mixedcos

Percentage Accurate: 66.8% → 97.4%

Time: 12.2s

Alternatives: 13

Speedup: 24.1×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% 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 79.9%

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

3. Simplified 79.9%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
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 97.7%

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

6. Applied egg-rr 97.7%

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

Alternative 2: 82.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 3e-46)
   (/ (/ (/ 1.0 (* c (* x s))) (* 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 <= 3e-46) {
		tmp = ((1.0 / (c * (x * s))) / (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 <= 3d-46) then
        tmp = ((1.0d0 / (c * (x * s))) / (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 <= 3e-46) {
		tmp = ((1.0 / (c * (x * s))) / (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 <= 3e-46:
		tmp = ((1.0 / (c * (x * s))) / (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 <= 3e-46)
		tmp = Float64(Float64(Float64(1.0 / Float64(c * Float64(x * s))) / Float64(x * 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 <= 3e-46)
		tmp = ((1.0 / (c * (x * s))) / (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, 3e-46], N[(N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $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 < 2.99999999999999987e-46

   1. Initial program 63.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. 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 79.4%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   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 96.8%

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

   6. Applied egg-rr 96.8%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

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

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

      21. unpow2 N/A

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

      22. *-lowering-*.f64 66.4%

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

   9. Simplified 66.4%

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

       1. inv-pow N/A

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

       2. *-commutative N/A

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

       3. unpow-prod-down N/A

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

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{1}{s \cdot c}}{x}}{x}}{s}}{c}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

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

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

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

       3. associate-/l/ N/A

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

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

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

       5. associate-*r* N/A

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

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

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

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

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

       8. *-lowering-*.f64 80.9%

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

   13. Applied egg-rr 80.9%

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

   if 2.99999999999999987e-46 < x

   1. Initial program 69.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 81.0%

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

   3. Simplified 81.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \frac{\cos \left(2 \cdot x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

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(cos(Float64(2.0 * x)) / Float64(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[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $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 79.9%

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

3. Simplified 79.9%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unswap-sqr N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 97.5%

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

6. Applied egg-rr 97.5%

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

Alternative 4: 73.7% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 5.2e-87)
   (/ 1.0 (* x (* x (* (* s c) (* s c)))))
   (/ 1.0 (* c (* s (* x (* s (* x c))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 5.2e-87) {
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))));
	} else {
		tmp = 1.0 / (c * (s * (x * (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 (c <= 5.2d-87) then
        tmp = 1.0d0 / (x * (x * ((s * c) * (s * c))))
    else
        tmp = 1.0d0 / (c * (s * (x * (s * (x * c)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 5.2e-87) {
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))));
	} else {
		tmp = 1.0 / (c * (s * (x * (s * (x * c)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 5.2e-87:
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))))
	else:
		tmp = 1.0 / (c * (s * (x * (s * (x * c)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 5.2e-87)
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(Float64(s * c) * Float64(s * c)))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(s * Float64(x * Float64(s * Float64(x * c))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 5.2e-87)
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))));
	else
		tmp = 1.0 / (c * (s * (x * (s * (x * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 5.2e-87], N[(1.0 / N[(x * N[(x * N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(s * N[(x * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.20000000000000005e-87

   1. Initial program 60.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 79.4%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   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 69.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 71.7%

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

   9. Applied egg-rr 71.7%

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

   if 5.20000000000000005e-87 < c

   1. Initial program 73.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 81.0%

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

   3. Simplified 81.0%

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

   5. 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 71.4%

      \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]
   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. swap-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. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 80.7%

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

   9. Applied egg-rr 80.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 7 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 7e-67)
   (/ 1.0 (* x (* x (* (* s c) (* s c)))))
   (/ 1.0 (* s (* x (* c (* c (* x s))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 7e-67) {
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))));
	} else {
		tmp = 1.0 / (s * (x * (c * (c * (x * 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 (c <= 7d-67) then
        tmp = 1.0d0 / (x * (x * ((s * c) * (s * c))))
    else
        tmp = 1.0d0 / (s * (x * (c * (c * (x * s)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 7e-67) {
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))));
	} else {
		tmp = 1.0 / (s * (x * (c * (c * (x * s)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 7e-67:
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))))
	else:
		tmp = 1.0 / (s * (x * (c * (c * (x * s)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 7e-67)
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(Float64(s * c) * Float64(s * c)))));
	else
		tmp = Float64(1.0 / Float64(s * Float64(x * Float64(c * Float64(c * Float64(x * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 7e-67)
		tmp = 1.0 / (x * (x * ((s * c) * (s * c))));
	else
		tmp = 1.0 / (s * (x * (c * (c * (x * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 7e-67], N[(1.0 / N[(x * N[(x * N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s * N[(x * N[(c * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 7.0000000000000001e-67

   1. Initial program 61.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 79.1%

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

   3. Simplified 79.1%

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

   5. 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 69.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 71.7%

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

   9. Applied egg-rr 71.7%

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

   if 7.0000000000000001e-67 < c

   1. Initial program 74.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 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 97.3%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

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

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

      21. unpow2 N/A

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

      22. *-lowering-*.f64 68.7%

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

   9. Simplified 68.7%

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

   10. Taylor expanded in c around 0

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

       10. *-lowering-*.f64 78.9%

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

   12. Simplified 78.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 7 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.4% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 1.3e-66)
   (/ 1.0 (* x (* x (* s (* c (* s c))))))
   (/ 1.0 (* s (* x (* c (* c (* x s))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 1.3e-66) {
		tmp = 1.0 / (x * (x * (s * (c * (s * c)))));
	} else {
		tmp = 1.0 / (s * (x * (c * (c * (x * 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 (c <= 1.3d-66) then
        tmp = 1.0d0 / (x * (x * (s * (c * (s * c)))))
    else
        tmp = 1.0d0 / (s * (x * (c * (c * (x * s)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 1.3e-66) {
		tmp = 1.0 / (x * (x * (s * (c * (s * c)))));
	} else {
		tmp = 1.0 / (s * (x * (c * (c * (x * s)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 1.3e-66:
		tmp = 1.0 / (x * (x * (s * (c * (s * c)))))
	else:
		tmp = 1.0 / (s * (x * (c * (c * (x * s)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 1.3e-66)
		tmp = Float64(1.0 / Float64(x * Float64(x * Float64(s * Float64(c * Float64(s * c))))));
	else
		tmp = Float64(1.0 / Float64(s * Float64(x * Float64(c * Float64(c * Float64(x * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 1.3e-66)
		tmp = 1.0 / (x * (x * (s * (c * (s * c)))));
	else
		tmp = 1.0 / (s * (x * (c * (c * (x * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 1.3e-66], N[(1.0 / N[(x * N[(x * N[(s * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s * N[(x * N[(c * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.2999999999999999e-66

   1. Initial program 61.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 79.1%

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

   3. Simplified 79.1%

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

   5. 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 69.6%

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

   if 1.2999999999999999e-66 < c

   1. Initial program 74.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 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 97.3%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

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

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

      21. unpow2 N/A

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

      22. *-lowering-*.f64 68.7%

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

   9. Simplified 68.7%

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

   10. Taylor expanded in c around 0

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

       10. *-lowering-*.f64 78.9%

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

   12. Simplified 78.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{s \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 8.5e+69)
   (/ 1.0 (* s (* c (* s (* c (* x x))))))
   (/ -2.0 (* (* s c) (* s c)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 8.5e+69) {
		tmp = 1.0 / (s * (c * (s * (c * (x * x)))));
	} else {
		tmp = -2.0 / ((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 (c <= 8.5d+69) then
        tmp = 1.0d0 / (s * (c * (s * (c * (x * x)))))
    else
        tmp = (-2.0d0) / ((s * c) * (s * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 8.5e+69) {
		tmp = 1.0 / (s * (c * (s * (c * (x * x)))));
	} else {
		tmp = -2.0 / ((s * c) * (s * c));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 8.5e+69:
		tmp = 1.0 / (s * (c * (s * (c * (x * x)))))
	else:
		tmp = -2.0 / ((s * c) * (s * c))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 8.5e+69)
		tmp = Float64(1.0 / Float64(s * Float64(c * Float64(s * Float64(c * Float64(x * x))))));
	else
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 8.5e+69)
		tmp = 1.0 / (s * (c * (s * (c * (x * x)))));
	else
		tmp = -2.0 / ((s * c) * (s * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 8.5e+69], N[(1.0 / N[(s * N[(c * N[(s * N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 8.5000000000000002e69

   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 78.2%

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

   3. Simplified 78.2%

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

      1. associate-/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 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

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

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

      21. unpow2 N/A

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

      22. *-lowering-*.f64 62.5%

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

   9. Simplified 62.5%

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

   if 8.5000000000000002e69 < c

   1. Initial program 74.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 86.9%

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

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   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 96.3%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 59.7%

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

   9. Simplified 59.7%

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

   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

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 50.4%

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

   12. Simplified 50.4%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. swap-sqr N/A

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

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

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

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

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

       6. *-lowering-*.f64 50.5%

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

   14. Applied egg-rr 50.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{s \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.6% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{2}{c}}{s \cdot \left(0 - s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 3e-36)
   (/ (/ 2.0 c) (* s (- 0.0 (* s c))))
   (/ -2.0 (* c (* c (* s s))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 3e-36) {
		tmp = (2.0 / c) / (s * (0.0 - (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 (s <= 3d-36) then
        tmp = (2.0d0 / c) / (s * (0.0d0 - (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 (s <= 3e-36) {
		tmp = (2.0 / c) / (s * (0.0 - (s * c)));
	} else {
		tmp = -2.0 / (c * (c * (s * s)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 3e-36:
		tmp = (2.0 / c) / (s * (0.0 - (s * c)))
	else:
		tmp = -2.0 / (c * (c * (s * s)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 3e-36)
		tmp = Float64(Float64(2.0 / c) / Float64(s * Float64(0.0 - 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 (s <= 3e-36)
		tmp = (2.0 / c) / (s * (0.0 - (s * c)));
	else
		tmp = -2.0 / (c * (c * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 3e-36], N[(N[(2.0 / c), $MachinePrecision] / N[(s * N[(0.0 - 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 s < 3.0000000000000002e-36

   1. Initial program 65.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 82.0%

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

   3. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   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 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 54.1%

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

   9. Simplified 54.1%

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

   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

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 18.0%

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

   12. Simplified 18.0%

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

       1. associate-/r* N/A

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

       2. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-2}{c}\right)}{\color{blue}{\mathsf{neg}\left(c \cdot \left(s \cdot s\right)\right)}} \]

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

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

       4. distribute-neg-frac N/A

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

       5. metadata-eval N/A

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

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

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

       7. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, c\right), \left(0 - \color{blue}{c \cdot \left(s \cdot s\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, c\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(c \cdot \left(s \cdot s\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, c\right), \mathsf{\_.f64}\left(0, \left(\left(s \cdot s\right) \cdot \color{blue}{c}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, c\right), \mathsf{\_.f64}\left(0, \left(s \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, c\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot c\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 25.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, c\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right)\right) \]

   14. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{\frac{\frac{2}{c}}{0 - s \cdot \left(s \cdot c\right)}} \]

   if 3.0000000000000002e-36 < s

   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 73.6%

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

   3. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   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 94.7%

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

   6. Applied egg-rr 94.7%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 59.5%

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

   9. Simplified 59.5%

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

   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

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 52.4%

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

   12. Simplified 52.4%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{2}{c}}{s \cdot \left(0 - s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.0% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\frac{-2}{\left(c \cdot c\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= s 1.95e+142)
   (/ -2.0 (* (* c c) (* s s)))
   (/ -2.0 (* c (* c (* s s))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (s <= 1.95e+142) {
		tmp = -2.0 / ((c * c) * (s * s));
	} 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 (s <= 1.95d+142) then
        tmp = (-2.0d0) / ((c * c) * (s * s))
    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 (s <= 1.95e+142) {
		tmp = -2.0 / ((c * c) * (s * s));
	} else {
		tmp = -2.0 / (c * (c * (s * s)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if s <= 1.95e+142:
		tmp = -2.0 / ((c * c) * (s * s))
	else:
		tmp = -2.0 / (c * (c * (s * s)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (s <= 1.95e+142)
		tmp = Float64(-2.0 / Float64(Float64(c * c) * Float64(s * s)));
	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 (s <= 1.95e+142)
		tmp = -2.0 / ((c * c) * (s * s));
	else
		tmp = -2.0 / (c * (c * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[s, 1.95e+142], N[(-2.0 / N[(N[(c * c), $MachinePrecision] * N[(s * s), $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 s < 1.95e142

   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 80.2%

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

   3. Simplified 80.2%

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

      1. associate-/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 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 53.5%

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

   9. Simplified 53.5%

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

   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

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 19.8%

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

   12. Simplified 19.8%

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

       1. remove-double-neg N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft-neg-in N/A

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

       4. distribute-lft-neg-in N/A

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

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

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

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

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

       7. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(0 - c \cdot c\right)\right), \left(s \cdot s\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot c\right)\right)\right), \left(s \cdot s\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(s \cdot s\right)\right)\right) \]

       10. *-lowering-*.f64 27.0%

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right) \]

   14. Applied egg-rr 27.0%

       \[\leadsto \frac{-2}{\color{blue}{\left(-\left(0 - c \cdot c\right)\right) \cdot \left(s \cdot s\right)}} \]

   if 1.95e142 < s

   1. Initial program 61.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 78.1%

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

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   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 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 65.9%

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

   9. Simplified 65.9%

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

   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

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

       3. associate-*l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 63.7%

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

   12. Simplified 63.7%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\frac{-2}{\left(c \cdot c\right) \cdot \left(s \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.4% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x c s) :precision binary64 (/ (/ (/ 1.0 (* c (* x s))) (* x s)) c))

C

double code(double x, double c, double s) {
	return ((1.0 / (c * (x * s))) / (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 / (c * (x * s))) / (x * s)) / c
end function

Java

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

Python

def code(x, c, s):
	return ((1.0 / (c * (x * s))) / (x * s)) / c

Julia

function code(x, c, s)
	return Float64(Float64(Float64(1.0 / Float64(c * Float64(x * s))) / Float64(x * s)) / c)
end

MATLAB

function tmp = code(x, c, s)
	tmp = ((1.0 / (c * (x * s))) / (x * s)) / c;
end

Wolfram

code[x_, c_, s_] := N[(N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $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 79.9%

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

3. Simplified 79.9%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
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 97.7%

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

6. Applied egg-rr 97.7%

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

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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

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

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

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

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

   21. unpow2 N/A

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

   22. *-lowering-*.f64 64.9%

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

9. Simplified 64.9%

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

    1. inv-pow N/A

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

    2. *-commutative N/A

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

    3. unpow-prod-down N/A

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

11. Applied egg-rr 75.6%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{1}{s \cdot c}}{x}}{x}}{s}}{c}} \]
12. Step-by-step derivation

    1. associate-/r* N/A

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

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

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

    3. associate-/l/ N/A

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

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

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

    5. associate-*r* N/A

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

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

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

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

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

    8. *-lowering-*.f64 77.2%

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

13. Applied egg-rr 77.2%

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

14. Final simplification 77.2%

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

Alternative 11: 78.1% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c)))) (/ 1.0 (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	return 1.0 / (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 = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $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 79.9%

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

3. Simplified 79.9%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unswap-sqr N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 97.5%

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

6. Applied egg-rr 97.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 78.5%

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

Alternative 12: 73.9% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* s (* x (* c (* c (* x s)))))))

C

double code(double x, double c, double s) {
	return 1.0 / (s * (x * (c * (c * (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 / (s * (x * (c * (c * (x * s)))))
end function

Java

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

Python

def code(x, c, s):
	return 1.0 / (s * (x * (c * (c * (x * s)))))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(s * Float64(x * Float64(c * Float64(c * Float64(x * s))))))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (s * (x * (c * (c * (x * s)))));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(s * N[(x * N[(c * N[(c * N[(x * s), $MachinePrecision]), $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 79.9%

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

3. Simplified 79.9%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
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 97.7%

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

6. Applied egg-rr 97.7%

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

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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

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

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

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

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

   21. unpow2 N/A

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

   22. *-lowering-*.f64 64.9%

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

9. Simplified 64.9%

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

10. Taylor expanded in c around 0

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

    1. unpow2 N/A

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

    2. associate-*r* N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

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

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

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

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

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

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

    10. *-lowering-*.f64 73.7%

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

12. Simplified 73.7%

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

13. Final simplification 73.7%

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

Alternative 13: 28.7% 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 79.9%

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

3. Simplified 79.9%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
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 97.7%

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

6. Applied egg-rr 97.7%

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

7. Taylor expanded in x around 0

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

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 55.4%

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

9. Simplified 55.4%

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

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

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

    3. associate-*l* N/A

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

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

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

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

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

    6. unpow2 N/A

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

    7. *-lowering-*.f64 26.6%

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

12. Simplified 26.6%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))