mixedcos

Percentage Accurate: 66.3% → 98.1%

Time: 13.5s

Alternatives: 13

Speedup: 24.1×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;{c\_m}^{2} \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{t\_0}{{\left(x \cdot \left(c\_m \cdot s\_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{c\_m \cdot \left(c\_m \cdot x\right)}}{s\_m \cdot \left(x \cdot s\_m\right)}\\ \end{array} \end{array} \]

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))))
   (if (<= (pow c_m 2.0) 4e+26)
     (/ t_0 (pow (* x (* c_m s_m)) 2.0))
     (/ (/ t_0 (* c_m (* c_m x))) (* s_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) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (pow(c_m, 2.0) <= 4e+26) {
		tmp = t_0 / pow((x * (c_m * s_m)), 2.0);
	} else {
		tmp = (t_0 / (c_m * (c_m * x))) / (s_m * (x * s_m));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    if ((c_m ** 2.0d0) <= 4d+26) then
        tmp = t_0 / ((x * (c_m * s_m)) ** 2.0d0)
    else
        tmp = (t_0 / (c_m * (c_m * x))) / (s_m * (x * s_m))
    end if
    code = tmp
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 = Math.cos((2.0 * x));
	double tmp;
	if (Math.pow(c_m, 2.0) <= 4e+26) {
		tmp = t_0 / Math.pow((x * (c_m * s_m)), 2.0);
	} else {
		tmp = (t_0 / (c_m * (c_m * x))) / (s_m * (x * 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))
	tmp = 0
	if math.pow(c_m, 2.0) <= 4e+26:
		tmp = t_0 / math.pow((x * (c_m * s_m)), 2.0)
	else:
		tmp = (t_0 / (c_m * (c_m * x))) / (s_m * (x * 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))
	tmp = 0.0
	if ((c_m ^ 2.0) <= 4e+26)
		tmp = Float64(t_0 / (Float64(x * Float64(c_m * s_m)) ^ 2.0));
	else
		tmp = Float64(Float64(t_0 / Float64(c_m * Float64(c_m * x))) / Float64(s_m * Float64(x * 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));
	tmp = 0.0;
	if ((c_m ^ 2.0) <= 4e+26)
		tmp = t_0 / ((x * (c_m * s_m)) ^ 2.0);
	else
		tmp = (t_0 / (c_m * (c_m * x))) / (s_m * (x * 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]}, If[LessEqual[N[Power[c$95$m, 2.0], $MachinePrecision], 4e+26], N[(t$95$0 / N[Power[N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(c$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 4.00000000000000019e26

   1. Initial program 67.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left(\left({s}^{2} \cdot x\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot {x}^{\color{blue}{2}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{2}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\left(\left(c \cdot s\right) \cdot x\right), \color{blue}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), x\right), 2\right)\right) \]

      9. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right), 2\right)\right) \]

   4. Applied egg-rr 97.9%

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

   if 4.00000000000000019e26 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 62.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left(\left({s}^{2} \cdot x\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot {x}^{\color{blue}{2}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{2}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\left(\left(c \cdot s\right) \cdot x\right), \color{blue}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), x\right), 2\right)\right) \]

      9. *-lowering-*.f64 94.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right), 2\right)\right) \]

   4. Applied egg-rr 94.4%

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

      1. *-rgt-identity N/A

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

      2. associate-*l* N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. times-frac N/A

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

      12. div-inv N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot c\right) \cdot x}\right), \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right) \]

   6. Applied egg-rr 80.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(c \cdot x\right)}}{s \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \frac{\frac{\frac{1}{s\_m}}{c\_m}}{x}\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+170}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(c\_m \cdot \left(x \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{s\_m}}{s\_m \cdot \left(\left(c\_m \cdot x\right) \cdot \left(c\_m \cdot x\right)\right)}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 s_m) c_m) x)))
   (if (<= x 5.8e-13)
     (* t_1 t_1)
     (if (<= x 7e+170)
       (/ t_0 (* s_m (* c_m (* x (* x (* c_m s_m))))))
       (/ (/ t_0 s_m) (* s_m (* (* c_m x) (* c_m 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) {
	double t_0 = cos((2.0 * x));
	double t_1 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 5.8e-13) {
		tmp = t_1 * t_1;
	} else if (x <= 7e+170) {
		tmp = t_0 / (s_m * (c_m * (x * (x * (c_m * s_m)))));
	} else {
		tmp = (t_0 / s_m) / (s_m * ((c_m * x) * (c_m * x)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    t_1 = ((1.0d0 / s_m) / c_m) / x
    if (x <= 5.8d-13) then
        tmp = t_1 * t_1
    else if (x <= 7d+170) then
        tmp = t_0 / (s_m * (c_m * (x * (x * (c_m * s_m)))))
    else
        tmp = (t_0 / s_m) / (s_m * ((c_m * x) * (c_m * x)))
    end if
    code = tmp
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 = Math.cos((2.0 * x));
	double t_1 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 5.8e-13) {
		tmp = t_1 * t_1;
	} else if (x <= 7e+170) {
		tmp = t_0 / (s_m * (c_m * (x * (x * (c_m * s_m)))));
	} else {
		tmp = (t_0 / s_m) / (s_m * ((c_m * x) * (c_m * x)));
	}
	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 = ((1.0 / s_m) / c_m) / x
	tmp = 0
	if x <= 5.8e-13:
		tmp = t_1 * t_1
	elif x <= 7e+170:
		tmp = t_0 / (s_m * (c_m * (x * (x * (c_m * s_m)))))
	else:
		tmp = (t_0 / s_m) / (s_m * ((c_m * x) * (c_m * x)))
	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(Float64(Float64(1.0 / s_m) / c_m) / x)
	tmp = 0.0
	if (x <= 5.8e-13)
		tmp = Float64(t_1 * t_1);
	elseif (x <= 7e+170)
		tmp = Float64(t_0 / Float64(s_m * Float64(c_m * Float64(x * Float64(x * Float64(c_m * s_m))))));
	else
		tmp = Float64(Float64(t_0 / s_m) / Float64(s_m * Float64(Float64(c_m * x) * Float64(c_m * x))));
	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 = ((1.0 / s_m) / c_m) / x;
	tmp = 0.0;
	if (x <= 5.8e-13)
		tmp = t_1 * t_1;
	elseif (x <= 7e+170)
		tmp = t_0 / (s_m * (c_m * (x * (x * (c_m * s_m)))));
	else
		tmp = (t_0 / s_m) / (s_m * ((c_m * x) * (c_m * x)));
	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[(N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 5.8e-13], N[(t$95$1 * t$95$1), $MachinePrecision], If[LessEqual[x, 7e+170], N[(t$95$0 / N[(s$95$m * N[(c$95$m * N[(x * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(s$95$m * N[(N[(c$95$m * x), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.7999999999999995e-13

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 78.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 73.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/l/ N/A

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

       11. div-inv N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{c \cdot s}}{x}\right), \color{blue}{\left(\frac{\frac{1}{c \cdot s}}{x}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), x\right), \left(\frac{\color{blue}{\frac{1}{c \cdot s}}}{x}\right)\right) \]

       16. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{c}\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s}\right), c\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \left(\frac{\frac{1}{c \cdot s}}{x}\right)\right) \]

       19. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 82.7%

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

   if 5.7999999999999995e-13 < x < 7.00000000000000011e170

   1. Initial program 63.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 80.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      8. *-lowering-*.f64 81.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   7. Applied egg-rr 81.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right) \cdot \color{blue}{s}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right) \cdot s\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right), \color{blue}{s}\right)\right) \]

   9. Applied egg-rr 92.9%

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

   if 7.00000000000000011e170 < x

   1. Initial program 52.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left(\left({s}^{2} \cdot x\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot {x}^{\color{blue}{2}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{2}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\left(\left(c \cdot s\right) \cdot x\right), \color{blue}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), x\right), 2\right)\right) \]

      9. *-lowering-*.f64 96.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right), 2\right)\right) \]

   4. Applied egg-rr 96.4%

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

      1. *-rgt-identity N/A

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

      2. associate-*l* N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. div-inv N/A

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

      14. associate-/l/ N/A

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

      15. associate-/l/ N/A

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

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{s}\right), \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right) \]

   6. Applied egg-rr 84.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right) \]

      10. *-lowering-*.f64 77.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), s\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right) \]

   8. Applied egg-rr 77.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{c}}{x} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+170}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{s}}{s \cdot \left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \frac{\frac{\frac{1}{s\_m}}{c\_m}}{x}\\ t_2 := x \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(c\_m \cdot \left(x \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{s\_m}}{x \cdot \left(c\_m \cdot t\_2\right)}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 s_m) c_m) x))
        (t_2 (* x (* c_m s_m))))
   (if (<= x 5.8e-13)
     (* t_1 t_1)
     (if (<= x 5e+167)
       (/ t_0 (* s_m (* c_m (* x t_2))))
       (/ (/ t_0 s_m) (* x (* c_m t_2)))))))

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 = ((1.0 / s_m) / c_m) / x;
	double t_2 = x * (c_m * s_m);
	double tmp;
	if (x <= 5.8e-13) {
		tmp = t_1 * t_1;
	} else if (x <= 5e+167) {
		tmp = t_0 / (s_m * (c_m * (x * t_2)));
	} else {
		tmp = (t_0 / s_m) / (x * (c_m * t_2));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    t_1 = ((1.0d0 / s_m) / c_m) / x
    t_2 = x * (c_m * s_m)
    if (x <= 5.8d-13) then
        tmp = t_1 * t_1
    else if (x <= 5d+167) then
        tmp = t_0 / (s_m * (c_m * (x * t_2)))
    else
        tmp = (t_0 / s_m) / (x * (c_m * t_2))
    end if
    code = tmp
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 = Math.cos((2.0 * x));
	double t_1 = ((1.0 / s_m) / c_m) / x;
	double t_2 = x * (c_m * s_m);
	double tmp;
	if (x <= 5.8e-13) {
		tmp = t_1 * t_1;
	} else if (x <= 5e+167) {
		tmp = t_0 / (s_m * (c_m * (x * t_2)));
	} else {
		tmp = (t_0 / s_m) / (x * (c_m * t_2));
	}
	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 = ((1.0 / s_m) / c_m) / x
	t_2 = x * (c_m * s_m)
	tmp = 0
	if x <= 5.8e-13:
		tmp = t_1 * t_1
	elif x <= 5e+167:
		tmp = t_0 / (s_m * (c_m * (x * t_2)))
	else:
		tmp = (t_0 / s_m) / (x * (c_m * t_2))
	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(Float64(Float64(1.0 / s_m) / c_m) / x)
	t_2 = Float64(x * Float64(c_m * s_m))
	tmp = 0.0
	if (x <= 5.8e-13)
		tmp = Float64(t_1 * t_1);
	elseif (x <= 5e+167)
		tmp = Float64(t_0 / Float64(s_m * Float64(c_m * Float64(x * t_2))));
	else
		tmp = Float64(Float64(t_0 / s_m) / Float64(x * Float64(c_m * t_2)));
	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 = ((1.0 / s_m) / c_m) / x;
	t_2 = x * (c_m * s_m);
	tmp = 0.0;
	if (x <= 5.8e-13)
		tmp = t_1 * t_1;
	elseif (x <= 5e+167)
		tmp = t_0 / (s_m * (c_m * (x * t_2)));
	else
		tmp = (t_0 / s_m) / (x * (c_m * t_2));
	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[(N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.8e-13], N[(t$95$1 * t$95$1), $MachinePrecision], If[LessEqual[x, 5e+167], N[(t$95$0 / N[(s$95$m * N[(c$95$m * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(x * N[(c$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.7999999999999995e-13

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 78.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 73.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/l/ N/A

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

       11. div-inv N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{c \cdot s}}{x}\right), \color{blue}{\left(\frac{\frac{1}{c \cdot s}}{x}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), x\right), \left(\frac{\color{blue}{\frac{1}{c \cdot s}}}{x}\right)\right) \]

       16. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{c}\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s}\right), c\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \left(\frac{\frac{1}{c \cdot s}}{x}\right)\right) \]

       19. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 82.7%

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

   if 5.7999999999999995e-13 < x < 4.9999999999999997e167

   1. Initial program 66.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 82.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      8. *-lowering-*.f64 84.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   7. Applied egg-rr 84.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right) \cdot \color{blue}{s}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right) \cdot s\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right), \color{blue}{s}\right)\right) \]

   9. Applied egg-rr 95.0%

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

   if 4.9999999999999997e167 < x

   1. Initial program 49.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left(\left({s}^{2} \cdot x\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot {x}^{\color{blue}{2}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{2}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\left(\left(c \cdot s\right) \cdot x\right), \color{blue}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), x\right), 2\right)\right) \]

      9. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right), 2\right)\right) \]

   4. Applied egg-rr 96.6%

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

      1. *-rgt-identity N/A

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

      2. associate-*l* N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. associate-*l* N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. div-inv N/A

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

      14. associate-/l/ N/A

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

      15. associate-/l/ N/A

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

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{s}\right), \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right) \]

   6. Applied egg-rr 81.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{c}}{x} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{s}}{x \cdot \left(c \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \frac{\frac{\frac{1}{s\_m}}{c\_m}}{x}\\ t_2 := x \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+169}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(c\_m \cdot \left(x \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(x \cdot \left(c\_m \cdot t\_2\right)\right)}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 s_m) c_m) x))
        (t_2 (* x (* c_m s_m))))
   (if (<= x 5.8e-13)
     (* t_1 t_1)
     (if (<= x 1.05e+169)
       (/ t_0 (* s_m (* c_m (* x t_2))))
       (/ t_0 (* s_m (* x (* c_m t_2))))))))

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 = ((1.0 / s_m) / c_m) / x;
	double t_2 = x * (c_m * s_m);
	double tmp;
	if (x <= 5.8e-13) {
		tmp = t_1 * t_1;
	} else if (x <= 1.05e+169) {
		tmp = t_0 / (s_m * (c_m * (x * t_2)));
	} else {
		tmp = t_0 / (s_m * (x * (c_m * t_2)));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    t_1 = ((1.0d0 / s_m) / c_m) / x
    t_2 = x * (c_m * s_m)
    if (x <= 5.8d-13) then
        tmp = t_1 * t_1
    else if (x <= 1.05d+169) then
        tmp = t_0 / (s_m * (c_m * (x * t_2)))
    else
        tmp = t_0 / (s_m * (x * (c_m * t_2)))
    end if
    code = tmp
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 = Math.cos((2.0 * x));
	double t_1 = ((1.0 / s_m) / c_m) / x;
	double t_2 = x * (c_m * s_m);
	double tmp;
	if (x <= 5.8e-13) {
		tmp = t_1 * t_1;
	} else if (x <= 1.05e+169) {
		tmp = t_0 / (s_m * (c_m * (x * t_2)));
	} else {
		tmp = t_0 / (s_m * (x * (c_m * t_2)));
	}
	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 = ((1.0 / s_m) / c_m) / x
	t_2 = x * (c_m * s_m)
	tmp = 0
	if x <= 5.8e-13:
		tmp = t_1 * t_1
	elif x <= 1.05e+169:
		tmp = t_0 / (s_m * (c_m * (x * t_2)))
	else:
		tmp = t_0 / (s_m * (x * (c_m * t_2)))
	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(Float64(Float64(1.0 / s_m) / c_m) / x)
	t_2 = Float64(x * Float64(c_m * s_m))
	tmp = 0.0
	if (x <= 5.8e-13)
		tmp = Float64(t_1 * t_1);
	elseif (x <= 1.05e+169)
		tmp = Float64(t_0 / Float64(s_m * Float64(c_m * Float64(x * t_2))));
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(x * Float64(c_m * t_2))));
	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 = ((1.0 / s_m) / c_m) / x;
	t_2 = x * (c_m * s_m);
	tmp = 0.0;
	if (x <= 5.8e-13)
		tmp = t_1 * t_1;
	elseif (x <= 1.05e+169)
		tmp = t_0 / (s_m * (c_m * (x * t_2)));
	else
		tmp = t_0 / (s_m * (x * (c_m * t_2)));
	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[(N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.8e-13], N[(t$95$1 * t$95$1), $MachinePrecision], If[LessEqual[x, 1.05e+169], N[(t$95$0 / N[(s$95$m * N[(c$95$m * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(s$95$m * N[(x * N[(c$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.7999999999999995e-13

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 78.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 73.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/l/ N/A

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

       11. div-inv N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{c \cdot s}}{x}\right), \color{blue}{\left(\frac{\frac{1}{c \cdot s}}{x}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), x\right), \left(\frac{\color{blue}{\frac{1}{c \cdot s}}}{x}\right)\right) \]

       16. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{c}\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s}\right), c\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \left(\frac{\frac{1}{c \cdot s}}{x}\right)\right) \]

       19. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 82.7%

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

   if 5.7999999999999995e-13 < x < 1.0500000000000001e169

   1. Initial program 66.0%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 82.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      8. *-lowering-*.f64 84.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   7. Applied egg-rr 84.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right) \cdot \color{blue}{s}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right) \cdot s\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right), \color{blue}{s}\right)\right) \]

   9. Applied egg-rr 95.0%

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

   if 1.0500000000000001e169 < x

   1. Initial program 49.3%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left(\left({s}^{2} \cdot x\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot {x}^{\color{blue}{2}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{2}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\left(\left(c \cdot s\right) \cdot x\right), \color{blue}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), x\right), 2\right)\right) \]

      9. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right), 2\right)\right) \]

   4. Applied egg-rr 96.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      2. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(x + x\right), \left({\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}\right)\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left({\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}\right)\right) \]

      4. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left({\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x \cdot 2\right)\right), \left({\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left({\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      8. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      21. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      25. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)\right)\right) \]

      26. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(c \cdot s\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      27. *-lowering-*.f64 81.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 81.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{c}}{x} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+169}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \frac{\frac{\frac{1}{s\_m}}{c\_m}}{x}\\ \mathbf{if}\;x \leq 2.85 \cdot 10^{-34}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+197}:\\ \;\;\;\;\frac{t\_0}{x \cdot \left(x \cdot \left(c\_m \cdot \left(s\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(c\_m \cdot x\right) \cdot \left(c\_m \cdot \left(s\_m \cdot \left(x \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 s_m) c_m) x)))
   (if (<= x 2.85e-34)
     (* t_1 t_1)
     (if (<= x 3.2e+197)
       (/ t_0 (* x (* x (* c_m (* s_m (* c_m s_m))))))
       (/ t_0 (* (* c_m x) (* c_m (* s_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) {
	double t_0 = cos((2.0 * x));
	double t_1 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 2.85e-34) {
		tmp = t_1 * t_1;
	} else if (x <= 3.2e+197) {
		tmp = t_0 / (x * (x * (c_m * (s_m * (c_m * s_m)))));
	} else {
		tmp = t_0 / ((c_m * x) * (c_m * (s_m * (x * s_m))));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    t_1 = ((1.0d0 / s_m) / c_m) / x
    if (x <= 2.85d-34) then
        tmp = t_1 * t_1
    else if (x <= 3.2d+197) then
        tmp = t_0 / (x * (x * (c_m * (s_m * (c_m * s_m)))))
    else
        tmp = t_0 / ((c_m * x) * (c_m * (s_m * (x * s_m))))
    end if
    code = tmp
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 = Math.cos((2.0 * x));
	double t_1 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 2.85e-34) {
		tmp = t_1 * t_1;
	} else if (x <= 3.2e+197) {
		tmp = t_0 / (x * (x * (c_m * (s_m * (c_m * s_m)))));
	} else {
		tmp = t_0 / ((c_m * x) * (c_m * (s_m * (x * 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 = ((1.0 / s_m) / c_m) / x
	tmp = 0
	if x <= 2.85e-34:
		tmp = t_1 * t_1
	elif x <= 3.2e+197:
		tmp = t_0 / (x * (x * (c_m * (s_m * (c_m * s_m)))))
	else:
		tmp = t_0 / ((c_m * x) * (c_m * (s_m * (x * 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(Float64(Float64(1.0 / s_m) / c_m) / x)
	tmp = 0.0
	if (x <= 2.85e-34)
		tmp = Float64(t_1 * t_1);
	elseif (x <= 3.2e+197)
		tmp = Float64(t_0 / Float64(x * Float64(x * Float64(c_m * Float64(s_m * Float64(c_m * s_m))))));
	else
		tmp = Float64(t_0 / Float64(Float64(c_m * x) * Float64(c_m * Float64(s_m * Float64(x * 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 = ((1.0 / s_m) / c_m) / x;
	tmp = 0.0;
	if (x <= 2.85e-34)
		tmp = t_1 * t_1;
	elseif (x <= 3.2e+197)
		tmp = t_0 / (x * (x * (c_m * (s_m * (c_m * s_m)))));
	else
		tmp = t_0 / ((c_m * x) * (c_m * (s_m * (x * 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[(N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 2.85e-34], N[(t$95$1 * t$95$1), $MachinePrecision], If[LessEqual[x, 3.2e+197], N[(t$95$0 / N[(x * N[(x * N[(c$95$m * N[(s$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(c$95$m * x), $MachinePrecision] * N[(c$95$m * N[(s$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.84999999999999987e-34

   1. Initial program 67.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 71.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 71.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 78.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 73.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/l/ N/A

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

       11. div-inv N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{c \cdot s}}{x}\right), \color{blue}{\left(\frac{\frac{1}{c \cdot s}}{x}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), x\right), \left(\frac{\color{blue}{\frac{1}{c \cdot s}}}{x}\right)\right) \]

       16. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{c}\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s}\right), c\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \left(\frac{\frac{1}{c \cdot s}}{x}\right)\right) \]

       19. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 82.4%

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

   if 2.84999999999999987e-34 < x < 3.1999999999999998e197

   1. Initial program 64.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 84.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 84.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      8. *-lowering-*.f64 84.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   7. Applied egg-rr 84.1%

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

   if 3.1999999999999998e197 < x

   1. Initial program 50.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left(\left({s}^{2} \cdot x\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot {x}^{\color{blue}{2}}\right)\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{2}}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\left(\left(c \cdot s\right) \cdot x\right), \color{blue}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), x\right), 2\right)\right) \]

      9. *-lowering-*.f64 95.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right), 2\right)\right) \]

   4. Applied egg-rr 95.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      2. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(\color{blue}{s} \cdot \left(s \cdot x\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left(c \cdot x\right) \cdot c\right) \cdot \left(\color{blue}{s} \cdot \left(s \cdot x\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(c \cdot x\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(c \cdot x\right), \color{blue}{\left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, x\right), \left(\color{blue}{c} \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, x\right), \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, x\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 88.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, x\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{x}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 88.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{c}}{x} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+197}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\frac{1}{s\_m}}{c\_m}}{x}\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s\_m \cdot \left(c\_m \cdot \left(x \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 s_m) c_m) x)))
   (if (<= x 4.2e-13)
     (* t_0 t_0)
     (/ (cos (* 2.0 x)) (* s_m (* c_m (* x (* x (* 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 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 4.2e-13) {
		tmp = t_0 * t_0;
	} else {
		tmp = cos((2.0 * x)) / (s_m * (c_m * (x * (x * (c_m * s_m)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = ((1.0d0 / s_m) / c_m) / x
    if (x <= 4.2d-13) then
        tmp = t_0 * t_0
    else
        tmp = cos((2.0d0 * x)) / (s_m * (c_m * (x * (x * (c_m * s_m)))))
    end if
    code = tmp
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 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 4.2e-13) {
		tmp = t_0 * t_0;
	} else {
		tmp = Math.cos((2.0 * x)) / (s_m * (c_m * (x * (x * (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 = ((1.0 / s_m) / c_m) / x
	tmp = 0
	if x <= 4.2e-13:
		tmp = t_0 * t_0
	else:
		tmp = math.cos((2.0 * x)) / (s_m * (c_m * (x * (x * (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 = Float64(Float64(Float64(1.0 / s_m) / c_m) / x)
	tmp = 0.0
	if (x <= 4.2e-13)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(s_m * Float64(c_m * Float64(x * Float64(x * 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 = ((1.0 / s_m) / c_m) / x;
	tmp = 0.0;
	if (x <= 4.2e-13)
		tmp = t_0 * t_0;
	else
		tmp = cos((2.0 * x)) / (s_m * (c_m * (x * (x * (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[(N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 4.2e-13], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(c$95$m * N[(x * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999977e-13

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 78.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 73.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/l/ N/A

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

       11. div-inv N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{c \cdot s}}{x}\right), \color{blue}{\left(\frac{\frac{1}{c \cdot s}}{x}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), x\right), \left(\frac{\color{blue}{\frac{1}{c \cdot s}}}{x}\right)\right) \]

       16. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{c}\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s}\right), c\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \left(\frac{\frac{1}{c \cdot s}}{x}\right)\right) \]

       19. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 82.7%

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

   if 4.19999999999999977e-13 < x

   1. Initial program 58.7%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 78.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 78.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      8. *-lowering-*.f64 78.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   7. Applied egg-rr 78.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right) \cdot \color{blue}{s}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right) \cdot s\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)\right), \color{blue}{s}\right)\right) \]

   9. Applied egg-rr 87.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{c}}{x} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\frac{1}{s\_m}}{c\_m}}{x}\\ \mathbf{if}\;x \leq 5.2 \cdot 10^{-34}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c\_m \cdot \left(s\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 s_m) c_m) x)))
   (if (<= x 5.2e-34)
     (* t_0 t_0)
     (/ (cos (* 2.0 x)) (* x (* x (* c_m (* s_m (* 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 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 5.2e-34) {
		tmp = t_0 * t_0;
	} else {
		tmp = cos((2.0 * x)) / (x * (x * (c_m * (s_m * (c_m * s_m)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = ((1.0d0 / s_m) / c_m) / x
    if (x <= 5.2d-34) then
        tmp = t_0 * t_0
    else
        tmp = cos((2.0d0 * x)) / (x * (x * (c_m * (s_m * (c_m * s_m)))))
    end if
    code = tmp
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 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 5.2e-34) {
		tmp = t_0 * t_0;
	} else {
		tmp = Math.cos((2.0 * x)) / (x * (x * (c_m * (s_m * (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 = ((1.0 / s_m) / c_m) / x
	tmp = 0
	if x <= 5.2e-34:
		tmp = t_0 * t_0
	else:
		tmp = math.cos((2.0 * x)) / (x * (x * (c_m * (s_m * (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 = Float64(Float64(Float64(1.0 / s_m) / c_m) / x)
	tmp = 0.0
	if (x <= 5.2e-34)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(x * Float64(c_m * Float64(s_m * 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 = ((1.0 / s_m) / c_m) / x;
	tmp = 0.0;
	if (x <= 5.2e-34)
		tmp = t_0 * t_0;
	else
		tmp = cos((2.0 * x)) / (x * (x * (c_m * (s_m * (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[(N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 5.2e-34], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c$95$m * N[(s$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.1999999999999999e-34

   1. Initial program 67.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 71.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 71.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 78.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 73.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/l/ N/A

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

       11. div-inv N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{c \cdot s}}{x}\right), \color{blue}{\left(\frac{\frac{1}{c \cdot s}}{x}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), x\right), \left(\frac{\color{blue}{\frac{1}{c \cdot s}}}{x}\right)\right) \]

       16. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{c}\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s}\right), c\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \left(\frac{\frac{1}{c \cdot s}}{x}\right)\right) \]

       19. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 82.4%

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

   if 5.1999999999999999e-34 < x

   1. Initial program 60.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 79.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

      8. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   7. Applied egg-rr 79.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{c}}{x} \cdot \frac{\frac{\frac{1}{s}}{c}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\frac{1}{s\_m}}{c\_m}}{x}\\ \mathbf{if}\;x \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s\_m \cdot \left(c\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 s_m) c_m) x)))
   (if (<= x 2.1e-13)
     (* t_0 t_0)
     (/ (cos (* 2.0 x)) (* x (* x (* s_m (* c_m (* 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 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 2.1e-13) {
		tmp = t_0 * t_0;
	} else {
		tmp = cos((2.0 * x)) / (x * (x * (s_m * (c_m * (c_m * s_m)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = ((1.0d0 / s_m) / c_m) / x
    if (x <= 2.1d-13) then
        tmp = t_0 * t_0
    else
        tmp = cos((2.0d0 * x)) / (x * (x * (s_m * (c_m * (c_m * s_m)))))
    end if
    code = tmp
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 = ((1.0 / s_m) / c_m) / x;
	double tmp;
	if (x <= 2.1e-13) {
		tmp = t_0 * t_0;
	} else {
		tmp = Math.cos((2.0 * x)) / (x * (x * (s_m * (c_m * (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 = ((1.0 / s_m) / c_m) / x
	tmp = 0
	if x <= 2.1e-13:
		tmp = t_0 * t_0
	else:
		tmp = math.cos((2.0 * x)) / (x * (x * (s_m * (c_m * (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 = Float64(Float64(Float64(1.0 / s_m) / c_m) / x)
	tmp = 0.0
	if (x <= 2.1e-13)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(x * Float64(s_m * Float64(c_m * 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 = ((1.0 / s_m) / c_m) / x;
	tmp = 0.0;
	if (x <= 2.1e-13)
		tmp = t_0 * t_0;
	else
		tmp = cos((2.0 * x)) / (x * (x * (s_m * (c_m * (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[(N[(N[(1.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 2.1e-13], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(s$95$m * N[(c$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999989e-13

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 78.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 73.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. associate-/l/ N/A

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

       10. associate-/l/ N/A

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

       11. div-inv N/A

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

       12. frac-times N/A

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

       13. associate-*l/ N/A

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

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{c \cdot s}}{x}\right), \color{blue}{\left(\frac{\frac{1}{c \cdot s}}{x}\right)}\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), x\right), \left(\frac{\color{blue}{\frac{1}{c \cdot s}}}{x}\right)\right) \]

       16. associate-/l/ N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{c}\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s}\right), c\right), x\right), \left(\frac{\frac{\color{blue}{1}}{c \cdot s}}{x}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \left(\frac{\frac{1}{c \cdot s}}{x}\right)\right) \]

       19. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, s\right), c\right), x\right), \mathsf{/.f64}\left(\left(\frac{1}{c \cdot s}\right), \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 82.7%

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

   if 2.09999999999999989e-13 < x

   1. Initial program 58.7%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 78.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 78.8%

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

Alternative 9: 79.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c\_m \cdot \left(c\_m \cdot x\right)}}{x \cdot s\_m}}{s\_m}\\ \end{array} \end{array} \]

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 (* x (* c_m s_m))))
   (if (<= c_m 5e+21)
     (/ (/ 1.0 t_0) t_0)
     (/ (/ (/ 1.0 (* c_m (* c_m x))) (* x s_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 = x * (c_m * s_m);
	double tmp;
	if (c_m <= 5e+21) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = ((1.0 / (c_m * (c_m * x))) / (x * s_m)) / s_m;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = x * (c_m * s_m)
    if (c_m <= 5d+21) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = ((1.0d0 / (c_m * (c_m * x))) / (x * s_m)) / s_m
    end if
    code = tmp
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 = x * (c_m * s_m);
	double tmp;
	if (c_m <= 5e+21) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = ((1.0 / (c_m * (c_m * x))) / (x * s_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 = x * (c_m * s_m)
	tmp = 0
	if c_m <= 5e+21:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = ((1.0 / (c_m * (c_m * x))) / (x * s_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 = Float64(x * Float64(c_m * s_m))
	tmp = 0.0
	if (c_m <= 5e+21)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(c_m * Float64(c_m * x))) / Float64(x * s_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 = x * (c_m * s_m);
	tmp = 0.0;
	if (c_m <= 5e+21)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = ((1.0 / (c_m * (c_m * x))) / (x * s_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[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 5e+21], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / N[(c$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 5e21

   1. Initial program 66.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 64.0%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(c \cdot s\right) \cdot x}\right), \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(c \cdot s\right), x\right)\right), \left(\left(c \cdot \color{blue}{s}\right) \cdot x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right), \mathsf{*.f64}\left(\left(c \cdot s\right), \color{blue}{x}\right)\right) \]

      13. *-lowering-*.f64 72.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right) \]

   7. Applied egg-rr 72.3%

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

   if 5e21 < c

   1. Initial program 61.7%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{x \cdot s}\right), \color{blue}{s}\right) \]

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{{c}^{2} \cdot x}\right)}, \mathsf{*.f64}\left(s, x\right)\right), s\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot x\right)\right), \mathsf{*.f64}\left(s, x\right)\right), s\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot x\right)\right), \mathsf{*.f64}\left(s, x\right)\right), s\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(s, x\right)\right), s\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(x \cdot c\right)\right)\right), \mathsf{*.f64}\left(s, x\right)\right), s\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(x \cdot c\right)\right)\right), \mathsf{*.f64}\left(s, x\right)\right), s\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(s, x\right)\right), s\right) \]

      7. *-lowering-*.f64 78.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, x\right)\right)\right), \mathsf{*.f64}\left(s, x\right)\right), s\right) \]

   7. Simplified 78.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot \left(c \cdot x\right)}}{x \cdot s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.8% accurate, 17.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c\_m \cdot \left(c\_m \cdot \left(x \cdot s\_m\right)\right)\right)}}{s\_m}\\ \end{array} \end{array} \]

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 (* x (* c_m s_m))))
   (if (<= c_m 2e+22)
     (/ (/ 1.0 t_0) t_0)
     (/ (/ 1.0 (* x (* c_m (* c_m (* x s_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 = x * (c_m * s_m);
	double tmp;
	if (c_m <= 2e+22) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_m))))) / s_m;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = x * (c_m * s_m)
    if (c_m <= 2d+22) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = (1.0d0 / (x * (c_m * (c_m * (x * s_m))))) / s_m
    end if
    code = tmp
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 = x * (c_m * s_m);
	double tmp;
	if (c_m <= 2e+22) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_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 = x * (c_m * s_m)
	tmp = 0
	if c_m <= 2e+22:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_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 = Float64(x * Float64(c_m * s_m))
	tmp = 0.0
	if (c_m <= 2e+22)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(x * Float64(c_m * Float64(c_m * Float64(x * s_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 = x * (c_m * s_m);
	tmp = 0.0;
	if (c_m <= 2e+22)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_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[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 2e+22], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(x * N[(c$95$m * N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 2e22

   1. Initial program 66.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 64.0%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left(c \cdot s\right) \cdot x}\right), \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(c \cdot s\right), x\right)\right), \left(\left(c \cdot \color{blue}{s}\right) \cdot x\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right), \mathsf{*.f64}\left(\left(c \cdot s\right), \color{blue}{x}\right)\right) \]

      13. *-lowering-*.f64 72.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right)\right) \]

   7. Applied egg-rr 72.3%

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

   if 2e22 < c

   1. Initial program 61.7%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{x \cdot s}\right), \color{blue}{s}\right) \]

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{{c}^{2} \cdot \left(s \cdot {x}^{2}\right)}\right)}, s\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot \left(s \cdot {x}^{2}\right)\right)\right), s\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right), s\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot \left(\left(s \cdot x\right) \cdot x\right)\right)\right), s\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left({c}^{2} \cdot \left(s \cdot x\right)\right) \cdot x\right)\right), s\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot \left({c}^{2} \cdot \left(s \cdot x\right)\right)\right)\right), s\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot x\right)\right)\right)\right), s\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot x\right)\right)\right)\right), s\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right)\right), s\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(c \cdot \left(s \cdot x\right)\right)\right)\right)\right), s\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right)\right)\right), s\right) \]

      11. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right)\right)\right), s\right) \]

   7. Simplified 77.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.8% accurate, 17.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;c\_m \leq 0.055:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(\left(x \cdot \left(c\_m \cdot s\_m\right)\right) \cdot \left(x \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c\_m \cdot \left(c\_m \cdot \left(x \cdot s\_m\right)\right)\right)}}{s\_m}\\ \end{array} \end{array} \]

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
 (if (<= c_m 0.055)
   (/ 1.0 (* c_m (* (* x (* c_m s_m)) (* x s_m))))
   (/ (/ 1.0 (* x (* c_m (* c_m (* x s_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 tmp;
	if (c_m <= 0.055) {
		tmp = 1.0 / (c_m * ((x * (c_m * s_m)) * (x * s_m)));
	} else {
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_m))))) / s_m;
	}
	return tmp;
}

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) :: tmp
    if (c_m <= 0.055d0) then
        tmp = 1.0d0 / (c_m * ((x * (c_m * s_m)) * (x * s_m)))
    else
        tmp = (1.0d0 / (x * (c_m * (c_m * (x * s_m))))) / s_m
    end if
    code = tmp
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 tmp;
	if (c_m <= 0.055) {
		tmp = 1.0 / (c_m * ((x * (c_m * s_m)) * (x * s_m)));
	} else {
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_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):
	tmp = 0
	if c_m <= 0.055:
		tmp = 1.0 / (c_m * ((x * (c_m * s_m)) * (x * s_m)))
	else:
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_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)
	tmp = 0.0
	if (c_m <= 0.055)
		tmp = Float64(1.0 / Float64(c_m * Float64(Float64(x * Float64(c_m * s_m)) * Float64(x * s_m))));
	else
		tmp = Float64(Float64(1.0 / Float64(x * Float64(c_m * Float64(c_m * Float64(x * s_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)
	tmp = 0.0;
	if (c_m <= 0.055)
		tmp = 1.0 / (c_m * ((x * (c_m * s_m)) * (x * s_m)));
	else
		tmp = (1.0 / (x * (c_m * (c_m * (x * s_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_] := If[LessEqual[c$95$m, 0.055], N[(1.0 / N[(c$95$m * N[(N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * N[(c$95$m * N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 0.0550000000000000003

   1. Initial program 66.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

      18. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

   5. Simplified 64.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. unpow2 N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

      19. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

      23. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      24. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 69.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

      10. *-lowering-*.f64 64.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 64.5%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot s\right) \cdot \color{blue}{s}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right) \]

       3. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot c\right) \cdot s\right) \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(s \cdot x\right), \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(x \cdot s\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

       16. *-lowering-*.f64 70.7%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 70.7%

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

   if 0.0550000000000000003 < c

   1. Initial program 63.1%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{x \cdot s}\right), \color{blue}{s}\right) \]

   4. Applied egg-rr 74.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{{c}^{2} \cdot \left(s \cdot {x}^{2}\right)}\right)}, s\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot \left(s \cdot {x}^{2}\right)\right)\right), s\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right), s\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot \left(\left(s \cdot x\right) \cdot x\right)\right)\right), s\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left({c}^{2} \cdot \left(s \cdot x\right)\right) \cdot x\right)\right), s\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot \left({c}^{2} \cdot \left(s \cdot x\right)\right)\right)\right), s\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot x\right)\right)\right)\right), s\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot x\right)\right)\right)\right), s\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right)\right), s\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(c \cdot \left(s \cdot x\right)\right)\right)\right)\right), s\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right)\right)\right), s\right) \]

      11. *-lowering-*.f64 77.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right)\right)\right), s\right) \]

   7. Simplified 77.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 0.055:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.8% accurate, 24.1× speedup

Math

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

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 (* c_m (* (* x (* c_m s_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 / (c_m * ((x * (c_m * s_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 / (c_m * ((x * (c_m * s_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 / (c_m * ((x * (c_m * s_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 / (c_m * ((x * (c_m * s_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(c_m * Float64(Float64(x * Float64(c_m * s_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 / (c_m * ((x * (c_m * s_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[(c$95$m * N[(N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

   18. *-lowering-*.f64 65.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

5. Simplified 65.8%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. swap-sqr N/A

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

   6. unpow2 N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({\left(\left(c \cdot s\right) \cdot x\right)}^{2}\right)}\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left({\left(c \cdot \left(s \cdot x\right)\right)}^{2}\right)\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left({c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}\right)\right) \]

   10. pow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot {\color{blue}{\left(s \cdot x\right)}}^{2}\right)\right) \]

   11. pow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot x\right)\right)\right)\right) \]

   19. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right)}\right)\right)\right) \]

   22. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

   23. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

   24. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

7. Applied egg-rr 71.4%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(x \cdot \left(c \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \left(x \cdot c\right)\right), s\right)\right)\right) \]

   10. *-lowering-*.f64 67.0%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right)\right) \]

9. Applied egg-rr 67.0%

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

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot s\right) \cdot \color{blue}{s}\right)\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right) \]

    3. swap-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot c\right) \cdot s\right) \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right) \]

    9. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(s \cdot x\right), \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(x \cdot s\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

    16. *-lowering-*.f64 71.9%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right) \]

11. Applied egg-rr 71.9%

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

12. Final simplification 71.9%

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

Alternative 13: 25.4% accurate, 34.8× speedup

Math

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

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 (/ -2.0 (* s_m (* c_m (* 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) {
	return -2.0 / (s_m * (c_m * (c_m * 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 = (-2.0d0) / (s_m * (c_m * (c_m * 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 -2.0 / (s_m * (c_m * (c_m * 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 -2.0 / (s_m * (c_m * (c_m * 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(-2.0 / Float64(s_m * Float64(c_m * Float64(c_m * 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 = -2.0 / (s_m * (c_m * (c_m * 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[(-2.0 / N[(s$95$m * N[(c$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({s}^{2} \cdot \color{blue}{{c}^{2}}\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot s\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {c}^{2}\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot \color{blue}{c}\right)\right)\right)\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f64 81.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

5. Simplified 81.4%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), \color{blue}{x}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right), x\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), x\right)\right)\right) \]

   8. *-lowering-*.f64 80.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

7. Applied egg-rr 80.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + -2 \cdot {x}^{2}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-2 \cdot {x}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-2 \cdot \left(x \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(-2 \cdot x\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(-2 \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-2 \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot -2\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

   7. *-lowering-*.f64 50.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, -2\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, s\right)\right)\right), x\right)\right)\right) \]

10. Simplified 50.6%

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

11. Taylor expanded in x around inf

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

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

    9. *-lowering-*.f64 23.0%

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

13. Simplified 23.0%

    \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))