mixedcos

Percentage Accurate: 67.1% → 97.8%

Time: 15.9s

Alternatives: 8

Speedup: 24.1×

• 32.0% 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: 67.1% 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: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \frac{t\_0}{x}\\ \mathbf{if}\;\frac{t\_0}{{c\_m}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(x \cdot s\_m\right)} \cdot \frac{\frac{t\_1}{s\_m}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot c\_m} \cdot \frac{1}{s\_m}\right) \cdot \left(t\_1 \cdot \frac{1}{c\_m \cdot s\_m}\right)\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))) (t_1 (/ t_0 x)))
   (if (<= (/ t_0 (* (pow c_m 2.0) (* x (* x (pow s_m 2.0))))) INFINITY)
     (* (/ 1.0 (* c_m (* x s_m))) (/ (/ t_1 s_m) c_m))
     (* (* (/ 1.0 (* x c_m)) (/ 1.0 s_m)) (* t_1 (/ 1.0 (* c_m s_m)))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((2.0 * x));
	double t_1 = t_0 / x;
	double tmp;
	if ((t_0 / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))))) <= ((double) INFINITY)) {
		tmp = (1.0 / (c_m * (x * s_m))) * ((t_1 / s_m) / c_m);
	} else {
		tmp = ((1.0 / (x * c_m)) * (1.0 / s_m)) * (t_1 * (1.0 / (c_m * s_m)));
	}
	return tmp;
}

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((2.0 * x));
	double t_1 = t_0 / x;
	double tmp;
	if ((t_0 / (Math.pow(c_m, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / (c_m * (x * s_m))) * ((t_1 / s_m) / c_m);
	} else {
		tmp = ((1.0 / (x * c_m)) * (1.0 / s_m)) * (t_1 * (1.0 / (c_m * s_m)));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((2.0 * x))
	t_1 = t_0 / x
	tmp = 0
	if (t_0 / (math.pow(c_m, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= math.inf:
		tmp = (1.0 / (c_m * (x * s_m))) * ((t_1 / s_m) / c_m)
	else:
		tmp = ((1.0 / (x * c_m)) * (1.0 / s_m)) * (t_1 * (1.0 / (c_m * s_m)))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(2.0 * x))
	t_1 = Float64(t_0 / x)
	tmp = 0.0
	if (Float64(t_0 / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(1.0 / Float64(c_m * Float64(x * s_m))) * Float64(Float64(t_1 / s_m) / c_m));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(x * c_m)) * Float64(1.0 / s_m)) * Float64(t_1 * Float64(1.0 / Float64(c_m * s_m))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((2.0 * x));
	t_1 = t_0 / x;
	tmp = 0.0;
	if ((t_0 / ((c_m ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= Inf)
		tmp = (1.0 / (c_m * (x * s_m))) * ((t_1 / s_m) / c_m);
	else
		tmp = ((1.0 / (x * c_m)) * (1.0 / s_m)) * (t_1 * (1.0 / (c_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / x), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(x * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / s$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. add-sqr-sqrt 83.0%

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

      3. times-frac 83.0%

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

   4. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 96.8%

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

      4. times-frac 96.3%

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

      5. *-commutative 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. associate-*l/ 96.8%

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

      2. *-un-lft-identity 96.8%

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

      3. *-commutative 96.8%

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

      4. associate-/r* 99.8%

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

   8. Applied egg-rr 99.8%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

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

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

   4. Applied egg-rr 84.7%

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

      1. *-un-lft-identity 84.7%

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

      2. *-commutative 84.7%

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

      3. associate-*r* 84.8%

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

      4. times-frac 84.7%

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

      5. *-commutative 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. inv-pow 84.7%

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

      2. associate-*r* 90.4%

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

      3. unpow-prod-down 90.5%

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

   8. Applied egg-rr 90.5%

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

      1. unpow-1 90.5%

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

      2. unpow-1 90.5%

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

   10. Simplified 90.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s}}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot c} \cdot \frac{1}{s}\right) \cdot \left(\frac{\cos \left(2 \cdot x\right)}{x} \cdot \frac{1}{c \cdot s}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 2.6× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (* (/ 1.0 (* c_m (* x s_m))) (/ (/ 1.0 c_m) (* s_m (/ x (cos (* 2.0 x)))))))

C

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

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c_m * (x * s_m))) * ((1.0d0 / c_m) / (s_m * (x / cos((2.0d0 * x)))))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (x * s_m))) * ((1.0 / c_m) / (s_m * (x / Math.cos((2.0 * x)))));
}

Python

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

Julia

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

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (1.0 / (c_m * (x * s_m))) * ((1.0 / c_m) / (s_m * (x / cos((2.0 * x)))));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(s$95$m * N[(x / N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

   1. *-un-lft-identity 69.4%

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

   2. add-sqr-sqrt 69.4%

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

   3. times-frac 69.4%

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

4. Applied egg-rr 97.2%

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

   1. *-un-lft-identity 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 94.8%

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

   4. times-frac 94.4%

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

   5. *-commutative 94.4%

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

6. Applied egg-rr 94.4%

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

   1. *-commutative 94.4%

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

   2. clear-num 94.4%

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

   3. associate-/r* 94.8%

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

   4. frac-times 97.3%

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

   5. *-un-lft-identity 97.3%

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

8. Applied egg-rr 97.3%

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

9. Final simplification 97.3%

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

Alternative 3: 97.3% accurate, 2.7× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (* (/ 1.0 (* c_m (* x s_m))) (/ (/ (/ (cos (* 2.0 x)) x) s_m) c_m)))

C

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

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c_m * (x * s_m))) * (((cos((2.0d0 * x)) / x) / s_m) / c_m)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (x * s_m))) * (((Math.cos((2.0 * x)) / x) / s_m) / c_m);
}

Python

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

Julia

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

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (1.0 / (c_m * (x * s_m))) * (((cos((2.0 * x)) / x) / s_m) / c_m);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

   1. *-un-lft-identity 69.4%

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

   2. add-sqr-sqrt 69.4%

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

   3. times-frac 69.4%

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

4. Applied egg-rr 97.2%

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

   1. *-un-lft-identity 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-*r* 94.8%

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

   4. times-frac 94.4%

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

   5. *-commutative 94.4%

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

6. Applied egg-rr 94.4%

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

   1. associate-*l/ 94.8%

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

   2. *-un-lft-identity 94.8%

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

   3. *-commutative 94.8%

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

   4. associate-/r* 97.3%

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

8. Applied egg-rr 97.3%

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

9. Final simplification 97.3%

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

Alternative 4: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \frac{\frac{\cos \left(2 \cdot x\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m)))) (/ (/ (cos (* 2.0 x)) t_0) t_0)))

C

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

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x * s_m)
    code = (cos((2.0d0 * x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

   1. add-cbrt-cube 69.3%

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

   2. add-cbrt-cube 64.6%

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

   3. cbrt-undiv 64.6%

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

   4. pow3 64.6%

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

   5. pow3 64.6%

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

4. Applied egg-rr 76.4%

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

   1. cbrt-div 76.4%

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

   2. rem-cbrt-cube 76.4%

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

   3. rem-cbrt-cube 97.2%

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

   4. unpow2 97.2%

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

   5. associate-/r* 97.3%

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

   6. *-commutative 97.3%

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

6. Applied egg-rr 97.3%

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

7. Final simplification 97.3%

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

Alternative 5: 79.8% accurate, 20.9× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (* (/ 1.0 (* c_m (* x s_m))) (/ (/ (/ 1.0 x) s_m) c_m)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (x * s_m))) * (((1.0 / x) / s_m) / c_m);
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c_m * (x * s_m))) * (((1.0d0 / x) / s_m) / c_m)
end function

Java

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

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (1.0 / (c_m * (x * s_m))) * (((1.0 / x) / s_m) / c_m)

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / x), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

   1. *-un-lft-identity 69.4%

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

   2. add-sqr-sqrt 69.4%

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

   3. times-frac 69.4%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in x around 0 81.1%

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

   1. *-commutative 81.1%

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

   2. associate-/l/ 81.1%

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

   3. associate-/r* 81.2%

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

7. Simplified 81.2%

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

8. Final simplification 81.2%

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

Alternative 6: 74.9% accurate, 24.1× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* (* x c_m) (* s_m (* c_m (* x s_m))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / ((x * c_m) * (s_m * (c_m * (x * s_m))));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / ((x * c_m) * (s_m * (c_m * (x * s_m))))
end function

Java

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

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / ((x * c_m) * (s_m * (c_m * (x * s_m))))

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(x * c$95$m), $MachinePrecision] * N[(s$95$m * N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

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

3. Taylor expanded in x around 0 58.3%

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

   1. associate-/r* 57.9%

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

   2. *-commutative 57.9%

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

   3. unpow2 57.9%

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

   4. unpow2 57.9%

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

   5. swap-sqr 66.2%

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

   6. unpow2 66.2%

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

   7. associate-/r* 66.6%

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

   8. unpow2 66.6%

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

   9. unpow2 66.6%

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

   10. swap-sqr 81.0%

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

   11. unpow2 81.0%

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

   12. *-commutative 81.0%

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

5. Simplified 81.0%

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

   1. *-commutative 81.0%

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

   2. unpow2 81.0%

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

   3. associate-*r* 80.7%

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

   4. associate-*l* 79.2%

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

7. Applied egg-rr 79.2%

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

8. Final simplification 79.2%

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

Alternative 7: 79.7% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x * s_m)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

3. Taylor expanded in x around 0 58.3%

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

   1. associate-/r* 57.9%

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

   2. *-commutative 57.9%

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

   3. unpow2 57.9%

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

   4. unpow2 57.9%

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

   5. swap-sqr 66.2%

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

   6. unpow2 66.2%

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

   7. associate-/r* 66.6%

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

   8. unpow2 66.6%

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

   9. unpow2 66.6%

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

   10. swap-sqr 81.0%

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

   11. unpow2 81.0%

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

   12. *-commutative 81.0%

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

5. Simplified 81.0%

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

   1. *-commutative 81.0%

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

   2. unpow2 81.0%

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

7. Applied egg-rr 81.0%

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

8. Final simplification 81.0%

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

Alternative 8: 79.8% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m)))) (/ (/ 1.0 t_0) t_0)))

C

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

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x * s_m)
    code = (1.0d0 / t_0) / t_0
end function

Java

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

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	return (1.0 / t_0) / t_0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

   1. add-cbrt-cube 69.3%

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

   2. add-cbrt-cube 64.6%

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

   3. cbrt-undiv 64.6%

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

   4. pow3 64.6%

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

   5. pow3 64.6%

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

4. Applied egg-rr 76.4%

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

   1. cbrt-div 76.4%

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

   2. rem-cbrt-cube 76.4%

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

   3. rem-cbrt-cube 97.2%

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

   4. unpow2 97.2%

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

   5. associate-/r* 97.3%

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

   6. *-commutative 97.3%

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

6. Applied egg-rr 97.3%

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

7. Taylor expanded in x around 0 81.1%

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

8. Final simplification 81.1%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))