mixedcos

Percentage Accurate: 66.7% → 99.1%

Time: 15.8s

Alternatives: 7

Speedup: 24.1×

• 31.5% 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: 66.7% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ \begin{array}{l} t_0 := \frac{\frac{{c\_m}^{-0.5}}{x}}{s}\\ t\_0 \cdot \left(t\_0 \cdot \frac{\cos \left(x \cdot 2\right)}{c\_m}\right) \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (/ (/ (pow c_m -0.5) x) s)))
   (* t_0 (* t_0 (/ (cos (* x 2.0)) c_m)))))

C

c_m = fabs(c);
double code(double x, double c_m, double s) {
	double t_0 = (pow(c_m, -0.5) / x) / s;
	return t_0 * (t_0 * (cos((x * 2.0)) / c_m));
}

Fortran

c_m = abs(c)
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = ((c_m ** (-0.5d0)) / x) / s
    code = t_0 * (t_0 * (cos((x * 2.0d0)) / c_m))
end function

Java

c_m = Math.abs(c);
public static double code(double x, double c_m, double s) {
	double t_0 = (Math.pow(c_m, -0.5) / x) / s;
	return t_0 * (t_0 * (Math.cos((x * 2.0)) / c_m));
}

Python

c_m = math.fabs(c)
def code(x, c_m, s):
	t_0 = (math.pow(c_m, -0.5) / x) / s
	return t_0 * (t_0 * (math.cos((x * 2.0)) / c_m))

Julia

c_m = abs(c)
function code(x, c_m, s)
	t_0 = Float64(Float64((c_m ^ -0.5) / x) / s)
	return Float64(t_0 * Float64(t_0 * Float64(cos(Float64(x * 2.0)) / c_m)))
end

MATLAB

c_m = abs(c);
function tmp = code(x, c_m, s)
	t_0 = ((c_m ^ -0.5) / x) / s;
	tmp = t_0 * (t_0 * (cos((x * 2.0)) / c_m));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(N[(N[Power[c$95$m, -0.5], $MachinePrecision] / x), $MachinePrecision] / s), $MachinePrecision]}, N[(t$95$0 * N[(t$95$0 * N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.4%

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

   1. clear-num 97.4%

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

   2. un-div-inv 97.4%

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

   3. associate-/r* 97.5%

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

   4. *-commutative 97.5%

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

   5. *-un-lft-identity 97.5%

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

   6. times-frac 97.5%

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

   7. /-rgt-identity 97.5%

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

   8. *-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

   1. associate-/l/ 94.2%

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

   2. add-sqr-sqrt 43.2%

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

   3. times-frac 45.1%

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

   4. inv-pow 45.1%

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

   5. sqrt-pow1 45.2%

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

   6. metadata-eval 45.2%

      \[\leadsto \frac{{c}^{\color{blue}{-0.5}}}{\left(x \cdot s\right) \cdot \frac{c}{\cos \left(x \cdot 2\right)}} \cdot \frac{\sqrt{\frac{1}{c}}}{x \cdot s} \]

   7. clear-num 45.1%

      \[\leadsto \frac{{c}^{-0.5}}{\left(x \cdot s\right) \cdot \color{blue}{\frac{1}{\frac{\cos \left(x \cdot 2\right)}{c}}}} \cdot \frac{\sqrt{\frac{1}{c}}}{x \cdot s} \]

   8. un-div-inv 45.1%

      \[\leadsto \frac{{c}^{-0.5}}{\color{blue}{\frac{x \cdot s}{\frac{\cos \left(x \cdot 2\right)}{c}}}} \cdot \frac{\sqrt{\frac{1}{c}}}{x \cdot s} \]

   9. inv-pow 45.1%

      \[\leadsto \frac{{c}^{-0.5}}{\frac{x \cdot s}{\frac{\cos \left(x \cdot 2\right)}{c}}} \cdot \frac{\sqrt{\color{blue}{{c}^{-1}}}}{x \cdot s} \]

   10. sqrt-pow1 45.1%

       \[\leadsto \frac{{c}^{-0.5}}{\frac{x \cdot s}{\frac{\cos \left(x \cdot 2\right)}{c}}} \cdot \frac{\color{blue}{{c}^{\left(\frac{-1}{2}\right)}}}{x \cdot s} \]

   11. metadata-eval 45.1%

       \[\leadsto \frac{{c}^{-0.5}}{\frac{x \cdot s}{\frac{\cos \left(x \cdot 2\right)}{c}}} \cdot \frac{{c}^{\color{blue}{-0.5}}}{x \cdot s} \]

8. Applied egg-rr 45.1%

   \[\leadsto \color{blue}{\frac{{c}^{-0.5}}{\frac{x \cdot s}{\frac{\cos \left(x \cdot 2\right)}{c}}} \cdot \frac{{c}^{-0.5}}{x \cdot s}} \]
9. Step-by-step derivation

   1. *-commutative 45.1%

      \[\leadsto \color{blue}{\frac{{c}^{-0.5}}{x \cdot s} \cdot \frac{{c}^{-0.5}}{\frac{x \cdot s}{\frac{\cos \left(x \cdot 2\right)}{c}}}} \]

   2. associate-/r* 45.2%

      \[\leadsto \color{blue}{\frac{\frac{{c}^{-0.5}}{x}}{s}} \cdot \frac{{c}^{-0.5}}{\frac{x \cdot s}{\frac{\cos \left(x \cdot 2\right)}{c}}} \]

   3. associate-/r/ 45.2%

      \[\leadsto \frac{\frac{{c}^{-0.5}}{x}}{s} \cdot \color{blue}{\left(\frac{{c}^{-0.5}}{x \cdot s} \cdot \frac{\cos \left(x \cdot 2\right)}{c}\right)} \]

   4. associate-/r* 46.2%

      \[\leadsto \frac{\frac{{c}^{-0.5}}{x}}{s} \cdot \left(\color{blue}{\frac{\frac{{c}^{-0.5}}{x}}{s}} \cdot \frac{\cos \left(x \cdot 2\right)}{c}\right) \]

10. Simplified 46.2%

    \[\leadsto \color{blue}{\frac{\frac{{c}^{-0.5}}{x}}{s} \cdot \left(\frac{\frac{{c}^{-0.5}}{x}}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{c}\right)} \]

11. Final simplification 46.2%

    \[\leadsto \frac{\frac{{c}^{-0.5}}{x}}{s} \cdot \left(\frac{\frac{{c}^{-0.5}}{x}}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{c}\right) \]
12. Add Preprocessing

Alternative 2: 97.1% accurate, 2.7× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
(FPCore (x c_m s)
 :precision binary64
 (* (/ 1.0 (* c_m (* x s))) (/ (/ (/ (cos (* x 2.0)) s) x) c_m)))

C

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

Fortran

c_m = abs(c)
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = (1.0d0 / (c_m * (x * s))) * (((cos((x * 2.0d0)) / s) / x) / c_m)
end function

Java

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

Python

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

Julia

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

MATLAB

c_m = abs(c);
function tmp = code(x, c_m, s)
	tmp = (1.0 / (c_m * (x * s))) * (((cos((x * 2.0)) / s) / x) / c_m);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
code[x_, c$95$m_, s_] := N[(N[(1.0 / N[(c$95$m * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s), $MachinePrecision] / x), $MachinePrecision] / c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.4%

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

5. Taylor expanded in x around inf 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*r* 94.8%

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

   3. associate-/l/ 94.8%

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

   4. associate-/l/ 97.5%

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

   5. associate-/r* 97.4%

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

   6. associate-/l/ 97.6%

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

7. Simplified 97.6%

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

8. Final simplification 97.6%

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

Alternative 3: 97.3% accurate, 2.7× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
(FPCore (x c_m s)
 :precision binary64
 (/ (/ (/ 1.0 c_m) (* x s)) (* (* x s) (/ c_m (cos (* x 2.0))))))

C

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

Fortran

c_m = abs(c)
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = ((1.0d0 / c_m) / (x * s)) / ((x * s) * (c_m / cos((x * 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

c_m = abs(c);
function tmp = code(x, c_m, s)
	tmp = ((1.0 / c_m) / (x * s)) / ((x * s) * (c_m / cos((x * 2.0))));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
code[x_, c$95$m_, s_] := N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(c$95$m / N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.4%

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

   1. clear-num 97.4%

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

   2. un-div-inv 97.4%

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

   3. associate-/r* 97.5%

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

   4. *-commutative 97.5%

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

   5. *-un-lft-identity 97.5%

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

   6. times-frac 97.5%

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

   7. /-rgt-identity 97.5%

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

   8. *-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

7. Final simplification 97.5%

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

Alternative 4: 94.0% accurate, 2.7× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
(FPCore (x c_m s)
 :precision binary64
 (/ (/ (cos (* x 2.0)) c_m) (* (* x s) (* c_m (* x s)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
code[x_, c$95$m_, s_] := N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(c$95$m * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-/r* 97.5%

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

   3. frac-times 94.3%

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

   4. div-inv 94.2%

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

   5. *-commutative 94.2%

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

6. Applied egg-rr 94.2%

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

7. Final simplification 94.2%

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

Alternative 5: 97.2% accurate, 2.7× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* c_m (* x s)))) (/ (/ (cos (* x 2.0)) t_0) t_0)))

C

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

Fortran

c_m = abs(c)
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c_m * (x * s)
    code = (cos((x * 2.0d0)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

c_m = abs(c);
function tmp = code(x, c_m, s)
	t_0 = c_m * (x * s);
	tmp = (cos((x * 2.0)) / t_0) / t_0;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(c$95$m * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. div-inv 97.4%

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

   4. div-inv 97.4%

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

   5. *-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

7. Final simplification 97.4%

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

Alternative 6: 78.6% accurate, 24.1× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
(FPCore (x c_m s)
 :precision binary64
 (let* ((t_0 (* c_m (* x s)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

c_m = abs(c)
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c_m * (x * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
code[x_, c$95$m_, s_] := Block[{t$95$0 = N[(c$95$m * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0 56.5%

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

   1. associate-/r* 56.5%

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

   2. *-commutative 56.5%

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

   3. unpow2 56.5%

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

   4. unpow2 56.5%

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

   5. swap-sqr 68.0%

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

   6. unpow2 68.0%

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

   7. associate-/r* 68.0%

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

   8. unpow2 68.0%

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

   9. unpow2 68.0%

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

   10. swap-sqr 82.9%

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

   11. unpow2 82.9%

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

   12. *-commutative 82.9%

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

5. Simplified 82.9%

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

   1. *-commutative 82.9%

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

   2. pow2 82.9%

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

7. Applied egg-rr 82.9%

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

8. Final simplification 82.9%

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

Alternative 7: 78.7% accurate, 24.1× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
(FPCore (x c_m s)
 :precision binary64
 (/ (/ (/ 1.0 c_m) (* x s)) (* c_m (* x s))))

C

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

Fortran

c_m = abs(c)
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = ((1.0d0 / c_m) / (x * s)) / (c_m * (x * s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
code[x_, c$95$m_, s_] := N[(N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.4%

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

   1. clear-num 97.4%

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

   2. un-div-inv 97.4%

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

   3. associate-/r* 97.5%

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

   4. *-commutative 97.5%

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

   5. *-un-lft-identity 97.5%

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

   6. times-frac 97.5%

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

   7. /-rgt-identity 97.5%

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

   8. *-commutative 97.5%

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

6. Applied egg-rr 97.5%

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

7. Taylor expanded in x around 0 82.9%

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

8. Final simplification 82.9%

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

Reproduce

herbie shell --seed 2024055 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))