mixedcos

Percentage Accurate: 66.6% → 97.3%

Time: 10.6s

Alternatives: 9

Speedup: 24.1×

• 31.5% 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: 66.6% 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: 97.3% accurate, 1.5× speedup

Math

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

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 (* x 2.0)) (pow (* s (* x c)) -2.0)))

C

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

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
    code = cos((x * 2.0d0)) * ((s * (x * c)) ** (-2.0d0))
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((x * 2.0)) * Math.pow((s * (x * c)), -2.0);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 57.7%

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

   3. associate-*r* 58.1%

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

   4. unpow2 58.1%

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

   5. unswap-sqr 76.8%

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

   6. unpow2 76.8%

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

   7. swap-sqr 96.7%

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

   8. *-commutative 96.7%

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

   9. *-commutative 96.7%

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

   10. *-commutative 96.7%

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

   11. *-commutative 96.7%

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

3. Simplified 96.7%

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

   1. div-inv 96.7%

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

   2. *-commutative 96.7%

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

   3. pow2 96.7%

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

   4. pow-flip 97.0%

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

   5. metadata-eval 97.0%

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

5. Applied egg-rr 97.0%

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

6. Final simplification 97.0%

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

Alternative 2: 87.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-5}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= x 6e-5)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* s (* s (* x (* c (* x c))))))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 6e-5) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * 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) :: tmp
    if (x <= 6d-5) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (s * (s * (x * (c * (x * 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 tmp;
	if (x <= 6e-5) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	}
	return tmp;
}

Python

c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 6e-5:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))))
	return tmp

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 6e-5)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(s * Float64(x * Float64(c * Float64(x * 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)
	tmp = 0.0;
	if (x <= 6e-5)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * 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_] := If[LessEqual[x, 6e-5], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(s * N[(x * N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000015e-5

   1. Initial program 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-*r* 55.9%

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

      3. associate-*r* 56.4%

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

      4. unpow2 56.4%

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

      5. unswap-sqr 75.2%

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

      6. unpow2 75.2%

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

      7. swap-sqr 97.3%

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

      8. *-commutative 97.3%

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

      9. *-commutative 97.3%

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

      10. *-commutative 97.3%

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

      11. *-commutative 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in x around 0 53.5%

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

      1. unpow2 53.5%

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

      2. unpow2 53.5%

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

      3. *-commutative 53.5%

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

      4. unpow2 53.5%

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

   6. Simplified 53.5%

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

      1. inv-pow 53.5%

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

      2. unswap-sqr 70.3%

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

      3. unpow2 70.3%

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

      4. *-commutative 70.3%

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

      5. pow2 70.3%

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

      6. pow-prod-down 84.3%

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

      7. *-commutative 84.3%

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

      8. associate-*r* 83.7%

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

      9. pow2 83.7%

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

      10. pow-prod-down 83.8%

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

      11. pow-prod-up 83.8%

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

      12. metadata-eval 83.8%

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

   8. Applied egg-rr 83.8%

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

   9. Taylor expanded in s around 0 84.4%

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

   if 6.00000000000000015e-5 < x

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-*l* 63.4%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. unpow2 61.8%

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

      6. associate-*r* 63.8%

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

      7. associate-*r* 65.5%

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

      8. *-commutative 65.5%

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

      9. unpow2 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. unpow2 67.0%

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

      3. associate-*r* 77.3%

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

      4. unpow2 77.3%

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

      5. associate-*r* 87.0%

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

      6. *-commutative 87.0%

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

   6. Simplified 87.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-5}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 3: 94.8% accurate, 2.7× speedup

Math

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

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

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return cos((x * 2.0)) / ((s * (x * c)) * (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((x * 2.0d0)) / ((s * (x * c)) * (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((x * 2.0)) / ((s * (x * c)) * (c * (x * s)));
}

Python

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

Julia

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

Derivation

1. Initial program 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 57.7%

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

   3. associate-*r* 58.1%

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

   4. unpow2 58.1%

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

   5. unswap-sqr 76.8%

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

   6. unpow2 76.8%

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

   7. swap-sqr 96.7%

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

   8. *-commutative 96.7%

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

   9. *-commutative 96.7%

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

   10. *-commutative 96.7%

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

   11. *-commutative 96.7%

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

3. Simplified 96.7%

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

4. Taylor expanded in s around 0 94.1%

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

5. Final simplification 94.1%

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

Alternative 4: 97.0% accurate, 2.7× speedup

Math

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

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 (* s (* x c)))) (/ (cos (* x 2.0)) (* 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 = s * (x * c);
	return cos((x * 2.0)) / (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 = s * (x * c)
    code = cos((x * 2.0d0)) / (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 = s * (x * c);
	return Math.cos((x * 2.0)) / (t_0 * t_0);
}

Python

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

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	return Float64(cos(Float64(x * 2.0)) / 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 = s * (x * c);
	tmp = cos((x * 2.0)) / (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[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 57.7%

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

   3. associate-*r* 58.1%

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

   4. unpow2 58.1%

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

   5. unswap-sqr 76.8%

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

   6. unpow2 76.8%

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

   7. swap-sqr 96.7%

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

   8. *-commutative 96.7%

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

   9. *-commutative 96.7%

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

   10. *-commutative 96.7%

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

   11. *-commutative 96.7%

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

3. Simplified 96.7%

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

4. Final simplification 96.7%

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

Alternative 5: 97.3% accurate, 2.7× speedup

Math

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

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 (* s (* x c)))) (/ (/ (cos (* x 2.0)) 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 = s * (x * c);
	return (cos((x * 2.0)) / 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 = s * (x * c)
    code = (cos((x * 2.0d0)) / 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 = s * (x * c);
	return (Math.cos((x * 2.0)) / t_0) / t_0;
}

Python

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

Julia

c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	return Float64(Float64(cos(Float64(x * 2.0)) / 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 = s * (x * c);
	tmp = (cos((x * 2.0)) / 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[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 57.7%

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

   3. associate-*r* 58.1%

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

   4. unpow2 58.1%

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

   5. unswap-sqr 76.8%

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

   6. unpow2 76.8%

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

   7. swap-sqr 96.7%

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

   8. *-commutative 96.7%

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

   9. *-commutative 96.7%

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

   10. *-commutative 96.7%

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

   11. *-commutative 96.7%

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

3. Simplified 96.7%

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

   1. div-inv 96.7%

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

   2. *-commutative 96.7%

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

   3. pow2 96.7%

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

   4. pow-flip 97.0%

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

   5. metadata-eval 97.0%

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

5. Applied egg-rr 97.0%

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

   1. metadata-eval 97.0%

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

   2. pow-prod-up 96.9%

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

   3. unpow-1 96.9%

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

   4. unpow-1 96.9%

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

7. Applied egg-rr 96.9%

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

   1. associate-*r* 96.8%

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

   2. un-div-inv 96.9%

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

   3. un-div-inv 96.9%

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

9. Applied egg-rr 96.9%

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

10. Final simplification 96.9%

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

Alternative 6: 80.0% accurate, 3.0× speedup

Math

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

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 (pow (* c (* x s)) -2.0))

C

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

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
    code = (c * (x * s)) ** (-2.0d0)
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.pow((c * (x * s)), -2.0);
}

Python

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

Julia

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

MATLAB

c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = (c * (x * s)) ^ -2.0;
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_] := N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 57.7%

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

   3. associate-*r* 58.1%

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

   4. unpow2 58.1%

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

   5. unswap-sqr 76.8%

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

   6. unpow2 76.8%

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

   7. swap-sqr 96.7%

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

   8. *-commutative 96.7%

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

   9. *-commutative 96.7%

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

   10. *-commutative 96.7%

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

   11. *-commutative 96.7%

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

3. Simplified 96.7%

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

4. Taylor expanded in x around 0 52.0%

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

   1. unpow2 52.0%

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

   2. unpow2 52.0%

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

   3. *-commutative 52.0%

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

   4. unpow2 52.0%

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

6. Simplified 52.0%

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

   1. inv-pow 52.0%

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

   2. unswap-sqr 65.4%

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

   3. unpow2 65.4%

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

   4. *-commutative 65.4%

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

   5. pow2 65.4%

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

   6. pow-prod-down 77.3%

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

   7. *-commutative 77.3%

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

   8. associate-*r* 76.8%

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

   9. pow2 76.8%

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

   10. pow-prod-down 76.9%

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

   11. pow-prod-up 76.9%

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

   12. metadata-eval 76.9%

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

8. Applied egg-rr 76.9%

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

9. Taylor expanded in s around 0 77.4%

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

10. Final simplification 77.4%

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

Alternative 7: 79.9% accurate, 20.9× speedup

Math

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

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) (* x s)))) (* 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) / (x * s);
	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) / (x * s)
    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) / (x * s);
	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) / (x * s)
	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(Float64(1.0 / c) / Float64(x * s))
	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) / (x * s);
	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[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.1%

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

   1. associate-/r* 64.2%

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

   2. unpow2 64.2%

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

   3. *-commutative 64.2%

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

   4. unpow2 64.2%

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

3. Simplified 64.2%

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

4. Taylor expanded in x around 0 57.0%

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

   1. unpow2 57.0%

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

6. Simplified 57.0%

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

   1. add-sqr-sqrt 57.0%

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

   2. sqrt-div 57.0%

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

   3. sqrt-div 57.0%

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

   4. metadata-eval 57.0%

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

   5. sqrt-prod 26.5%

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

   6. add-sqr-sqrt 40.9%

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

   7. associate-*r* 35.5%

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

   8. sqrt-prod 35.5%

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

   9. sqrt-unprod 22.0%

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

   10. add-sqr-sqrt 40.7%

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

   11. sqrt-prod 21.7%

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

   12. add-sqr-sqrt 43.3%

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

   13. *-commutative 43.3%

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

   14. sqrt-div 43.3%

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

   15. sqrt-div 44.1%

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

   16. metadata-eval 44.1%

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

   17. sqrt-prod 22.3%

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

   18. add-sqr-sqrt 47.6%

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

   19. associate-*r* 42.1%

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

   20. sqrt-prod 43.2%

       \[\leadsto \frac{\frac{1}{c}}{s \cdot x} \cdot \frac{\frac{1}{c}}{\color{blue}{\sqrt{x \cdot x} \cdot \sqrt{s \cdot s}}} \]

8. Applied egg-rr 77.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{c}}{s \cdot x} \cdot \frac{\frac{1}{c}}{s \cdot x}} \]

9. Final simplification 77.4%

   \[\leadsto \frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s} \]

Alternative 8: 74.9% accurate, 24.1× speedup

Math

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

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 (* (* x c) (* c (* x s))))))

C

c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return 1.0 / (s * ((x * c) * (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 = 1.0d0 / (s * ((x * c) * (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 1.0 / (s * ((x * c) * (c * (x * s))));
}

Python

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

Julia

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

Derivation

1. Initial program 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 57.7%

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

   3. associate-*r* 58.1%

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

   4. unpow2 58.1%

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

   5. unswap-sqr 76.8%

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

   6. unpow2 76.8%

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

   7. swap-sqr 96.7%

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

   8. *-commutative 96.7%

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

   9. *-commutative 96.7%

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

   10. *-commutative 96.7%

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

   11. *-commutative 96.7%

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

3. Simplified 96.7%

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

4. Taylor expanded in x around 0 52.0%

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

   1. unpow2 52.0%

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

   2. unpow2 52.0%

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

   3. *-commutative 52.0%

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

   4. unpow2 52.0%

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

6. Simplified 52.0%

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

   1. inv-pow 52.0%

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

   2. unswap-sqr 65.4%

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

   3. unpow2 65.4%

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

   4. *-commutative 65.4%

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

   5. pow2 65.4%

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

   6. pow-prod-down 77.3%

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

   7. *-commutative 77.3%

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

   8. associate-*r* 76.8%

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

   9. pow2 76.8%

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

   10. pow-prod-down 76.9%

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

   11. pow-prod-up 76.9%

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

   12. metadata-eval 76.9%

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

8. Applied egg-rr 76.9%

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

   1. metadata-eval 76.9%

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

   2. pow-prod-up 76.9%

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

   3. pow-prod-down 76.8%

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

   4. pow2 76.8%

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

   5. associate-*r* 77.3%

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

   6. *-commutative 77.3%

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

   7. pow-prod-down 65.4%

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

   8. pow2 65.4%

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

   9. *-commutative 65.4%

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

   10. unpow2 65.4%

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

   11. unswap-sqr 52.0%

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

   12. inv-pow 52.0%

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

   13. unswap-sqr 65.4%

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

   14. unpow2 65.4%

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

   15. *-commutative 65.4%

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

   16. pow2 65.4%

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

   17. pow-prod-down 77.3%

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

   18. *-commutative 77.3%

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

   19. associate-*r* 76.8%

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

10. Applied egg-rr 76.8%

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

    1. unpow2 76.8%

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

    2. associate-*r* 76.3%

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

    3. *-commutative 76.3%

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

    4. associate-*l* 75.2%

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

    5. *-commutative 75.2%

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

12. Applied egg-rr 75.2%

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

13. Final simplification 75.2%

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

Alternative 9: 79.9% 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 65.1%

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

   1. *-commutative 65.1%

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

   2. associate-*r* 57.7%

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

   3. associate-*r* 58.1%

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

   4. unpow2 58.1%

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

   5. unswap-sqr 76.8%

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

   6. unpow2 76.8%

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

   7. swap-sqr 96.7%

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

   8. *-commutative 96.7%

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

   9. *-commutative 96.7%

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

   10. *-commutative 96.7%

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

   11. *-commutative 96.7%

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

3. Simplified 96.7%

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

4. Taylor expanded in x around 0 52.0%

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

   1. unpow2 52.0%

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

   2. unpow2 52.0%

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

   3. *-commutative 52.0%

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

   4. unpow2 52.0%

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

6. Simplified 52.0%

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

   1. inv-pow 52.0%

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

   2. unswap-sqr 65.4%

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

   3. unpow2 65.4%

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

   4. *-commutative 65.4%

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

   5. pow2 65.4%

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

   6. pow-prod-down 77.3%

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

   7. *-commutative 77.3%

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

   8. associate-*r* 76.8%

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

   9. pow2 76.8%

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

   10. pow-prod-down 76.9%

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

   11. pow-prod-up 76.9%

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

   12. metadata-eval 76.9%

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

8. Applied egg-rr 76.9%

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

   1. metadata-eval 76.9%

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

   2. pow-prod-up 76.9%

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

   3. pow-prod-down 76.8%

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

   4. pow2 76.8%

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

   5. associate-*r* 77.3%

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

   6. *-commutative 77.3%

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

   7. pow-prod-down 65.4%

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

   8. pow2 65.4%

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

   9. *-commutative 65.4%

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

   10. unpow2 65.4%

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

   11. unswap-sqr 52.0%

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

   12. inv-pow 52.0%

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

   13. unswap-sqr 65.4%

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

   14. unpow2 65.4%

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

   15. *-commutative 65.4%

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

   16. pow2 65.4%

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

   17. pow-prod-down 77.3%

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

   18. *-commutative 77.3%

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

   19. associate-*r* 76.8%

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

10. Applied egg-rr 76.8%

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

    1. unpow2 76.8%

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

    2. associate-*r* 76.3%

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

    3. *-commutative 76.3%

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

    4. *-commutative 76.3%

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

    5. associate-*r* 77.3%

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

    6. *-commutative 77.3%

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

    7. *-commutative 77.3%

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

12. Applied egg-rr 77.3%

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

13. Final simplification 77.3%

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

Reproduce

herbie shell --seed 2023229 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))