c \cdot s\right)\right)\right)}} \cdot \frac{\cos \left(2 \cdot x\right)}{\sqrt{s \cdot \left(\left(x \cdot x\right) \cdot \left(0 + c \cdot \left(c \cdot s\right)\right)\right)}}} \]

   5. Simplified 93.7%

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

      [Start] 96.5	\[ \frac{1}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \cdot \frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l/ [=>] 96.5	\[ \color{blue}{\frac{1 \cdot \frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}} \]
      *-lft-identity [=>] 96.5	\[ \frac{\color{blue}{\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*r* [=>] 91.2	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot \sqrt{s}\right)} \cdot \sqrt{s}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 91.2	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot c\right) \cdot \left(\sqrt{s} \cdot \sqrt{s}\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      unpow1/2 [<=] 91.2	\[ \frac{\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left(\color{blue}{{s}^{0.5}} \cdot \sqrt{s}\right)}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      metadata-eval [<=] 91.2	\[ \frac{\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left({s}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{s}\right)}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      unpow1/2 [<=] 91.2	\[ \frac{\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left({s}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{s}^{0.5}}\right)}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      metadata-eval [<=] 91.2	\[ \frac{\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left({s}^{\left(\frac{1}{2}\right)} \cdot {s}^{\color{blue}{\left(\frac{1}{2}\right)}}\right)}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      sqr-pow [<=] 91.5	\[ \frac{\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \color{blue}{{s}^{1}}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      unpow1 [=>] 91.5	\[ \frac{\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \color{blue}{s}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*r* [<=] 93.2	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{x \cdot \left(c \cdot s\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      *-commutative [<=] 93.2	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot x}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      *-commutative [=>] 93.2	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot c\right)} \cdot x}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*l* [=>] 91.5	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{s \cdot \left(c \cdot x\right)}}}{\left(x \cdot \left(c \cdot \sqrt{s}\right)\right) \cdot \sqrt{s}} \]
      associate-*r* [=>] 93.4	\[ \frac{\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \sqrt{s}\right)} \cdot \sqrt{s}} \]
      associate-*l* [=>] 93.3	\[ \frac{\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\sqrt{s} \cdot \sqrt{s}\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{+220}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot s} \cdot \frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot s\right)}\\ \mathbf{elif}\;c \leq -6.6 \cdot 10^{-200} \lor \neg \left(c \leq 9 \cdot 10^{-124}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{-1}{x}}{s}}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot x\right)}}{s \cdot \left(c \cdot x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	7757

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x + x\right)\\ t_2 := x \cdot \left(c \cdot s\right)\\ t_3 := s \cdot \left(c \cdot x\right)\\ \mathbf{if}\;c \leq -8 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{t_1}{t_2}}{t_2}\\ \mathbf{elif}\;c \leq -7.8 \cdot 10^{-199} \lor \neg \left(c \leq 1.12 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_3}}{t_3}\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	7757

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x + x\right)\\ t_2 := s \cdot \left(c \cdot x\right)\\ \mathbf{if}\;c \leq -7.2 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot s} \cdot \frac{t_1}{x \cdot \left(c \cdot s\right)}\\ \mathbf{elif}\;c \leq -1.75 \cdot 10^{-199} \lor \neg \left(c \leq 1.12 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_2}}{t_2}\\ \end{array} \]

Alternative 3
Accuracy	85.2%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -0.000155 \lor \neg \left(x \leq 0.0042\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(c \cdot \left(x \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{c}}{x}}{s}}{s \cdot \left(c \cdot x\right)}\\ \end{array} \]

Alternative 4
Accuracy	93.6%
Cost	7625

\[\begin{array}{l} t_0 := s \cdot \left(c \cdot x\right)\\ \mathbf{if}\;x \leq -6.3 \cdot 10^{-15} \lor \neg \left(x \leq 5 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{c}}{x}}{s}}{t_0}\\ \end{array} \]

Alternative 5
Accuracy	96.5%
Cost	7624

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := x \cdot \left(c \cdot s\right)\\ t_2 := s \cdot \left(c \cdot x\right)\\ \mathbf{if}\;c \leq -4.5 \cdot 10^{+266}:\\ \;\;\;\;\frac{1}{\frac{t_1}{\frac{1}{t_1}}}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_2 \cdot t_2}\\ \end{array} \]

Alternative 6
Accuracy	95.8%
Cost	7360

\[\begin{array}{l} t_0 := s \cdot \left(c \cdot x\right)\\ \frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0} \end{array} \]

Alternative 7
Accuracy	69.4%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;s \leq -3.4 \cdot 10^{-227} \lor \neg \left(s \leq 7.5 \cdot 10^{-201}\right):\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	73.8%
Cost	1097

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;s \leq -6 \cdot 10^{-223} \lor \neg \left(s \leq 6.2 \cdot 10^{-222}\right):\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	75.2%
Cost	964

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := s \cdot \left(c \cdot x\right)\\ \mathbf{if}\;s \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 10
Accuracy	74.1%
Cost	964

\[\begin{array}{l} t_0 := s \cdot \left(c \cdot x\right)\\ \mathbf{if}\;s \leq 8.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	74.2%
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 7 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{c \cdot x}}{s \cdot \left(c \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	74.3%
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{c}}{x}}{s}}{s \cdot \left(c \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	39.2%
Cost	832

\[\frac{0}{\left(s \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right)} \]

Alternative 14
Accuracy	44.5%
Cost	832

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

Alternative 15
Accuracy	58.3%
Cost	832

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

Reproduce

herbie shell --seed 2023146 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))