mixedcos

Percentage Accurate: 66.7% → 98.1%

Time: 17.2s

Alternatives: 11

Speedup: 24.1×

• 31.4% 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: 66.7% 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: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;{s}^{2} \leq 10^{+110}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{\frac{t_0}{x}}{s}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s \cdot c}{\frac{1}{x}}} \cdot \frac{t_0}{x \cdot \left(s \cdot c\right)}\\ \end{array} \end{array} \]

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 (cos (* 2.0 x))))
   (if (<= (pow s 2.0) 1e+110)
     (* (/ 1.0 (* c (* s x))) (/ (/ (/ t_0 x) s) c))
     (* (/ 1.0 (/ (* s c) (/ 1.0 x))) (/ t_0 (* x (* s c)))))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (pow(s, 2.0) <= 1e+110) {
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c);
	} else {
		tmp = (1.0 / ((s * c) / (1.0 / x))) * (t_0 / (x * (s * c)));
	}
	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 = cos((2.0d0 * x))
    if ((s ** 2.0d0) <= 1d+110) then
        tmp = (1.0d0 / (c * (s * x))) * (((t_0 / x) / s) / c)
    else
        tmp = (1.0d0 / ((s * c) / (1.0d0 / x))) * (t_0 / (x * (s * c)))
    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 = Math.cos((2.0 * x));
	double tmp;
	if (Math.pow(s, 2.0) <= 1e+110) {
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c);
	} else {
		tmp = (1.0 / ((s * c) / (1.0 / x))) * (t_0 / (x * (s * c)));
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if math.pow(s, 2.0) <= 1e+110:
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c)
	else:
		tmp = (1.0 / ((s * c) / (1.0 / x))) * (t_0 / (x * (s * c)))
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if ((s ^ 2.0) <= 1e+110)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(s * x))) * Float64(Float64(Float64(t_0 / x) / s) / c));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(s * c) / Float64(1.0 / x))) * Float64(t_0 / Float64(x * Float64(s * c))));
	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 = cos((2.0 * x));
	tmp = 0.0;
	if ((s ^ 2.0) <= 1e+110)
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c);
	else
		tmp = (1.0 / ((s * c) / (1.0 / x))) * (t_0 / (x * (s * c)));
	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[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[s, 2.0], $MachinePrecision], 1e+110], N[(N[(1.0 / N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / x), $MachinePrecision] / s), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(s * c), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s 2) < 1e110

   1. Initial program 65.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in x around inf 97.2%

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

      1. associate-/r* 97.2%

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

      2. *-commutative 97.2%

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

      3. *-commutative 97.2%

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

      4. *-rgt-identity 97.2%

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

      5. associate-*r/ 96.4%

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

      6. associate-*l/ 96.3%

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

      7. associate-*r/ 96.3%

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

      8. *-rgt-identity 96.3%

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

      9. *-commutative 96.3%

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

   6. Simplified 96.3%

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

      1. *-un-lft-identity 96.3%

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

      2. times-frac 96.4%

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

   8. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.5%

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

      2. *-lft-identity 96.5%

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

   10. Simplified 96.5%

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

   if 1e110 < (pow.f64 s 2)

   1. Initial program 70.1%

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

   3. Applied egg-rr 96.4%

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

      1. /-rgt-identity 96.4%

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

      2. *-commutative 96.4%

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

      3. associate-*r* 95.6%

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

      4. associate-/l* 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in x around inf 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-*r* 98.8%

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

   8. Simplified 98.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 10^{+110}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s \cdot c}{\frac{1}{x}}} \cdot \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot c\right)}\\ \end{array} \]

Alternative 2: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;{s}^{2} \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{\frac{t_0}{x}}{s}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1 \cdot t_1}\\ \end{array} \end{array} \]

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 (cos (* 2.0 x))) (t_1 (* x (* s c))))
   (if (<= (pow s 2.0) 2e+43)
     (* (/ 1.0 (* c (* s x))) (/ (/ (/ t_0 x) s) c))
     (* t_0 (/ 1.0 (* t_1 t_1))))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double t_1 = x * (s * c);
	double tmp;
	if (pow(s, 2.0) <= 2e+43) {
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c);
	} else {
		tmp = t_0 * (1.0 / (t_1 * t_1));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    t_1 = x * (s * c)
    if ((s ** 2.0d0) <= 2d+43) then
        tmp = (1.0d0 / (c * (s * x))) * (((t_0 / x) / s) / c)
    else
        tmp = t_0 * (1.0d0 / (t_1 * t_1))
    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 = Math.cos((2.0 * x));
	double t_1 = x * (s * c);
	double tmp;
	if (Math.pow(s, 2.0) <= 2e+43) {
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c);
	} else {
		tmp = t_0 * (1.0 / (t_1 * t_1));
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((2.0 * x))
	t_1 = x * (s * c)
	tmp = 0
	if math.pow(s, 2.0) <= 2e+43:
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c)
	else:
		tmp = t_0 * (1.0 / (t_1 * t_1))
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	t_1 = Float64(x * Float64(s * c))
	tmp = 0.0
	if ((s ^ 2.0) <= 2e+43)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(s * x))) * Float64(Float64(Float64(t_0 / x) / s) / c));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(t_1 * t_1)));
	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 = cos((2.0 * x));
	t_1 = x * (s * c);
	tmp = 0.0;
	if ((s ^ 2.0) <= 2e+43)
		tmp = (1.0 / (c * (s * x))) * (((t_0 / x) / s) / c);
	else
		tmp = t_0 * (1.0 / (t_1 * t_1));
	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[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[s, 2.0], $MachinePrecision], 2e+43], N[(N[(1.0 / N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / x), $MachinePrecision] / s), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s 2) < 2.00000000000000003e43

   1. Initial program 63.6%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in x around inf 97.0%

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

      1. associate-/r* 97.0%

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

      2. *-commutative 97.0%

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

      3. *-commutative 97.0%

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

      4. *-rgt-identity 97.0%

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

      5. associate-*r/ 96.1%

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

      6. associate-*l/ 96.0%

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

      7. associate-*r/ 96.0%

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

      8. *-rgt-identity 96.0%

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

      9. *-commutative 96.0%

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

   6. Simplified 96.0%

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

      1. *-un-lft-identity 96.0%

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

      2. times-frac 96.1%

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

   8. Applied egg-rr 96.1%

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

      1. associate-*l/ 96.2%

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

      2. *-lft-identity 96.2%

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

   10. Simplified 96.2%

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

   if 2.00000000000000003e43 < (pow.f64 s 2)

   1. Initial program 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. associate-*r* 70.8%

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

      3. times-frac 70.8%

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

      4. *-commutative 70.8%

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

      5. associate-*r* 68.5%

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

      6. pow-prod-down 83.1%

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

   3. Applied egg-rr 83.1%

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

   5. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(x \cdot 2\right) \cdot {\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 95.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x \cdot 2\right) \cdot {\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 96.8%

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

      3. *-commutative 96.8%

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

      4. associate-*r* 98.9%

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

   7. Simplified 98.9%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

      3. sqrt-pow1 98.9%

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

      4. metadata-eval 98.9%

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

      5. unpow-1 98.9%

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

      6. associate-/l/ 98.8%

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

      7. clear-num 98.8%

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

      8. pow2 98.8%

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

      9. frac-2neg 98.8%

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

      10. metadata-eval 98.8%

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

      11. frac-2neg 98.8%

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

      12. metadata-eval 98.8%

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

      13. frac-times 98.0%

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

      14. metadata-eval 98.0%

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

      15. associate-/r/ 98.1%

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

      16. /-rgt-identity 98.1%

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

      17. distribute-rgt-neg-in 98.1%

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

      18. associate-/r/ 98.1%

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

      19. /-rgt-identity 98.1%

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

   9. Applied egg-rr 98.1%

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

4. Final simplification 97.1%

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

Alternative 3: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \cos \left(2 \cdot x\right) \cdot \frac{\frac{\frac{1}{c}}{s \cdot x}}{c \cdot \left(s \cdot x\right)} \end{array} \]

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)) (/ (/ (/ 1.0 c) (* s x)) (* c (* s x)))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return cos((2.0 * x)) * (((1.0 / c) / (s * x)) / (c * (s * x)));
}

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)) * (((1.0d0 / c) / (s * x)) / (c * (s * x)))
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)) * (((1.0 / c) / (s * x)) / (c * (s * x)));
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return math.cos((2.0 * x)) * (((1.0 / c) / (s * x)) / (c * (s * x)))

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) * Float64(Float64(Float64(1.0 / c) / Float64(s * x)) / Float64(c * Float64(s * x))))
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)) * (((1.0 / c) / (s * x)) / (c * (s * x)));
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[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(1.0 / c), $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision] / N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

   1. *-un-lft-identity 67.3%

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

   2. associate-*r* 67.5%

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

   3. times-frac 67.7%

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

   4. *-commutative 67.7%

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

   5. associate-*r* 65.3%

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

   6. pow-prod-down 82.6%

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

3. Applied egg-rr 82.6%

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

5. Applied egg-rr 73.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos \left(x \cdot 2\right) \cdot {\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 87.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x \cdot 2\right) \cdot {\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\right)\right)} \]

   2. expm1-log1p 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-*r* 96.8%

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

7. Simplified 96.8%

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

   1. metadata-eval 96.8%

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

   2. pow-prod-up 96.7%

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

   3. pow-prod-down 96.2%

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

   4. associate-*l* 94.3%

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

   5. *-commutative 94.3%

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

   6. associate-*r* 91.7%

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

   7. inv-pow 91.7%

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

   8. associate-*r* 94.3%

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

   9. *-commutative 94.3%

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

   10. associate-*l* 96.2%

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

   11. associate-/r* 96.8%

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

   12. metadata-eval 96.8%

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

   13. associate-*l* 94.6%

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

   14. *-commutative 94.6%

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

   15. frac-times 94.0%

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

   16. associate-*l/ 94.6%

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

   17. *-un-lft-identity 94.6%

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

   18. associate-*l* 96.9%

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

9. Applied egg-rr 96.9%

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

10. Final simplification 96.9%

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

Alternative 4: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \frac{1}{c \cdot \left(s \cdot x\right)} \cdot \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s}}{c} \end{array} \]

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
 (* (/ 1.0 (* c (* s x))) (/ (/ (/ (cos (* 2.0 x)) x) s) c)))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return (1.0 / (c * (s * x))) * (((cos((2.0 * x)) / x) / s) / c);
}

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 = (1.0d0 / (c * (s * x))) * (((cos((2.0d0 * x)) / x) / s) / c)
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 (1.0 / (c * (s * x))) * (((Math.cos((2.0 * x)) / x) / s) / c);
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return (1.0 / (c * (s * x))) * (((math.cos((2.0 * x)) / x) / s) / c)

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(Float64(1.0 / Float64(c * Float64(s * x))) * Float64(Float64(Float64(cos(Float64(2.0 * x)) / x) / s) / c))
end

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = (1.0 / (c * (s * x))) * (((cos((2.0 * x)) / x) / s) / c);
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[(1.0 / N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / s), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

3. Applied egg-rr 96.8%

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

4. Taylor expanded in x around inf 96.8%

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

   1. associate-/r* 96.9%

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

   2. *-commutative 96.9%

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

   3. *-commutative 96.9%

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

   4. *-rgt-identity 96.9%

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

   5. associate-*r/ 96.4%

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

   6. associate-*l/ 96.4%

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

   7. associate-*r/ 96.4%

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

   8. *-rgt-identity 96.4%

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

   9. *-commutative 96.4%

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

6. Simplified 96.4%

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

   1. *-un-lft-identity 96.4%

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

   2. times-frac 96.4%

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

8. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.5%

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

   2. *-lft-identity 96.5%

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

10. Simplified 96.5%

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

11. Final simplification 96.5%

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

Alternative 5: 96.8% accurate, 2.7× speedup

Math

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

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 (* s x)))) (/ (/ (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 * (s * x);
	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 * (s * x)
    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 * (s * x);
	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 * (s * x)
	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(s * x))
	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 * (s * x);
	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[(s * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 67.3%

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

   1. *-un-lft-identity 67.3%

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

   2. associate-*r* 67.5%

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

   3. times-frac 67.7%

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

   4. *-commutative 67.7%

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

   5. associate-*r* 65.3%

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

   6. pow-prod-down 82.6%

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

3. Applied egg-rr 82.6%

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

   1. clear-num 82.6%

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

   2. frac-times 82.2%

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

   3. *-un-lft-identity 82.2%

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

   4. /-rgt-identity 82.2%

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

   5. associate-*l* 75.1%

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

   6. unpow2 75.1%

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

   7. swap-sqr 96.2%

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

   8. associate-*r* 94.3%

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

   9. *-commutative 94.3%

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

   10. associate-*r* 96.5%

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

   11. *-commutative 96.5%

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

   12. associate-/r* 96.9%

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

   13. *-commutative 96.9%

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

5. Applied egg-rr 96.9%

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

6. Final simplification 96.9%

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

Alternative 6: 79.7% accurate, 20.9× speedup

Math

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

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 (/ 1.0 (* c (* s 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 = 1.0 / (c * (s * x));
	return 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 = 1.0d0 / (c * (s * x))
    code = 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 = 1.0 / (c * (s * x));
	return t_0 * t_0;
}

Python

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

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(1.0 / Float64(c * Float64(s * x)))
	return 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 = 1.0 / (c * (s * x));
	tmp = 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[(1.0 / N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 67.3%

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

3. Applied egg-rr 96.8%

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

4. Taylor expanded in x around 0 78.1%

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

5. Final simplification 78.1%

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

Alternative 7: 79.7% accurate, 20.9× speedup

Math

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

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
 (* (/ 1.0 (* c (* s x))) (/ (/ (/ 1.0 s) x) c)))

C

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

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 = (1.0d0 / (c * (s * x))) * (((1.0d0 / s) / x) / c)
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 (1.0 / (c * (s * x))) * (((1.0 / s) / x) / c);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 67.3%

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

3. Applied egg-rr 96.8%

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

4. Taylor expanded in x around 0 78.1%

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

   1. associate-/r* 78.1%

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

   2. *-commutative 78.1%

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

   3. *-lft-identity 78.1%

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

   4. associate-*l/ 77.6%

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

   5. associate-*r/ 77.6%

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

   6. *-rgt-identity 77.6%

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

   7. associate-/l/ 77.6%

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

6. Simplified 77.6%

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

7. Final simplification 77.6%

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

Alternative 8: 79.7% accurate, 20.9× speedup

Math

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

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
 (* (/ 1.0 (* c (* s x))) (/ (/ (/ 1.0 x) s) c)))

C

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

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 = (1.0d0 / (c * (s * x))) * (((1.0d0 / x) / s) / c)
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 (1.0 / (c * (s * x))) * (((1.0 / x) / s) / c);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 67.3%

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

3. Applied egg-rr 96.8%

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

4. Taylor expanded in x around inf 96.8%

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

   1. associate-/r* 96.9%

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

   2. *-commutative 96.9%

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

   3. *-commutative 96.9%

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

   4. *-rgt-identity 96.9%

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

   5. associate-*r/ 96.4%

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

   6. associate-*l/ 96.4%

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

   7. associate-*r/ 96.4%

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

   8. *-rgt-identity 96.4%

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

   9. *-commutative 96.4%

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

6. Simplified 96.4%

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

   1. *-un-lft-identity 96.4%

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

   2. times-frac 96.4%

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

8. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.5%

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

   2. *-lft-identity 96.5%

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

10. Simplified 96.5%

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

11. Taylor expanded in x around 0 77.6%

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

12. Final simplification 77.6%

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

Alternative 9: 76.6% accurate, 24.1× speedup

Math

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

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 (/ 1.0 (* (* s c) (* x (* c (* s x))))))

C

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

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 = 1.0d0 / ((s * c) * (x * (c * (s * x))))
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 1.0 / ((s * c) * (x * (c * (s * x))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0 54.0%

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

   1. unpow2 54.0%

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

   2. *-commutative 54.0%

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

   3. unpow2 54.0%

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

   4. unpow2 54.0%

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

   5. swap-sqr 66.0%

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

   6. swap-sqr 78.0%

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

   7. unpow2 78.0%

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

   8. *-commutative 78.0%

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

4. Simplified 78.0%

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

   1. unpow2 78.0%

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

   2. associate-*r* 77.6%

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

   3. *-commutative 77.6%

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

   4. associate-*l* 76.7%

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

6. Applied egg-rr 76.7%

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

7. Final simplification 76.7%

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

Alternative 10: 76.7% accurate, 24.1× speedup

Math

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

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 (/ 1.0 (* (* s c) (* x (* x (* s c))))))

C

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

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 = 1.0d0 / ((s * c) * (x * (x * (s * c))))
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 1.0 / ((s * c) * (x * (x * (s * c))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0 54.0%

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

   1. unpow2 54.0%

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

   2. *-commutative 54.0%

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

   3. unpow2 54.0%

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

   4. unpow2 54.0%

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

   5. swap-sqr 66.0%

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

   6. swap-sqr 78.0%

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

   7. unpow2 78.0%

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

   8. *-commutative 78.0%

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

4. Simplified 78.0%

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

   1. unpow2 78.0%

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

   2. associate-*r* 77.6%

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

   3. *-commutative 77.6%

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

   4. associate-*l* 76.7%

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

6. Applied egg-rr 76.7%

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

7. Taylor expanded in c around 0 76.7%

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

   1. associate-*r* 77.6%

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

9. Simplified 77.6%

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

10. Final simplification 77.6%

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

Alternative 11: 79.6% accurate, 24.1× speedup

Math

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

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 (* s x)))) (/ 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 * (s * x);
	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 * (s * x)
    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 * (s * x);
	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 * (s * x)
	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(s * x))
	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 * (s * x);
	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[(s * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0 54.0%

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

   1. unpow2 54.0%

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

   2. *-commutative 54.0%

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

   3. unpow2 54.0%

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

   4. unpow2 54.0%

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

   5. swap-sqr 66.0%

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

   6. swap-sqr 78.0%

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

   7. unpow2 78.0%

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

   8. *-commutative 78.0%

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

4. Simplified 78.0%

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

   1. *-commutative 78.0%

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

   2. unpow2 78.0%

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

6. Applied egg-rr 78.0%

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

7. Final simplification 78.0%

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

Reproduce

herbie shell --seed 2023320 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))