mixedcos

Percentage Accurate: 67.3% → 97.4%

Time: 14.1s

Alternatives: 21

Speedup: 24.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.3% 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: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

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

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

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

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   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. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

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

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

   18. *-lowering-*.f64 82.3%

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

3. Simplified 82.3%

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

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. count-2 N/A

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

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

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

   13. count-2 N/A

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

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

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

   15. *-commutative N/A

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

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

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

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

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

   18. *-commutative N/A

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

   19. *-lowering-*.f64 97.8%

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

6. Applied egg-rr 97.8%

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

Alternative 2: 71.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{\frac{1}{c}}{s}}{x}}{t\_0}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot t\_0}}{x \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c))))
   (if (<= x 9.5e-6)
     (/ (/ (* (+ 1.0 (* (* x x) -2.0)) (/ (/ 1.0 c) s)) x) t_0)
     (if (<= x 3.9e+130)
       (/ (cos (* 2.0 x)) (* c (* c (* s (* s (* x x))))))
       (/ (/ 1.0 (* s t_0)) (* x c))))))

C

double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 9.5e-6) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else if (x <= 3.9e+130) {
		tmp = cos((2.0 * x)) / (c * (c * (s * (s * (x * x)))));
	} else {
		tmp = (1.0 / (s * t_0)) / (x * c);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (s * c)
    if (x <= 9.5d-6) then
        tmp = (((1.0d0 + ((x * x) * (-2.0d0))) * ((1.0d0 / c) / s)) / x) / t_0
    else if (x <= 3.9d+130) then
        tmp = cos((2.0d0 * x)) / (c * (c * (s * (s * (x * x)))))
    else
        tmp = (1.0d0 / (s * t_0)) / (x * c)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 9.5e-6) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else if (x <= 3.9e+130) {
		tmp = Math.cos((2.0 * x)) / (c * (c * (s * (s * (x * x)))));
	} else {
		tmp = (1.0 / (s * t_0)) / (x * c);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (s * c)
	tmp = 0
	if x <= 9.5e-6:
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0
	elif x <= 3.9e+130:
		tmp = math.cos((2.0 * x)) / (c * (c * (s * (s * (x * x)))))
	else:
		tmp = (1.0 / (s * t_0)) / (x * c)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (x <= 9.5e-6)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * -2.0)) * Float64(Float64(1.0 / c) / s)) / x) / t_0);
	elseif (x <= 3.9e+130)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(c * Float64(c * Float64(s * Float64(s * Float64(x * x))))));
	else
		tmp = Float64(Float64(1.0 / Float64(s * t_0)) / Float64(x * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 0.0;
	if (x <= 9.5e-6)
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	elseif (x <= 3.9e+130)
		tmp = cos((2.0 * x)) / (c * (c * (s * (s * (x * x)))));
	else
		tmp = (1.0 / (s * t_0)) / (x * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9.5e-6], N[(N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] / s), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 3.9e+130], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(c * N[(c * N[(s * N[(s * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(s * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.5000000000000005e-6

   1. Initial program 62.9%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.4%

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

   3. Simplified 81.4%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

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

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

   9. Simplified 67.8%

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

   if 9.5000000000000005e-6 < x < 3.9000000000000002e130

   1. Initial program 79.1%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   6. 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. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 83.7%

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

   7. Simplified 83.7%

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

   if 3.9000000000000002e130 < x

   1. Initial program 61.1%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 76.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. associate-/l/ N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f64 79.9%

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

   9. Applied egg-rr 79.9%

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

Alternative 3: 76.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;x \leq 0.00041:\\ \;\;\;\;\frac{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{\frac{1}{c}}{s}}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{s}}{x \cdot \left(c \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c))))
   (if (<= x 0.00041)
     (/ (/ (* (+ 1.0 (* (* x x) -2.0)) (/ (/ 1.0 c) s)) x) t_0)
     (/ (/ (cos (* 2.0 x)) s) (* x (* c t_0))))))

C

double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 0.00041) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else {
		tmp = (cos((2.0 * x)) / s) / (x * (c * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (s * c)
    if (x <= 0.00041d0) then
        tmp = (((1.0d0 + ((x * x) * (-2.0d0))) * ((1.0d0 / c) / s)) / x) / t_0
    else
        tmp = (cos((2.0d0 * x)) / s) / (x * (c * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 0.00041) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else {
		tmp = (Math.cos((2.0 * x)) / s) / (x * (c * t_0));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (s * c)
	tmp = 0
	if x <= 0.00041:
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0
	else:
		tmp = (math.cos((2.0 * x)) / s) / (x * (c * t_0))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (x <= 0.00041)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * -2.0)) * Float64(Float64(1.0 / c) / s)) / x) / t_0);
	else
		tmp = Float64(Float64(cos(Float64(2.0 * x)) / s) / Float64(x * Float64(c * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 0.0;
	if (x <= 0.00041)
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	else
		tmp = (cos((2.0 * x)) / s) / (x * (c * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.00041], N[(N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] / s), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / s), $MachinePrecision] / N[(x * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0999999999999999e-4

   1. Initial program 62.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.4%

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

   3. Simplified 81.4%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

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

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

   9. Simplified 67.9%

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

   if 4.0999999999999999e-4 < x

   1. Initial program 71.4%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 84.8%

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

   3. Simplified 84.8%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. associate-*r* N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r* N/A

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

      6. associate-/l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/r* N/A

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

      10. associate-*l/ N/A

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

      11. div-inv N/A

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

      12. associate-/l/ N/A

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

      13. associate-*r* N/A

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

   8. Applied egg-rr 94.2%

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

Alternative 4: 76.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;x \leq 0.00038:\\ \;\;\;\;\frac{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{\frac{1}{c}}{s}}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(x \cdot \left(c \cdot t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c))))
   (if (<= x 0.00038)
     (/ (/ (* (+ 1.0 (* (* x x) -2.0)) (/ (/ 1.0 c) s)) x) t_0)
     (/ (cos (* 2.0 x)) (* s (* x (* c t_0)))))))

C

double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 0.00038) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else {
		tmp = cos((2.0 * x)) / (s * (x * (c * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (s * c)
    if (x <= 0.00038d0) then
        tmp = (((1.0d0 + ((x * x) * (-2.0d0))) * ((1.0d0 / c) / s)) / x) / t_0
    else
        tmp = cos((2.0d0 * x)) / (s * (x * (c * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 0.00038) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else {
		tmp = Math.cos((2.0 * x)) / (s * (x * (c * t_0)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (s * c)
	tmp = 0
	if x <= 0.00038:
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0
	else:
		tmp = math.cos((2.0 * x)) / (s * (x * (c * t_0)))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (x <= 0.00038)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * -2.0)) * Float64(Float64(1.0 / c) / s)) / x) / t_0);
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(s * Float64(x * Float64(c * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 0.0;
	if (x <= 0.00038)
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	else
		tmp = cos((2.0 * x)) / (s * (x * (c * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.00038], N[(N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] / s), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(s * N[(x * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8000000000000002e-4

   1. Initial program 62.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.4%

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

   3. Simplified 81.4%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

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

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

   9. Simplified 67.9%

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

   if 3.8000000000000002e-4 < x

   1. Initial program 71.4%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   6. 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. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 79.8%

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

   7. Simplified 79.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

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

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

      19. *-lowering-*.f64 93.1%

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

   9. Applied egg-rr 93.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00038:\\ \;\;\;\;\frac{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{\frac{1}{c}}{s}}{x}}{x \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{\left(x \cdot s\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 4.6e-47)
   (/ (/ 1.0 (* x s)) (* (* x s) (* c c)))
   (/ (cos (* 2.0 x)) (* x (* (* x c) (* s (* s c)))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.6e-47) {
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c));
	} else {
		tmp = cos((2.0 * x)) / (x * ((x * c) * (s * (s * c))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 4.6d-47) then
        tmp = (1.0d0 / (x * s)) / ((x * s) * (c * c))
    else
        tmp = cos((2.0d0 * x)) / (x * ((x * c) * (s * (s * c))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.6e-47) {
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c));
	} else {
		tmp = Math.cos((2.0 * x)) / (x * ((x * c) * (s * (s * c))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 4.6e-47:
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c))
	else:
		tmp = math.cos((2.0 * x)) / (x * ((x * c) * (s * (s * c))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 4.6e-47)
		tmp = Float64(Float64(1.0 / Float64(x * s)) / Float64(Float64(x * s) * Float64(c * c)));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(Float64(x * c) * Float64(s * Float64(s * c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4.6e-47)
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c));
	else
		tmp = cos((2.0 * x)) / (x * ((x * c) * (s * (s * c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 4.6e-47], N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(x * c), $MachinePrecision] * N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.59999999999999964e-47

   1. Initial program 62.5%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 80.9%

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

   3. Simplified 80.9%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 66.1%

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

   9. Simplified 66.1%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-/l/ N/A

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

       9. div-inv N/A

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

       10. associate-/r* N/A

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

       11. associate-/r* N/A

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

       12. frac-times N/A

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

       13. div-inv N/A

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

       14. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

       20. *-lowering-*.f64 70.8%

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

   11. Applied egg-rr 70.8%

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

   if 4.59999999999999964e-47 < x

   1. Initial program 71.1%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 85.9%

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

   3. Simplified 85.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 88.1%

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

   6. Applied egg-rr 88.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{\left(x \cdot s\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{\left(x \cdot s\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 7.5e-48)
   (/ (/ 1.0 (* x s)) (* (* x s) (* c c)))
   (/ (cos (* 2.0 x)) (* x (* c (* s (* x (* s c))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 7.5e-48) {
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c));
	} else {
		tmp = cos((2.0 * x)) / (x * (c * (s * (x * (s * c)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 7.5d-48) then
        tmp = (1.0d0 / (x * s)) / ((x * s) * (c * c))
    else
        tmp = cos((2.0d0 * x)) / (x * (c * (s * (x * (s * c)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 7.5e-48) {
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c));
	} else {
		tmp = Math.cos((2.0 * x)) / (x * (c * (s * (x * (s * c)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 7.5e-48:
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c))
	else:
		tmp = math.cos((2.0 * x)) / (x * (c * (s * (x * (s * c)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 7.5e-48)
		tmp = Float64(Float64(1.0 / Float64(x * s)) / Float64(Float64(x * s) * Float64(c * c)));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(c * Float64(s * Float64(x * Float64(s * c))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 7.5e-48)
		tmp = (1.0 / (x * s)) / ((x * s) * (c * c));
	else
		tmp = cos((2.0 * x)) / (x * (c * (s * (x * (s * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 7.5e-48], N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(c * N[(s * N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.50000000000000042e-48

   1. Initial program 62.5%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 80.9%

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

   3. Simplified 80.9%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 66.1%

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

   9. Simplified 66.1%

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-/l/ N/A

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

       9. div-inv N/A

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

       10. associate-/r* N/A

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

       11. associate-/r* N/A

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

       12. frac-times N/A

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

       13. div-inv N/A

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

       14. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

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

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

       20. *-lowering-*.f64 70.8%

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

   11. Applied egg-rr 70.8%

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

   if 7.50000000000000042e-48 < x

   1. Initial program 71.1%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 85.9%

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

   3. Simplified 85.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 92.8%

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

   6. Applied egg-rr 92.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{\left(x \cdot s\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 7e-5)
   (/ (/ (* (+ 1.0 (* (* x x) -2.0)) (/ (/ 1.0 c) s)) x) (* x (* s c)))
   (/ (cos (* 2.0 x)) (* x (* x (* s (* c (* s c))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 7e-5) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / (x * (s * c));
	} else {
		tmp = cos((2.0 * x)) / (x * (x * (s * (c * (s * c)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 7d-5) then
        tmp = (((1.0d0 + ((x * x) * (-2.0d0))) * ((1.0d0 / c) / s)) / x) / (x * (s * c))
    else
        tmp = cos((2.0d0 * x)) / (x * (x * (s * (c * (s * c)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 7e-5) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / (x * (s * c));
	} else {
		tmp = Math.cos((2.0 * x)) / (x * (x * (s * (c * (s * c)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 7e-5:
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / (x * (s * c))
	else:
		tmp = math.cos((2.0 * x)) / (x * (x * (s * (c * (s * c)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 7e-5)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * -2.0)) * Float64(Float64(1.0 / c) / s)) / x) / Float64(x * Float64(s * c)));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(x * Float64(s * Float64(c * Float64(s * c))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 7e-5)
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / (x * (s * c));
	else
		tmp = cos((2.0 * x)) / (x * (x * (s * (c * (s * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 7e-5], N[(N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] / s), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(s * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.9999999999999994e-5

   1. Initial program 62.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.4%

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

   3. Simplified 81.4%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. count-2 N/A

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

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

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

      13. count-2 N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

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

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

   9. Simplified 67.9%

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

   if 6.9999999999999994e-5 < x

   1. Initial program 71.4%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      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. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 84.8%

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

   3. Simplified 84.8%

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

4. Final simplification 72.4%

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

Alternative 8: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \frac{\frac{\cos \left(2 \cdot x\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c)))) (/ (/ (cos (* 2.0 x)) t_0) t_0)))

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = x * (s * c)
    code = (cos((2.0d0 * x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	return Float64(Float64(cos(Float64(2.0 * x)) / t_0) / t_0)
end

MATLAB

function tmp = code(x, c, s)
	t_0 = x * (s * c);
	tmp = (cos((2.0 * x)) / t_0) / t_0;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   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. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 82.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(\color{blue}{c} \cdot s\right)\right)} \]

   4. unswap-sqr N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{\color{blue}{x \cdot \left(c \cdot s\right)}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   8. count-2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   10. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   17. *-lowering-*.f64 97.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(s \cdot c\right)}} \]
7. Add Preprocessing

Alternative 9: 68.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{\frac{1}{c}}{s}}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c))))
   (if (<= x 1.85e+77)
     (/ (/ (* (+ 1.0 (* (* x x) -2.0)) (/ (/ 1.0 c) s)) x) t_0)
     (/ (/ 1.0 (* s (* x c))) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 1.85e+77) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else {
		tmp = (1.0 / (s * (x * c))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (s * c)
    if (x <= 1.85d+77) then
        tmp = (((1.0d0 + ((x * x) * (-2.0d0))) * ((1.0d0 / c) / s)) / x) / t_0
    else
        tmp = (1.0d0 / (s * (x * c))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (x <= 1.85e+77) {
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	} else {
		tmp = (1.0 / (s * (x * c))) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (s * c)
	tmp = 0
	if x <= 1.85e+77:
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0
	else:
		tmp = (1.0 / (s * (x * c))) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (x <= 1.85e+77)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * -2.0)) * Float64(Float64(1.0 / c) / s)) / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(s * Float64(x * c))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 0.0;
	if (x <= 1.85e+77)
		tmp = (((1.0 + ((x * x) * -2.0)) * ((1.0 / c) / s)) / x) / t_0;
	else
		tmp = (1.0 / (s * (x * c))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85e+77], N[(N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] / s), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.84999999999999997e77

   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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 82.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 98.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 67.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{\frac{1}{c}}{s}}{x}}}{x \cdot \left(s \cdot c\right)} \]

   if 1.84999999999999997e77 < x

   1. Initial program 65.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 83.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 96.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(c \cdot s\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot c\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      6. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 75.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{s \cdot \left(c \cdot x\right)}}}{x \cdot \left(s \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{\frac{1}{c}}{s}}{x}}{x \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1}{\frac{x \cdot s}{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}}}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 1.2e+76)
   (/ (/ 1.0 (/ (* x s) (/ (+ 1.0 (* (* x x) -2.0)) (* c c)))) (* x s))
   (/ (/ 1.0 (* s (* x c))) (* x (* s c)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.2e+76) {
		tmp = (1.0 / ((x * s) / ((1.0 + ((x * x) * -2.0)) / (c * c)))) / (x * s);
	} else {
		tmp = (1.0 / (s * (x * c))) / (x * (s * c));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 1.2d+76) then
        tmp = (1.0d0 / ((x * s) / ((1.0d0 + ((x * x) * (-2.0d0))) / (c * c)))) / (x * s)
    else
        tmp = (1.0d0 / (s * (x * c))) / (x * (s * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.2e+76) {
		tmp = (1.0 / ((x * s) / ((1.0 + ((x * x) * -2.0)) / (c * c)))) / (x * s);
	} else {
		tmp = (1.0 / (s * (x * c))) / (x * (s * c));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 1.2e+76:
		tmp = (1.0 / ((x * s) / ((1.0 + ((x * x) * -2.0)) / (c * c)))) / (x * s)
	else:
		tmp = (1.0 / (s * (x * c))) / (x * (s * c))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 1.2e+76)
		tmp = Float64(Float64(1.0 / Float64(Float64(x * s) / Float64(Float64(1.0 + Float64(Float64(x * x) * -2.0)) / Float64(c * c)))) / Float64(x * s));
	else
		tmp = Float64(Float64(1.0 / Float64(s * Float64(x * c))) / Float64(x * Float64(s * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1.2e+76)
		tmp = (1.0 / ((x * s) / ((1.0 + ((x * x) * -2.0)) / (c * c)))) / (x * s);
	else
		tmp = (1.0 / (s * (x * c))) / (x * (s * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 1.2e+76], N[(N[(1.0 / N[(N[(x * s), $MachinePrecision] / N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e76

   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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 82.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 55.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{\left(c \cdot c\right) \cdot s} \cdot \left(1 + \left(x \cdot x\right) \cdot -2\right)}{x}\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{1}{s}}{c \cdot c} \cdot \left(1 + \left(x \cdot x\right) \cdot -2\right)}{x}\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{1}{s} \cdot \left(1 + \left(x \cdot x\right) \cdot -2\right)}{c \cdot c}}{x}\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{s} \cdot \left(1 + \left(x \cdot x\right) \cdot -2\right)}{x \cdot \left(c \cdot c\right)}\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{s} \cdot \left(1 + \left(x \cdot x\right) \cdot -2\right)\right), \left(x \cdot \left(c \cdot c\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{s}\right), \left(1 + \left(x \cdot x\right) \cdot -2\right)\right), \left(x \cdot \left(c \cdot c\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, s\right), \left(1 + \left(x \cdot x\right) \cdot -2\right)\right), \left(x \cdot \left(c \cdot c\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, s\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot -2\right)\right)\right), \left(x \cdot \left(c \cdot c\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, s\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), -2\right)\right)\right), \left(x \cdot \left(c \cdot c\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, s\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right), \left(x \cdot \left(c \cdot c\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, s\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot c\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       12. *-lowering-*.f64 55.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, s\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

   11. Applied egg-rr 55.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{s} \cdot \left(1 + \left(x \cdot x\right) \cdot -2\right)}{x \cdot \left(c \cdot c\right)}}}{x \cdot s} \]
   12. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{s}}{x} \cdot \frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c} \cdot \frac{\frac{1}{s}}{x}\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c} \cdot \frac{1}{x \cdot s}\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       4. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}}{x \cdot s}\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{x \cdot s}{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}}}\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{x \cdot s}{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot s\right), \left(\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{/.f64}\left(\left(1 + \left(x \cdot x\right) \cdot -2\right), \left(c \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot -2\right)\right), \left(c \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-2 \cdot \left(x \cdot x\right)\right)\right), \left(c \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-2, \left(x \cdot x\right)\right)\right), \left(c \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(c \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       14. *-lowering-*.f64 57.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

   13. Applied egg-rr 57.5%

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x \cdot s}{\frac{1 + -2 \cdot \left(x \cdot x\right)}{c \cdot c}}}}}{x \cdot s} \]

   if 1.2e76 < x

   1. Initial program 65.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 83.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 96.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(c \cdot s\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot c\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      6. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 75.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{s \cdot \left(c \cdot x\right)}}}{x \cdot \left(s \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1}{\frac{x \cdot s}{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}}}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s} \cdot \frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 1.2e+76)
   (/ (* (/ 1.0 (* x s)) (/ (+ 1.0 (* (* x x) -2.0)) (* c c))) (* x s))
   (/ (/ 1.0 (* s (* x c))) (* x (* s c)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.2e+76) {
		tmp = ((1.0 / (x * s)) * ((1.0 + ((x * x) * -2.0)) / (c * c))) / (x * s);
	} else {
		tmp = (1.0 / (s * (x * c))) / (x * (s * c));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 1.2d+76) then
        tmp = ((1.0d0 / (x * s)) * ((1.0d0 + ((x * x) * (-2.0d0))) / (c * c))) / (x * s)
    else
        tmp = (1.0d0 / (s * (x * c))) / (x * (s * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.2e+76) {
		tmp = ((1.0 / (x * s)) * ((1.0 + ((x * x) * -2.0)) / (c * c))) / (x * s);
	} else {
		tmp = (1.0 / (s * (x * c))) / (x * (s * c));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 1.2e+76:
		tmp = ((1.0 / (x * s)) * ((1.0 + ((x * x) * -2.0)) / (c * c))) / (x * s)
	else:
		tmp = (1.0 / (s * (x * c))) / (x * (s * c))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 1.2e+76)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) * Float64(Float64(1.0 + Float64(Float64(x * x) * -2.0)) / Float64(c * c))) / Float64(x * s));
	else
		tmp = Float64(Float64(1.0 / Float64(s * Float64(x * c))) / Float64(x * Float64(s * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1.2e+76)
		tmp = ((1.0 / (x * s)) * ((1.0 + ((x * x) * -2.0)) / (c * c))) / (x * s);
	else
		tmp = (1.0 / (s * (x * c))) / (x * (s * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 1.2e+76], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e76

   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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 82.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 55.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot 1}{\left(c \cdot c\right) \cdot s}}{x}\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       2. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c} \cdot \frac{1}{s}}{x}\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c} \cdot \frac{\frac{1}{s}}{x}\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       4. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c} \cdot \frac{1}{x \cdot s}\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}\right), \left(\frac{1}{x \cdot s}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(x \cdot x\right) \cdot -2\right), \left(c \cdot c\right)\right), \left(\frac{1}{x \cdot s}\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot -2\right)\right), \left(c \cdot c\right)\right), \left(\frac{1}{x \cdot s}\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), -2\right)\right), \left(c \cdot c\right)\right), \left(\frac{1}{x \cdot s}\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \left(c \cdot c\right)\right), \left(\frac{1}{x \cdot s}\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \left(\frac{1}{x \cdot s}\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{/.f64}\left(1, \left(x \cdot s\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

       12. *-lowering-*.f64 57.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(c, c\right)\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, s\right)\right)\right), \mathsf{*.f64}\left(x, s\right)\right) \]

   11. Applied egg-rr 57.5%

       \[\leadsto \frac{\color{blue}{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c} \cdot \frac{1}{x \cdot s}}}{x \cdot s} \]

   if 1.2e76 < x

   1. Initial program 65.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 83.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 96.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(c \cdot s\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot c\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      6. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 75.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{s \cdot \left(c \cdot x\right)}}}{x \cdot \left(s \cdot c\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s} \cdot \frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.3% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2.7e+16)
   (/ 1.0 (* c (* c (* (* x s) (* x s)))))
   (if (<= x 1.65e+87)
     (/ -2.0 (* c (* c (* s s))))
     (/ 1.0 (* s (* (* x (* s c)) (* x c)))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else if (x <= 1.65e+87) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (s * ((x * (s * c)) * (x * c)));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 2.7d+16) then
        tmp = 1.0d0 / (c * (c * ((x * s) * (x * s))))
    else if (x <= 1.65d+87) then
        tmp = (-2.0d0) / (c * (c * (s * s)))
    else
        tmp = 1.0d0 / (s * ((x * (s * c)) * (x * c)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else if (x <= 1.65e+87) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (s * ((x * (s * c)) * (x * c)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2.7e+16:
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))))
	elif x <= 1.65e+87:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = 1.0 / (s * ((x * (s * c)) * (x * c)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2.7e+16)
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(Float64(x * s) * Float64(x * s)))));
	elseif (x <= 1.65e+87)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(s * Float64(Float64(x * Float64(s * c)) * Float64(x * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.7e+16)
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	elseif (x <= 1.65e+87)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = 1.0 / (s * ((x * (s * c)) * (x * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2.7e+16], N[(1.0 / N[(c * N[(c * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+87], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s * N[(N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.7e16

   1. Initial program 63.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 65.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

       3. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(x \cdot s\right), \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(\color{blue}{x} \cdot s\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 74.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 74.5%

       \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\right)} \]

   if 2.7e16 < x < 1.6500000000000001e87

   1. Initial program 81.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. 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 70.2%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 70.2%

       \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]

   13. Taylor expanded in s around 0

       \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right) \]
   14. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot c\right) \cdot {\color{blue}{s}}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 69.9%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   15. Simplified 69.9%

       \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if 1.6500000000000001e87 < x

   1. Initial program 65.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 74.4%

      \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)\right)\right)\right) \]

      4. unswap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{s}\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{s}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)\right), \color{blue}{s}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(s \cdot c\right)\right), \left(x \cdot c\right)\right), s\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(s \cdot c\right)\right), \left(x \cdot c\right)\right), s\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot c\right)\right), s\right)\right) \]

      12. *-lowering-*.f64 76.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, c\right)\right), s\right)\right) \]

   9. Applied egg-rr 76.9%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)\right) \cdot s}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.3% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+87}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot s\right) \cdot \left(c \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2.7e+16)
   (/ 1.0 (* c (* c (* (* x s) (* x s)))))
   (if (<= x 1.35e+87)
     (/ -2.0 (* c (* c (* s s))))
     (/ 1.0 (* (* s s) (* c (* x (* x c))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else if (x <= 1.35e+87) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / ((s * s) * (c * (x * (x * c))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 2.7d+16) then
        tmp = 1.0d0 / (c * (c * ((x * s) * (x * s))))
    else if (x <= 1.35d+87) then
        tmp = (-2.0d0) / (c * (c * (s * s)))
    else
        tmp = 1.0d0 / ((s * s) * (c * (x * (x * c))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else if (x <= 1.35e+87) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / ((s * s) * (c * (x * (x * c))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2.7e+16:
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))))
	elif x <= 1.35e+87:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = 1.0 / ((s * s) * (c * (x * (x * c))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2.7e+16)
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(Float64(x * s) * Float64(x * s)))));
	elseif (x <= 1.35e+87)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(Float64(s * s) * Float64(c * Float64(x * Float64(x * c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.7e+16)
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	elseif (x <= 1.35e+87)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = 1.0 / ((s * s) * (c * (x * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2.7e+16], N[(1.0 / N[(c * N[(c * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+87], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(s * s), $MachinePrecision] * N[(c * N[(x * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.7e16

   1. Initial program 63.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 65.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

       3. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(x \cdot s\right), \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(\color{blue}{x} \cdot s\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 74.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 74.5%

       \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\right)} \]

   if 2.7e16 < x < 1.35000000000000003e87

   1. Initial program 81.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. 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 70.2%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 70.2%

       \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]

   13. Taylor expanded in s around 0

       \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right) \]
   14. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot c\right) \cdot {\color{blue}{s}}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 69.9%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   15. Simplified 69.9%

       \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if 1.35000000000000003e87 < x

   1. Initial program 65.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 74.4%

      \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot \left(c \cdot s\right)\right) \cdot s\right) \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot \left(c \cdot s\right)\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot x\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(c \cdot c\right) \cdot s\right) \cdot \left(\color{blue}{s} \cdot \left(x \cdot x\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right) \cdot c\right) \cdot \color{blue}{c}\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot c\right) \cdot c\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(s \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)\right) \cdot c\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot c\right) \cdot c\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(s \cdot s\right), \color{blue}{\left(\left(\left(x \cdot x\right) \cdot c\right) \cdot c\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \left(\color{blue}{\left(\left(x \cdot x\right) \cdot c\right)} \cdot c\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot c\right), \color{blue}{c}\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot c\right)\right), c\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot c\right)\right), c\right)\right)\right) \]

      17. *-lowering-*.f64 69.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, c\right)\right), c\right)\right)\right) \]

   9. Applied egg-rr 69.6%

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \left(\left(x \cdot \left(x \cdot c\right)\right) \cdot c\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+87}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot s\right) \cdot \left(c \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.3% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2.7e+16)
   (/ 1.0 (* c (* c (* (* x s) (* x s)))))
   (if (<= x 1.65e+87)
     (/ -2.0 (* c (* c (* s s))))
     (/ 1.0 (* x (* c (* s (* s (* x c)))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else if (x <= 1.65e+87) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (x * (c * (s * (s * (x * c)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 2.7d+16) then
        tmp = 1.0d0 / (c * (c * ((x * s) * (x * s))))
    else if (x <= 1.65d+87) then
        tmp = (-2.0d0) / (c * (c * (s * s)))
    else
        tmp = 1.0d0 / (x * (c * (s * (s * (x * c)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.7e+16) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else if (x <= 1.65e+87) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (x * (c * (s * (s * (x * c)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2.7e+16:
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))))
	elif x <= 1.65e+87:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = 1.0 / (x * (c * (s * (s * (x * c)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2.7e+16)
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(Float64(x * s) * Float64(x * s)))));
	elseif (x <= 1.65e+87)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(c * Float64(s * Float64(s * Float64(x * c))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.7e+16)
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	elseif (x <= 1.65e+87)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = 1.0 / (x * (c * (s * (s * (x * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2.7e+16], N[(1.0 / N[(c * N[(c * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+87], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(c * N[(s * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.7e16

   1. Initial program 63.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 81.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 65.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

       3. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(x \cdot s\right), \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(\color{blue}{x} \cdot s\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 74.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 74.5%

       \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\right)} \]

   if 2.7e16 < x < 1.6500000000000001e87

   1. Initial program 81.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 70.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. 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 70.2%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 70.2%

       \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]

   13. Taylor expanded in s around 0

       \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right) \]
   14. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot c\right) \cdot {\color{blue}{s}}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 69.9%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   15. Simplified 69.9%

       \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if 1.6500000000000001e87 < x

   1. Initial program 65.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 74.4%

      \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot x\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot x\right)\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot x\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot x\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(\left({s}^{2} \cdot x\right) \cdot \color{blue}{c}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(\left(\left(s \cdot s\right) \cdot x\right) \cdot c\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot c\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot c\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(s \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(\left(c \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot x\right)\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(s \cdot \color{blue}{\left(c \cdot x\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left(c \cdot x\right)}\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 76.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 76.9%

       \[\leadsto \frac{1}{x \cdot \color{blue}{\left(c \cdot \left(s \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.7% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{0 - 2}{s \cdot \left(c \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 9.2e-29)
   (/ (- 0.0 2.0) (* s (* c (* s c))))
   (/ 1.0 (* c (* c (* s (* s (* x x))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 9.2e-29) {
		tmp = (0.0 - 2.0) / (s * (c * (s * c)));
	} else {
		tmp = 1.0 / (c * (c * (s * (s * (x * x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 9.2d-29) then
        tmp = (0.0d0 - 2.0d0) / (s * (c * (s * c)))
    else
        tmp = 1.0d0 / (c * (c * (s * (s * (x * x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 9.2e-29) {
		tmp = (0.0 - 2.0) / (s * (c * (s * c)));
	} else {
		tmp = 1.0 / (c * (c * (s * (s * (x * x)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 9.2e-29:
		tmp = (0.0 - 2.0) / (s * (c * (s * c)))
	else:
		tmp = 1.0 / (c * (c * (s * (s * (x * x)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 9.2e-29)
		tmp = Float64(Float64(0.0 - 2.0) / Float64(s * Float64(c * Float64(s * c))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(s * Float64(s * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 9.2e-29)
		tmp = (0.0 - 2.0) / (s * (c * (s * c)));
	else
		tmp = 1.0 / (c * (c * (s * (s * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 9.2e-29], N[(N[(0.0 - 2.0), $MachinePrecision] / N[(s * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(c * N[(s * N[(s * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 9.19999999999999965e-29

   1. Initial program 62.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 48.6%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. 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 24.2%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 24.2%

       \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]
   13. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(-2\right)}{\color{blue}{\mathsf{neg}\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{neg}\left(\color{blue}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}\right)} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\mathsf{neg}\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(0 - \color{blue}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 40.7%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 40.7%

       \[\leadsto \color{blue}{\frac{2}{0 - s \cdot \left(c \cdot \left(s \cdot c\right)\right)}} \]

   if 9.19999999999999965e-29 < s

   1. Initial program 70.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 88.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      19. *-lowering-*.f64 98.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 76.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   9. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{0 - 2}{s \cdot \left(c \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.1% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;\frac{0 - 2}{s \cdot \left(c \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 0.78)
   (/ (- 0.0 2.0) (* s (* c (* s c))))
   (/ -2.0 (* c (* c (* s s))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.78) {
		tmp = (0.0 - 2.0) / (s * (c * (s * c)));
	} else {
		tmp = -2.0 / (c * (c * (s * s)));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.78d0) then
        tmp = (0.0d0 - 2.0d0) / (s * (c * (s * c)))
    else
        tmp = (-2.0d0) / (c * (c * (s * s)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.78) {
		tmp = (0.0 - 2.0) / (s * (c * (s * c)));
	} else {
		tmp = -2.0 / (c * (c * (s * s)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 0.78:
		tmp = (0.0 - 2.0) / (s * (c * (s * c)))
	else:
		tmp = -2.0 / (c * (c * (s * s)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 0.78)
		tmp = Float64(Float64(0.0 - 2.0) / Float64(s * Float64(c * Float64(s * c))));
	else
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.78)
		tmp = (0.0 - 2.0) / (s * (c * (s * c)));
	else
		tmp = -2.0 / (c * (c * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 0.78], N[(N[(0.0 - 2.0), $MachinePrecision] / N[(s * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 62.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 81.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 54.5%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. 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 23.2%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 23.2%

       \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]
   13. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(-2\right)}{\color{blue}{\mathsf{neg}\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{neg}\left(\color{blue}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}\right)} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\mathsf{neg}\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(0 - \color{blue}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 42.5%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 42.5%

       \[\leadsto \color{blue}{\frac{2}{0 - s \cdot \left(c \cdot \left(s \cdot c\right)\right)}} \]

   if 0.78000000000000003 < x

   1. Initial program 71.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      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. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 84.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

      10. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

   6. Applied egg-rr 84.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

   9. Simplified 30.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. 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 39.1%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 39.1%

       \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]

   13. Taylor expanded in s around 0

       \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right) \]
   14. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot c\right) \cdot {\color{blue}{s}}^{2}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 41.7%

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   15. Simplified 41.7%

       \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;\frac{0 - 2}{s \cdot \left(c \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 77.8% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \frac{x}{\frac{1}{s \cdot c}}} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ 1.0 (* (* x (* s c)) (/ x (/ 1.0 (* s c))))))

C

double code(double x, double c, double s) {
	return 1.0 / ((x * (s * c)) * (x / (1.0 / (s * c))));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((x * (s * c)) * (x / (1.0d0 / (s * c))))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((x * (s * c)) * (x / (1.0 / (s * c))));
}

Python

def code(x, c, s):
	return 1.0 / ((x * (s * c)) * (x / (1.0 / (s * c))))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(x * Float64(s * c)) * Float64(x / Float64(1.0 / Float64(s * c)))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((x * (s * c)) * (x / (1.0 / (s * c))));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(x / N[(1.0 / N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   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. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 82.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   19. *-lowering-*.f64 97.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 64.6%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

9. Simplified 64.6%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{c} \cdot c\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot x\right) \cdot \left(s \cdot s\right)\right) \cdot \left(\color{blue}{c} \cdot c\right)\right)\right) \]

    5. swap-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right) \cdot \left(\color{blue}{c} \cdot c\right)\right)\right) \]

    6. swap-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(x \cdot s\right) \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot c\right)}\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot c\right)\right)\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right) \]

    9. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)}\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(\color{blue}{x} \cdot \left(s \cdot c\right)\right)\right)\right)\right) \]

    11. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)}\right)\right)\right) \]

    12. /-rgt-identity N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x}{1} \cdot \left(\color{blue}{c} \cdot \left(s \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)\right)\right)\right) \]

    13. associate-/r/ N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x}{\color{blue}{\frac{1}{c \cdot \left(s \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)}}}\right)\right) \]

    14. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x}{\frac{1}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}}}\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x}{\frac{1}{\left(s \cdot c\right) \cdot \left(\color{blue}{x} \cdot \left(s \cdot c\right)\right)}}\right)\right) \]

    16. associate-/r* N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x}{\frac{\frac{1}{s \cdot c}}{\color{blue}{x \cdot \left(s \cdot c\right)}}}\right)\right) \]

    17. inv-pow N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x}{\frac{{\left(s \cdot c\right)}^{-1}}{\color{blue}{x} \cdot \left(s \cdot c\right)}}\right)\right) \]

    18. associate-/r/ N/A

        \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x}{{\left(s \cdot c\right)}^{-1}} \cdot \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}\right)\right) \]

    19. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{x}{{\left(s \cdot c\right)}^{-1}}\right), \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}\right)\right) \]

11. Applied egg-rr 78.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{1}{s \cdot c}} \cdot \left(x \cdot \left(s \cdot c\right)\right)}} \]

12. Final simplification 78.2%

    \[\leadsto \frac{1}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \frac{x}{\frac{1}{s \cdot c}}} \]
13. Add Preprocessing

Alternative 18: 77.8% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \frac{s \cdot c}{\frac{1}{x}}} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ 1.0 (* (* x (* s c)) (/ (* s c) (/ 1.0 x)))))

C

double code(double x, double c, double s) {
	return 1.0 / ((x * (s * c)) * ((s * c) / (1.0 / 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 = 1.0d0 / ((x * (s * c)) * ((s * c) / (1.0d0 / x)))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((x * (s * c)) * ((s * c) / (1.0 / x)));
}

Python

def code(x, c, s):
	return 1.0 / ((x * (s * c)) * ((s * c) / (1.0 / x)))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(x * Float64(s * c)) * Float64(Float64(s * c) / Float64(1.0 / x))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((x * (s * c)) * ((s * c) / (1.0 / x)));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(N[(s * c), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   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. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 82.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 69.1%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. remove-double-div N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}}}\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\frac{1}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}{\color{blue}{\frac{1}{x}}}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}{\frac{1}{x}}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}{\frac{1}{x}}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)}{\frac{\color{blue}{1}}{x}}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\frac{s \cdot c}{\frac{1}{x}}}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot \left(s \cdot c\right)\right), \color{blue}{\left(\frac{s \cdot c}{\frac{1}{x}}\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(s \cdot c\right)\right), \left(\frac{\color{blue}{s \cdot c}}{\frac{1}{x}}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \left(\frac{s \cdot \color{blue}{c}}{\frac{1}{x}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \mathsf{/.f64}\left(\left(s \cdot c\right), \color{blue}{\left(\frac{1}{x}\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(s, c\right), \left(\frac{\color{blue}{1}}{x}\right)\right)\right)\right) \]

   13. /-lowering-/.f64 78.2%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(s, c\right), \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right)\right)\right) \]

9. Applied egg-rr 78.2%

   \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \frac{s \cdot c}{\frac{1}{x}}}} \]
10. Add Preprocessing

Alternative 19: 76.9% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(s \cdot c\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ (/ 1.0 (* s (* x c))) (* x (* s c))))

C

double code(double x, double c, double s) {
	return (1.0 / (s * (x * c))) / (x * (s * c));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = (1.0d0 / (s * (x * c))) / (x * (s * c))
end function

Java

public static double code(double x, double c, double s) {
	return (1.0 / (s * (x * c))) / (x * (s * c));
}

Python

def code(x, c, s):
	return (1.0 / (s * (x * c))) / (x * (s * c))

Julia

function code(x, c, s)
	return Float64(Float64(1.0 / Float64(s * Float64(x * c))) / Float64(x * Float64(s * c)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = (1.0 / (s * (x * c))) / (x * (s * c));
end

Wolfram

code[x_, c_, s_] := N[(N[(1.0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   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. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 82.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   19. *-lowering-*.f64 97.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(c \cdot s\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot c\right) \cdot x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   6. *-lowering-*.f64 77.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

9. Simplified 77.5%

   \[\leadsto \frac{\color{blue}{\frac{1}{s \cdot \left(c \cdot x\right)}}}{x \cdot \left(s \cdot c\right)} \]

10. Final simplification 77.5%

    \[\leadsto \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(s \cdot c\right)} \]
11. Add Preprocessing

Alternative 20: 72.3% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* c (* c (* (* x s) (* x s))))))

C

double code(double x, double c, double s) {
	return 1.0 / (c * (c * ((x * s) * (x * s))));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / (c * (c * ((x * s) * (x * s))))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (c * (c * ((x * s) * (x * s))));
}

Python

def code(x, c, s):
	return 1.0 / (c * (c * ((x * s) * (x * s))))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(c * Float64(c * Float64(Float64(x * s) * Float64(x * s)))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(c * N[(c * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   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. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 82.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\color{blue}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x}}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{c \cdot s}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x}\right), \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(c \cdot s\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \left(s \cdot c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   19. *-lowering-*.f64 97.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), x\right), \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x}}{s \cdot c}}{x \cdot \left(s \cdot c\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 64.6%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

9. Simplified 64.6%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

    3. swap-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(x \cdot s\right), \color{blue}{\left(x \cdot s\right)}\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(\color{blue}{x} \cdot s\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f64 72.0%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(x, \color{blue}{s}\right)\right)\right)\right)\right) \]

11. Applied egg-rr 72.0%

    \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\right)} \]
12. Add Preprocessing

Alternative 21: 28.8% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ -2.0 (* c (* c (* s s)))))

C

double code(double x, double c, double s) {
	return -2.0 / (c * (c * (s * s)));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = (-2.0d0) / (c * (c * (s * s)))
end function

Java

public static double code(double x, double c, double s) {
	return -2.0 / (c * (c * (s * s)));
}

Python

def code(x, c, s):
	return -2.0 / (c * (c * (s * s)))

Julia

function code(x, c, s)
	return Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = -2.0 / (c * (c * (s * s)));
end

Wolfram

code[x_, c_, s_] := N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   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. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 82.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}}{\color{blue}{x}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}}{x} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}}{x} \]

   4. swap-sqr N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(s \cdot s\right) \cdot \left(c \cdot c\right)\right)}}{x} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot \left(c \cdot c\right)\right)}}{x} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}{x} \]

   7. associate-*r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot {s}^{2}\right) \cdot {c}^{2}}}{x} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}{x} \]

   9. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x} \]

   10. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{{s}^{2}}}{x} \]

   11. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\color{blue}{{s}^{2} \cdot x}} \]

   12. unpow2 N/A

       \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{\left(s \cdot s\right) \cdot x} \]

   13. associate-*l* N/A

       \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \color{blue}{\left(s \cdot x\right)}} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s \cdot \left(x \cdot \color{blue}{s}\right)} \]

   15. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}}{\color{blue}{x \cdot s}} \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x}}{s}\right), \color{blue}{\left(x \cdot s\right)}\right) \]

6. Applied egg-rr 77.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot c\right)}}{s}}{x \cdot s}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}}{x}\right)}, \mathsf{*.f64}\left(x, s\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot s} + \frac{1}{{c}^{2} \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{x}, s\right)\right) \]

9. Simplified 48.2%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{\left(c \cdot c\right) \cdot s}}{x}}}{x \cdot s} \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
11. 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 27.4%

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right) \]

12. Simplified 27.4%

    \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]

13. Taylor expanded in s around 0

    \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right) \]
14. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot c\right) \cdot {\color{blue}{s}}^{2}\right)\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot {s}^{2}\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

    6. *-lowering-*.f64 31.1%

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

15. Simplified 31.1%

    \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))