mixedcos

Percentage Accurate: 66.9% → 98.2%

Time: 11.3s

Alternatives: 5

Speedup: 24.1×

• 31.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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

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));
}

Python

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

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

MATLAB

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

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));
}

Python

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

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

MATLAB

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

Alternative 1: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;{c}^{2} \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot s} \cdot \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= (pow c 2.0) 4e-54)
     (* (/ (/ 1.0 x) (* c s)) (/ (cos (* 2.0 x)) (* x (* c s))))
     (/ (/ 1.0 t_0) t_0))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (pow(c, 2.0) <= 4e-54) {
		tmp = ((1.0 / x) / (c * s)) * (cos((2.0 * x)) / (x * (c * s)));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this 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) :: tmp
    t_0 = c * (x * s)
    if ((c ** 2.0d0) <= 4d-54) then
        tmp = ((1.0d0 / x) / (c * s)) * (cos((2.0d0 * x)) / (x * (c * s)))
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (Math.pow(c, 2.0) <= 4e-54) {
		tmp = ((1.0 / x) / (c * s)) * (Math.cos((2.0 * x)) / (x * (c * s)));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if math.pow(c, 2.0) <= 4e-54:
		tmp = ((1.0 / x) / (c * s)) * (math.cos((2.0 * x)) / (x * (c * s)))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if ((c ^ 2.0) <= 4e-54)
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(c * s)) * Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(c * s))));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if ((c ^ 2.0) <= 4e-54)
		tmp = ((1.0 / x) / (c * s)) * (cos((2.0 * x)) / (x * (c * s)));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[c, 2.0], $MachinePrecision], 4e-54], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c 2) < 4.0000000000000001e-54

   1. Initial program 72.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. *-un-lft-identity 72.3%

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

      2. associate-*r* 75.1%

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

      3. times-frac 75.1%

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

      4. *-commutative 75.1%

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

      5. associate-*r* 72.0%

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

      6. *-commutative 72.0%

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

      7. pow-prod-down 88.2%

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

   3. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{\frac{1}{{\left(s \cdot c\right)}^{2} \cdot x} \cdot \frac{\cos \left(2 \cdot x\right)}{x}} \]
   4. Step-by-step derivation

      1. frac-times 88.3%

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

      2. *-un-lft-identity 88.3%

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

      3. clear-num 88.3%

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

      4. associate-*l* 81.5%

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

      5. pow2 81.5%

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

      6. unpow-prod-down 98.5%

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

      7. *-commutative 98.5%

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

      8. associate-*r* 97.6%

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

      9. *-commutative 97.6%

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

      10. *-commutative 97.6%

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

      11. associate-*l* 98.5%

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

      12. *-commutative 98.5%

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

      13. *-commutative 98.5%

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

   5. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}{\cos \left(x \cdot 2\right)}}} \]
   6. Step-by-step derivation

      1. clear-num 98.5%

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

      2. *-un-lft-identity 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-*r* 97.6%

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

      5. pow2 97.6%

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

      6. times-frac 97.6%

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

      7. *-commutative 97.6%

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

      8. *-commutative 97.6%

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

      9. associate-/l/ 97.7%

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

      10. *-un-lft-identity 97.7%

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

      11. times-frac 96.3%

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

      12. associate-/r* 96.3%

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

      13. div-inv 96.6%

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

      14. associate-*r* 98.6%

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

      15. *-commutative 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 4.0000000000000001e-54 < (pow.f64 c 2)

   1. Initial program 58.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. add-cbrt-cube 58.4%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. add-cbrt-cube 53.3%

         \[\leadsto \frac{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

      3. cbrt-undiv 53.3%

         \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

      4. pow3 53.3%

         \[\leadsto \sqrt[3]{\frac{\color{blue}{{\cos \left(2 \cdot x\right)}^{3}}}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}} \]

      5. pow3 53.3%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

      6. *-commutative 53.3%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

      7. associate-*r* 49.8%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

      8. unpow2 49.8%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

      9. pow-prod-down 60.9%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{{\left(x \cdot s\right)}^{2}}\right)}^{3}}} \]

   3. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}}} \]
   4. Step-by-step derivation

      1. unpow2 60.9%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}} \]

      2. rem-square-sqrt 60.9%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\left(c \cdot c\right) \cdot \color{blue}{\left(\sqrt{{\left(x \cdot s\right)}^{2}} \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}\right)}^{3}}} \]

      3. swap-sqr 66.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right) \cdot \left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)\right)}}^{3}}} \]

      4. unpow2 66.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}^{2}\right)}}^{3}}} \]

      5. unpow2 66.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \sqrt{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}}\right)}^{2}\right)}^{3}}} \]

      6. rem-sqrt-square 69.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \color{blue}{\left|x \cdot s\right|}\right)}^{2}\right)}^{3}}} \]

      7. *-commutative 69.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|\color{blue}{s \cdot x}\right|\right)}^{2}\right)}^{3}}} \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]
   6. Step-by-step derivation

      1. pow1 69.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{1}\right)}}^{2}\right)}^{3}}} \]

      2. metadata-eval 69.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left({\left(c \cdot \left|s \cdot x\right|\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{2}\right)}^{3}}} \]

      3. sqrt-pow1 69.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{2}\right)}^{3}}} \]

      4. pow2 69.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}} \cdot \sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{3}}} \]

      5. add-sqr-sqrt 69.2%

         \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}}^{3}}} \]

      6. cbrt-div 69.2%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\cos \left(2 \cdot x\right)}^{3}}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]

      7. rem-cbrt-cube 69.2%

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}} \]

      8. rem-cbrt-cube 96.7%

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

      9. unpow2 96.7%

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

      10. associate-/l/ 97.4%

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

   7. Applied egg-rr 97.4%

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

   8. Taylor expanded in x around 0 83.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{c \cdot \left(s \cdot x\right)}}}{c \cdot \left(s \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot s} \cdot \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 2: 96.9% accurate, 2.7× speedup

Math

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

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ (/ (cos (* 2.0 x)) t_0) t_0)))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (cos((2.0 * x)) / t_0) / t_0;
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this 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
    t_0 = c * (x * s)
    code = (cos((2.0d0 * x)) / t_0) / t_0
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (Math.cos((2.0 * x)) / t_0) / t_0;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return (math.cos((2.0 * x)) / t_0) / t_0

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(Float64(cos(Float64(2.0 * x)) / t_0) / t_0)
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = (cos((2.0 * x)) / t_0) / t_0;
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.7%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. add-cbrt-cube 65.7%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. add-cbrt-cube 60.5%

      \[\leadsto \frac{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

   3. cbrt-undiv 60.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

   4. pow3 60.5%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{{\cos \left(2 \cdot x\right)}^{3}}}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}} \]

   5. pow3 60.5%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

   6. *-commutative 60.5%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

   7. associate-*r* 55.3%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

   8. unpow2 55.3%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

   9. pow-prod-down 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{{\left(x \cdot s\right)}^{2}}\right)}^{3}}} \]

3. Applied egg-rr 65.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}}} \]
4. Step-by-step derivation

   1. unpow2 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}} \]

   2. rem-square-sqrt 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\left(c \cdot c\right) \cdot \color{blue}{\left(\sqrt{{\left(x \cdot s\right)}^{2}} \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}\right)}^{3}}} \]

   3. swap-sqr 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right) \cdot \left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)\right)}}^{3}}} \]

   4. unpow2 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}^{2}\right)}}^{3}}} \]

   5. unpow2 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \sqrt{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}}\right)}^{2}\right)}^{3}}} \]

   6. rem-sqrt-square 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \color{blue}{\left|x \cdot s\right|}\right)}^{2}\right)}^{3}}} \]

   7. *-commutative 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|\color{blue}{s \cdot x}\right|\right)}^{2}\right)}^{3}}} \]

5. Simplified 74.0%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]
6. Step-by-step derivation

   1. pow1 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{1}\right)}}^{2}\right)}^{3}}} \]

   2. metadata-eval 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left({\left(c \cdot \left|s \cdot x\right|\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{2}\right)}^{3}}} \]

   3. sqrt-pow1 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{2}\right)}^{3}}} \]

   4. pow2 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}} \cdot \sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{3}}} \]

   5. add-sqr-sqrt 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}}^{3}}} \]

   6. cbrt-div 74.0%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\cos \left(2 \cdot x\right)}^{3}}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]

   7. rem-cbrt-cube 74.0%

      \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}} \]

   8. rem-cbrt-cube 97.2%

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

   9. unpow2 97.2%

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

   10. associate-/l/ 97.5%

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

7. Applied egg-rr 97.5%

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

8. Taylor expanded in x around inf 97.5%

   \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s \cdot x\right)}}}{c \cdot \left(s \cdot x\right)} \]
9. Step-by-step derivation

   1. *-commutative 97.5%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

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

Alternative 3: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)} \end{array} \]

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (/ (/ (/ (cos (* 2.0 x)) c) (* x s)) (* c (* x s))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return ((cos((2.0 * x)) / c) / (x * s)) / (c * (x * s));
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
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) / (x * s)) / (c * (x * s))
end function

Java

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

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return ((math.cos((2.0 * x)) / c) / (x * s)) / (c * (x * s))

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(Float64(Float64(cos(Float64(2.0 * x)) / c) / Float64(x * s)) / Float64(c * Float64(x * s)))
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = ((cos((2.0 * x)) / c) / (x * s)) / (c * (x * s));
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. add-cbrt-cube 65.7%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. add-cbrt-cube 60.5%

      \[\leadsto \frac{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

   3. cbrt-undiv 60.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

   4. pow3 60.5%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{{\cos \left(2 \cdot x\right)}^{3}}}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}} \]

   5. pow3 60.5%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

   6. *-commutative 60.5%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

   7. associate-*r* 55.3%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

   8. unpow2 55.3%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

   9. pow-prod-down 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{{\left(x \cdot s\right)}^{2}}\right)}^{3}}} \]

3. Applied egg-rr 65.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}}} \]
4. Step-by-step derivation

   1. unpow2 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}} \]

   2. rem-square-sqrt 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\left(c \cdot c\right) \cdot \color{blue}{\left(\sqrt{{\left(x \cdot s\right)}^{2}} \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}\right)}^{3}}} \]

   3. swap-sqr 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right) \cdot \left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)\right)}}^{3}}} \]

   4. unpow2 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}^{2}\right)}}^{3}}} \]

   5. unpow2 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \sqrt{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}}\right)}^{2}\right)}^{3}}} \]

   6. rem-sqrt-square 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \color{blue}{\left|x \cdot s\right|}\right)}^{2}\right)}^{3}}} \]

   7. *-commutative 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|\color{blue}{s \cdot x}\right|\right)}^{2}\right)}^{3}}} \]

5. Simplified 74.0%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]
6. Step-by-step derivation

   1. pow1 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{1}\right)}}^{2}\right)}^{3}}} \]

   2. metadata-eval 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left({\left(c \cdot \left|s \cdot x\right|\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{2}\right)}^{3}}} \]

   3. sqrt-pow1 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{2}\right)}^{3}}} \]

   4. pow2 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}} \cdot \sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{3}}} \]

   5. add-sqr-sqrt 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}}^{3}}} \]

   6. cbrt-div 74.0%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\cos \left(2 \cdot x\right)}^{3}}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]

   7. rem-cbrt-cube 74.0%

      \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}} \]

   8. rem-cbrt-cube 97.2%

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

   9. unpow2 97.2%

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

   10. associate-/l/ 97.5%

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

7. Applied egg-rr 97.5%

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

8. Final simplification 97.5%

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

Alternative 4: 79.4% accurate, 24.1× speedup

Math

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

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ 1.0 (* t_0 t_0))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this 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
    t_0 = c * (x * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return 1.0 / (t_0 * t_0)

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.7%

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

2. Taylor expanded in x around 0 55.2%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
3. Step-by-step derivation

   1. *-commutative 55.2%

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

   2. unpow2 55.2%

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

   3. unpow2 55.2%

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

   4. swap-sqr 67.9%

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

   5. unpow2 67.9%

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

   6. unpow2 67.9%

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

   7. rem-square-sqrt 67.9%

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

   8. swap-sqr 74.2%

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

   9. unpow2 74.2%

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

   10. unpow2 74.2%

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

   11. rem-sqrt-square 78.1%

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

   12. *-commutative 78.1%

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

4. Simplified 78.1%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}} \]
5. Step-by-step derivation

   1. pow1 78.1%

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

   2. metadata-eval 78.1%

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

   3. sqrt-pow1 78.1%

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

   4. pow2 78.1%

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

   5. add-sqr-sqrt 78.1%

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

   6. unpow2 78.1%

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

   7. add-sqr-sqrt 40.1%

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

   8. fabs-sqr 40.1%

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

   9. add-sqr-sqrt 54.4%

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

   10. add-sqr-sqrt 34.2%

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

   11. fabs-sqr 34.2%

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

   12. add-sqr-sqrt 78.1%

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

6. Applied egg-rr 78.1%

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

7. Final simplification 78.1%

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

Alternative 5: 79.5% accurate, 24.1× speedup

Math

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

FPCore

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ (/ 1.0 t_0) t_0)))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) / t_0;
}

Fortran

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this 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
    t_0 = c * (x * s)
    code = (1.0d0 / t_0) / t_0
end function

Java

c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) / t_0;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return (1.0 / t_0) / t_0

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.7%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. add-cbrt-cube 65.7%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. add-cbrt-cube 60.5%

      \[\leadsto \frac{\sqrt[3]{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}}{\color{blue}{\sqrt[3]{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

   3. cbrt-undiv 60.5%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\cos \left(2 \cdot x\right) \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}}} \]

   4. pow3 60.5%

      \[\leadsto \sqrt[3]{\frac{\color{blue}{{\cos \left(2 \cdot x\right)}^{3}}}{\left(\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \cdot \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}} \]

   5. pow3 60.5%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{\color{blue}{{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}^{3}}}} \]

   6. *-commutative 60.5%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\right)}^{3}}} \]

   7. associate-*r* 55.3%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}\right)}^{3}}} \]

   8. unpow2 55.3%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \left(\color{blue}{{x}^{2}} \cdot {s}^{2}\right)\right)}^{3}}} \]

   9. pow-prod-down 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot \color{blue}{{\left(x \cdot s\right)}^{2}}\right)}^{3}}} \]

3. Applied egg-rr 65.8%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({c}^{2} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}}} \]
4. Step-by-step derivation

   1. unpow2 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\color{blue}{\left(c \cdot c\right)} \cdot {\left(x \cdot s\right)}^{2}\right)}^{3}}} \]

   2. rem-square-sqrt 65.8%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left(\left(c \cdot c\right) \cdot \color{blue}{\left(\sqrt{{\left(x \cdot s\right)}^{2}} \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}\right)}^{3}}} \]

   3. swap-sqr 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right) \cdot \left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)\right)}}^{3}}} \]

   4. unpow2 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \sqrt{{\left(x \cdot s\right)}^{2}}\right)}^{2}\right)}}^{3}}} \]

   5. unpow2 70.4%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \sqrt{\color{blue}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}}\right)}^{2}\right)}^{3}}} \]

   6. rem-sqrt-square 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \color{blue}{\left|x \cdot s\right|}\right)}^{2}\right)}^{3}}} \]

   7. *-commutative 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|\color{blue}{s \cdot x}\right|\right)}^{2}\right)}^{3}}} \]

5. Simplified 74.0%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]
6. Step-by-step derivation

   1. pow1 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{1}\right)}}^{2}\right)}^{3}}} \]

   2. metadata-eval 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\left({\left(c \cdot \left|s \cdot x\right|\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{2}\right)}^{3}}} \]

   3. sqrt-pow1 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\left({\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{2}\right)}^{3}}} \]

   4. pow2 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left(\sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}} \cdot \sqrt{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)}}^{3}}} \]

   5. add-sqr-sqrt 74.0%

      \[\leadsto \sqrt[3]{\frac{{\cos \left(2 \cdot x\right)}^{3}}{{\color{blue}{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}}^{3}}} \]

   6. cbrt-div 74.0%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\cos \left(2 \cdot x\right)}^{3}}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}}} \]

   7. rem-cbrt-cube 74.0%

      \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{\sqrt[3]{{\left({\left(c \cdot \left|s \cdot x\right|\right)}^{2}\right)}^{3}}} \]

   8. rem-cbrt-cube 97.2%

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

   9. unpow2 97.2%

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

   10. associate-/l/ 97.5%

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

7. Applied egg-rr 97.5%

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

8. Taylor expanded in x around 0 78.2%

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

9. Final simplification 78.2%

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

Reproduce

herbie shell --seed 2023307 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))