mixedcos

Percentage Accurate: 67.8% → 97.0%

Time: 15.1s

Alternatives: 10

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.8% 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: 97.0% accurate, 2.7× speedup

Math

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

FPCore

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 (* 2.0 x)) (* t_0 t_0))))

C

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

Fortran

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((2.0d0 * x)) / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.8%

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

   1. associate-/r* 64.9%

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

   2. remove-double-neg 64.9%

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

   3. distribute-lft-neg-out 64.9%

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

   4. distribute-lft-neg-out 64.9%

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

   5. distribute-rgt-neg-out 64.9%

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

   6. associate-/l/ 64.8%

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

   7. distribute-rgt-neg-out 64.8%

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

   8. distribute-lft-neg-out 64.8%

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

   9. associate-*l* 66.6%

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

   10. distribute-lft-neg-in 66.6%

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

   11. distribute-lft-neg-out 66.6%

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

   12. remove-double-neg 66.6%

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

   13. associate-*r* 66.5%

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

   14. *-commutative 66.5%

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

   15. associate-*r* 65.6%

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

3. Simplified 71.7%

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

4. Taylor expanded in x around 0 57.9%

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

   1. unpow2 57.9%

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

   2. associate-*r* 57.9%

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

   3. unpow2 57.9%

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

   4. unpow2 57.9%

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

   5. swap-sqr 71.6%

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

   6. swap-sqr 95.2%

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

   7. *-commutative 95.2%

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

   8. *-commutative 95.2%

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

   9. unpow2 95.2%

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

   10. *-commutative 95.2%

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

   11. *-commutative 95.2%

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

   12. associate-*l* 97.0%

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

6. Simplified 97.0%

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

   1. unpow2 97.0%

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

8. Applied egg-rr 97.0%

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

9. Final simplification 97.0%

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

Alternative 2: 85.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 1.05e-13)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* 2.0 x)) (* x (* s (* (* x c) (* s c)))))))

C

assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.05e-13) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((2.0 * x)) / (x * (s * ((x * c) * (s * c))));
	}
	return tmp;
}

Fortran

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 <= 1.05d-13) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((2.0d0 * x)) / (x * (s * ((x * c) * (s * c))))
    end if
    code = tmp
end function

Java

assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.05e-13) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((2.0 * x)) / (x * (s * ((x * c) * (s * c))));
	}
	return tmp;
}

Python

[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 1.05e-13:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((2.0 * x)) / (x * (s * ((x * c) * (s * c))))
	return tmp

Julia

c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 1.05e-13)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(s * Float64(Float64(x * c) * Float64(s * c)))));
	end
	return tmp
end

MATLAB

c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1.05e-13)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((2.0 * x)) / (x * (s * ((x * c) * (s * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 1.05e-13], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(s * N[(N[(x * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04999999999999994e-13

   1. Initial program 64.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. remove-double-neg 64.2%

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

      3. distribute-lft-neg-out 64.2%

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

      4. distribute-lft-neg-out 64.2%

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

      5. distribute-rgt-neg-out 64.2%

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

      6. associate-/l/ 64.2%

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

      7. distribute-rgt-neg-out 64.2%

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

      8. distribute-lft-neg-out 64.2%

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

      9. associate-*l* 66.2%

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

      10. distribute-lft-neg-in 66.2%

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

      11. distribute-lft-neg-out 66.2%

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

      12. remove-double-neg 66.2%

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

      13. associate-*r* 65.9%

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

      14. *-commutative 65.9%

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

      15. associate-*r* 64.3%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around 0 57.3%

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

      1. unpow2 57.3%

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

      2. associate-*r* 56.9%

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

      3. unpow2 56.9%

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

      4. unpow2 56.9%

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

      5. swap-sqr 70.0%

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

      6. swap-sqr 94.6%

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

      7. *-commutative 94.6%

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

      8. *-commutative 94.6%

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

      9. unpow2 94.6%

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

      10. *-commutative 94.6%

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

      11. *-commutative 94.6%

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

      12. associate-*l* 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in x around 0 54.2%

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

      1. unpow2 54.2%

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

      2. unpow2 54.2%

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

      3. unpow2 54.2%

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

      4. associate-*r* 54.1%

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

      5. *-commutative 54.1%

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

      6. swap-sqr 66.2%

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

      7. swap-sqr 86.3%

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

      8. associate-*r* 85.0%

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

      9. associate-*r* 88.3%

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

      10. associate-/r* 88.2%

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

      11. *-lft-identity 88.2%

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

      12. associate-*l/ 88.2%

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

      13. unpow-1 88.2%

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

      14. unpow-1 88.2%

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

      15. pow-sqr 88.3%

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

      16. associate-*r* 86.3%

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

      17. *-commutative 86.3%

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

      18. associate-*r* 89.1%

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

      19. metadata-eval 89.1%

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

   9. Simplified 89.1%

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

   if 1.04999999999999994e-13 < x

   1. Initial program 66.7%

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

      1. associate-/r* 67.1%

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

      2. remove-double-neg 67.1%

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

      3. distribute-lft-neg-out 67.1%

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

      4. distribute-lft-neg-out 67.1%

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

      5. distribute-rgt-neg-out 67.1%

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

      6. associate-/l/ 66.7%

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

      7. distribute-rgt-neg-out 66.7%

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

      8. distribute-lft-neg-out 66.7%

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

      9. associate-*l* 68.1%

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

      10. distribute-lft-neg-in 68.1%

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

      11. distribute-lft-neg-out 68.1%

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

      12. remove-double-neg 68.1%

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

      13. associate-*r* 68.1%

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

      14. *-commutative 68.1%

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

      15. associate-*r* 69.7%

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

   3. Simplified 77.6%

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

      1. add-sqr-sqrt 77.6%

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

      2. pow2 77.6%

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

      3. associate-*r* 69.7%

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

      4. swap-sqr 85.4%

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

      5. *-commutative 85.4%

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

      6. sqrt-prod 85.3%

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

      7. sqrt-prod 48.5%

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

      8. add-sqr-sqrt 90.7%

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

   5. Applied egg-rr 90.7%

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

      1. unpow2 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*r* 90.7%

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

      4. associate-*r* 90.7%

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

      5. *-commutative 90.7%

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

      6. add-sqr-sqrt 90.7%

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

      7. associate-*r* 89.3%

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

      8. *-commutative 89.3%

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

   7. Applied egg-rr 89.3%

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

      1. associate-*l* 86.4%

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

      2. *-commutative 86.4%

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

   9. Simplified 86.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 3: 86.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-59}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right) \cdot \left(s \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 1e-59)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* 2.0 x)) (* (* x (* s (* x c))) (* s c)))))

C

assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 1e-59) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((2.0 * x)) / ((x * (s * (x * c))) * (s * c));
	}
	return tmp;
}

Fortran

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 <= 1d-59) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((2.0d0 * x)) / ((x * (s * (x * c))) * (s * c))
    end if
    code = tmp
end function

Java

assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1e-59) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((2.0 * x)) / ((x * (s * (x * c))) * (s * c));
	}
	return tmp;
}

Python

[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 1e-59:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((2.0 * x)) / ((x * (s * (x * c))) * (s * c))
	return tmp

Julia

c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 1e-59)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(Float64(x * Float64(s * Float64(x * c))) * Float64(s * c)));
	end
	return tmp
end

MATLAB

c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1e-59)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((2.0 * x)) / ((x * (s * (x * c))) * (s * c));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 1e-59], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(x * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-59

   1. Initial program 64.6%

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

      1. associate-/r* 64.6%

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

      2. remove-double-neg 64.6%

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

      3. distribute-lft-neg-out 64.6%

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

      4. distribute-lft-neg-out 64.6%

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

      5. distribute-rgt-neg-out 64.6%

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

      6. associate-/l/ 64.6%

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

      7. distribute-rgt-neg-out 64.6%

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

      8. distribute-lft-neg-out 64.6%

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

      9. associate-*l* 66.6%

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

      10. distribute-lft-neg-in 66.6%

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

      11. distribute-lft-neg-out 66.6%

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

      12. remove-double-neg 66.6%

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

      13. associate-*r* 66.4%

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

      14. *-commutative 66.4%

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

      15. associate-*r* 64.7%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in x around 0 57.5%

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

      1. unpow2 57.5%

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

      2. associate-*r* 57.0%

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

      3. unpow2 57.0%

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

      4. unpow2 57.0%

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

      5. swap-sqr 69.0%

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

      6. swap-sqr 94.4%

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

      7. *-commutative 94.4%

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

      8. *-commutative 94.4%

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

      9. unpow2 94.4%

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

      10. *-commutative 94.4%

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

      11. *-commutative 94.4%

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

      12. associate-*l* 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around 0 54.2%

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

      1. unpow2 54.2%

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

      2. unpow2 54.2%

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

      3. unpow2 54.2%

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

      4. associate-*r* 54.2%

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

      5. *-commutative 54.2%

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

      6. swap-sqr 65.1%

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

      7. swap-sqr 85.9%

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

      8. associate-*r* 84.6%

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

      9. associate-*r* 87.9%

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

      10. associate-/r* 87.9%

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

      11. *-lft-identity 87.9%

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

      12. associate-*l/ 87.8%

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

      13. unpow-1 87.8%

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

      14. unpow-1 87.8%

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

      15. pow-sqr 88.0%

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

      16. associate-*r* 85.9%

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

      17. *-commutative 85.9%

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

      18. associate-*r* 88.8%

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

      19. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if 1e-59 < x

   1. Initial program 65.4%

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

      1. associate-/r* 65.8%

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

      2. remove-double-neg 65.8%

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

      3. distribute-lft-neg-out 65.8%

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

      4. distribute-lft-neg-out 65.8%

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

      5. distribute-rgt-neg-out 65.8%

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

      6. associate-/l/ 65.4%

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

      7. distribute-rgt-neg-out 65.4%

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

      8. distribute-lft-neg-out 65.4%

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

      9. associate-*l* 66.7%

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

      10. distribute-lft-neg-in 66.7%

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

      11. distribute-lft-neg-out 66.7%

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

      12. remove-double-neg 66.7%

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

      13. associate-*r* 66.7%

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

      14. *-commutative 66.7%

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

      15. associate-*r* 68.2%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around 0 59.0%

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

      1. unpow2 59.0%

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

      2. associate-*r* 60.4%

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

      3. unpow2 60.4%

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

      4. unpow2 60.4%

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

      5. swap-sqr 78.8%

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

      6. swap-sqr 97.5%

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

      7. *-commutative 97.5%

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

      8. *-commutative 97.5%

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

      9. unpow2 97.5%

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

      10. *-commutative 97.5%

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

      11. *-commutative 97.5%

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

      12. associate-*l* 97.9%

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

   6. Simplified 97.9%

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

      1. unpow2 68.1%

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

      2. associate-*r* 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*l* 67.3%

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

      5. *-commutative 67.3%

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

   8. Applied egg-rr 93.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-59}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right) \cdot \left(s \cdot c\right)}\\ \end{array} \]

Alternative 4: 94.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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)) / ((s * (x * c)) * (c * (x * s)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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[(s * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

   1. associate-/r* 64.9%

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

   2. remove-double-neg 64.9%

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

   3. distribute-lft-neg-out 64.9%

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

   4. distribute-lft-neg-out 64.9%

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

   5. distribute-rgt-neg-out 64.9%

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

   6. associate-/l/ 64.8%

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

   7. distribute-rgt-neg-out 64.8%

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

   8. distribute-lft-neg-out 64.8%

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

   9. associate-*l* 66.6%

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

   10. distribute-lft-neg-in 66.6%

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

   11. distribute-lft-neg-out 66.6%

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

   12. remove-double-neg 66.6%

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

   13. associate-*r* 66.5%

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

   14. *-commutative 66.5%

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

   15. associate-*r* 65.6%

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

3. Simplified 71.7%

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

4. Taylor expanded in x around 0 57.9%

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

   1. unpow2 57.9%

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

   2. associate-*r* 57.9%

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

   3. unpow2 57.9%

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

   4. unpow2 57.9%

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

   5. swap-sqr 71.6%

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

   6. swap-sqr 95.2%

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

   7. *-commutative 95.2%

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

   8. *-commutative 95.2%

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

   9. unpow2 95.2%

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

   10. *-commutative 95.2%

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

   11. *-commutative 95.2%

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

   12. associate-*l* 97.0%

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

6. Simplified 97.0%

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

   1. unpow2 97.0%

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

8. Applied egg-rr 97.0%

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

9. Taylor expanded in s around 0 96.0%

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

10. Final simplification 96.0%

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

Alternative 5: 79.0% accurate, 3.0× speedup

Math

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

FPCore

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

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

Fortran

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

assert c < s;
public static double code(double x, double c, double s) {
	return Math.pow((c * (x * s)), -2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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 64.8%

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

   1. associate-/r* 64.9%

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

   2. remove-double-neg 64.9%

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

   3. distribute-lft-neg-out 64.9%

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

   4. distribute-lft-neg-out 64.9%

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

   5. distribute-rgt-neg-out 64.9%

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

   6. associate-/l/ 64.8%

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

   7. distribute-rgt-neg-out 64.8%

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

   8. distribute-lft-neg-out 64.8%

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

   9. associate-*l* 66.6%

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

   10. distribute-lft-neg-in 66.6%

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

   11. distribute-lft-neg-out 66.6%

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

   12. remove-double-neg 66.6%

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

   13. associate-*r* 66.5%

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

   14. *-commutative 66.5%

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

   15. associate-*r* 65.6%

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

3. Simplified 71.7%

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

4. Taylor expanded in x around 0 57.9%

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

   1. unpow2 57.9%

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

   2. associate-*r* 57.9%

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

   3. unpow2 57.9%

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

   4. unpow2 57.9%

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

   5. swap-sqr 71.6%

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

   6. swap-sqr 95.2%

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

   7. *-commutative 95.2%

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

   8. *-commutative 95.2%

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

   9. unpow2 95.2%

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

   10. *-commutative 95.2%

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

   11. *-commutative 95.2%

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

   12. associate-*l* 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in x around 0 53.5%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
8. 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. unpow2 53.5%

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

   4. associate-*r* 53.5%

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

   5. *-commutative 53.5%

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

   6. swap-sqr 64.2%

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

   7. swap-sqr 81.2%

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

   8. associate-*r* 80.2%

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

   9. associate-*r* 82.7%

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

   10. associate-/r* 82.7%

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

   11. *-lft-identity 82.7%

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

   12. associate-*l/ 82.7%

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

   13. unpow-1 82.7%

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

   14. unpow-1 82.7%

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

   15. pow-sqr 82.8%

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

   16. associate-*r* 81.3%

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

   17. *-commutative 81.3%

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

   18. associate-*r* 83.4%

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

   19. metadata-eval 83.4%

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

9. Simplified 83.4%

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

10. Final simplification 83.4%

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

Alternative 6: 75.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 / (x * (s * ((x * c) * (s * c))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.8%

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

   1. associate-/r* 64.9%

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

   2. remove-double-neg 64.9%

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

   3. distribute-lft-neg-out 64.9%

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

   4. distribute-lft-neg-out 64.9%

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

   5. distribute-rgt-neg-out 64.9%

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

   6. associate-/l/ 64.8%

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

   7. distribute-rgt-neg-out 64.8%

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

   8. distribute-lft-neg-out 64.8%

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

   9. associate-*l* 66.6%

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

   10. distribute-lft-neg-in 66.6%

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

   11. distribute-lft-neg-out 66.6%

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

   12. remove-double-neg 66.6%

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

   13. associate-*r* 66.5%

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

   14. *-commutative 66.5%

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

   15. associate-*r* 65.6%

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

3. Simplified 71.7%

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

   1. add-sqr-sqrt 40.0%

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

   2. pow2 40.0%

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

   3. associate-*r* 36.5%

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

   4. swap-sqr 46.3%

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

   5. *-commutative 46.3%

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

   6. sqrt-prod 46.3%

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

   7. sqrt-prod 26.4%

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

   8. add-sqr-sqrt 50.2%

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

5. Applied egg-rr 50.2%

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

   1. unpow2 50.2%

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

   2. *-commutative 50.2%

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

   3. associate-*r* 50.2%

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

   4. associate-*r* 50.2%

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

   5. *-commutative 50.2%

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

   6. add-sqr-sqrt 90.0%

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

   7. associate-*r* 88.4%

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

   8. *-commutative 88.4%

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

7. Applied egg-rr 88.4%

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

   1. associate-*l* 86.4%

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

   2. *-commutative 86.4%

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

9. Simplified 86.4%

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

10. Taylor expanded in x around 0 76.3%

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

11. Final simplification 76.3%

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

Alternative 7: 76.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 / ((x * (s * (x * c))) * (s * c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.8%

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

   1. associate-/r* 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 58.0%

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

   4. unpow2 58.0%

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

   5. unpow2 58.0%

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

   6. associate-*r* 63.3%

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

   7. associate-/r* 65.8%

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

   8. associate-/l/ 66.1%

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

   9. associate-/l/ 65.9%

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

   10. *-commutative 65.9%

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

   11. associate-*l* 63.3%

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

   12. unpow2 63.3%

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

   13. associate-*l* 57.9%

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

   14. unpow2 57.9%

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

   15. unswap-sqr 71.6%

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

   16. *-commutative 71.6%

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

   17. *-commutative 71.6%

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

3. Simplified 71.6%

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

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. associate-/r* 53.5%

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

   2. unpow2 53.5%

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

   3. associate-/r* 53.5%

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

   4. associate-/r* 53.5%

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

   5. unpow2 53.5%

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

   6. unpow2 53.5%

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

   7. swap-sqr 64.2%

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

   8. unpow2 64.2%

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

   9. associate-/r* 64.2%

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

   10. *-commutative 64.2%

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

   11. unpow2 64.2%

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

   12. swap-sqr 81.2%

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

   13. unpow2 81.2%

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

   14. *-commutative 81.2%

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

   15. *-commutative 81.2%

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

   16. associate-*l* 82.7%

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

6. Simplified 82.7%

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

   1. unpow2 82.7%

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

   2. associate-*r* 80.2%

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

   3. *-commutative 80.2%

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

   4. associate-*l* 77.8%

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

   5. *-commutative 77.8%

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

8. Applied egg-rr 77.8%

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

9. Final simplification 77.8%

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

Alternative 8: 79.0% accurate, 24.1× speedup

Math

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

FPCore

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)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

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 = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.8%

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

   1. associate-/r* 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 58.0%

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

   4. unpow2 58.0%

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

   5. unpow2 58.0%

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

   6. associate-*r* 63.3%

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

   7. associate-/r* 65.8%

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

   8. associate-/l/ 66.1%

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

   9. associate-/l/ 65.9%

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

   10. *-commutative 65.9%

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

   11. associate-*l* 63.3%

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

   12. unpow2 63.3%

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

   13. associate-*l* 57.9%

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

   14. unpow2 57.9%

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

   15. unswap-sqr 71.6%

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

   16. *-commutative 71.6%

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

   17. *-commutative 71.6%

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

3. Simplified 71.6%

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

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. associate-/r* 53.5%

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

   2. unpow2 53.5%

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

   3. associate-/r* 53.5%

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

   4. associate-/r* 53.5%

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

   5. unpow2 53.5%

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

   6. unpow2 53.5%

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

   7. swap-sqr 64.2%

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

   8. unpow2 64.2%

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

   9. associate-/r* 64.2%

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

   10. *-commutative 64.2%

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

   11. unpow2 64.2%

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

   12. swap-sqr 81.2%

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

   13. unpow2 81.2%

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

   14. *-commutative 81.2%

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

   15. *-commutative 81.2%

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

   16. associate-*l* 82.7%

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

6. Simplified 82.7%

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

   1. unpow2 97.0%

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

8. Applied egg-rr 82.7%

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

9. Final simplification 82.7%

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

Alternative 9: 78.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 * (x * s))) / (s * (x * c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.8%

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

   1. associate-/r* 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 58.0%

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

   4. unpow2 58.0%

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

   5. unpow2 58.0%

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

   6. associate-*r* 63.3%

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

   7. associate-/r* 65.8%

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

   8. associate-/l/ 66.1%

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

   9. associate-/l/ 65.9%

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

   10. *-commutative 65.9%

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

   11. associate-*l* 63.3%

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

   12. unpow2 63.3%

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

   13. associate-*l* 57.9%

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

   14. unpow2 57.9%

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

   15. unswap-sqr 71.6%

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

   16. *-commutative 71.6%

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

   17. *-commutative 71.6%

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

3. Simplified 71.6%

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

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. associate-/r* 53.5%

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

   2. unpow2 53.5%

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

   3. associate-/r* 53.5%

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

   4. associate-/r* 53.5%

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

   5. unpow2 53.5%

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

   6. unpow2 53.5%

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

   7. swap-sqr 64.2%

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

   8. unpow2 64.2%

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

   9. associate-/r* 64.2%

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

   10. *-commutative 64.2%

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

   11. unpow2 64.2%

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

   12. swap-sqr 81.2%

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

   13. unpow2 81.2%

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

   14. *-commutative 81.2%

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

   15. *-commutative 81.2%

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

   16. associate-*l* 82.7%

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

6. Simplified 82.7%

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

   1. pow-flip 82.8%

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

   2. add-sqr-sqrt 50.5%

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

   3. unpow-prod-down 50.5%

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

   4. metadata-eval 50.5%

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

   5. metadata-eval 50.5%

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

8. Applied egg-rr 50.5%

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

   1. pow-sqr 50.6%

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

   2. metadata-eval 50.6%

      \[\leadsto {\left(\sqrt{s \cdot \left(c \cdot x\right)}\right)}^{\color{blue}{-4}} \]

10. Simplified 50.6%

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

    1. sqrt-pow2 82.8%

       \[\leadsto \color{blue}{{\left(s \cdot \left(c \cdot x\right)\right)}^{\left(\frac{-4}{2}\right)}} \]

    2. metadata-eval 82.8%

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

    3. metadata-eval 82.8%

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

    4. pow-div 82.7%

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

    5. inv-pow 82.7%

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

    6. pow1 82.7%

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

12. Applied egg-rr 82.7%

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

13. Taylor expanded in s around 0 82.5%

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

14. Final simplification 82.5%

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

Alternative 10: 28.4% accurate, 34.8× speedup

Math

\[\begin{array}{l} [c, s] = \mathsf{sort}([c, s])\\ \\ \frac{\frac{-2}{c \cdot c}}{s \cdot s} \end{array} \]

FPCore

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

C

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

Fortran

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 = ((-2.0d0) / (c * c)) / (s * s)
end function

Java

assert c < s;
public static double code(double x, double c, double s) {
	return (-2.0 / (c * c)) / (s * s);
}

Python

[c, s] = sort([c, s])
def code(x, c, s):
	return (-2.0 / (c * c)) / (s * s)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.8%

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

   1. associate-/r* 64.9%

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

   2. *-commutative 64.9%

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

   3. associate-*l* 58.0%

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

   4. unpow2 58.0%

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

   5. unpow2 58.0%

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

   6. associate-*r* 63.3%

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

   7. associate-/r* 65.8%

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

   8. associate-/l/ 66.1%

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

   9. associate-/l/ 65.9%

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

   10. *-commutative 65.9%

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

   11. associate-*l* 63.3%

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

   12. unpow2 63.3%

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

   13. associate-*l* 57.9%

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

   14. unpow2 57.9%

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

   15. unswap-sqr 71.6%

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

   16. *-commutative 71.6%

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

   17. *-commutative 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in x around 0 52.2%

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

   1. unpow2 52.2%

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

6. Simplified 52.2%

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

7. Taylor expanded in x around inf 25.4%

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
8. Step-by-step derivation

   1. associate-/r* 25.1%

      \[\leadsto \color{blue}{\frac{\frac{-2}{{c}^{2}}}{{s}^{2}}} \]

   2. unpow2 25.1%

      \[\leadsto \frac{\frac{-2}{\color{blue}{c \cdot c}}}{{s}^{2}} \]

   3. unpow2 25.1%

      \[\leadsto \frac{\frac{-2}{c \cdot c}}{\color{blue}{s \cdot s}} \]

9. Simplified 25.1%

   \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot c}}{s \cdot s}} \]

10. Final simplification 25.1%

    \[\leadsto \frac{\frac{-2}{c \cdot c}}{s \cdot s} \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))