mixedcos

Percentage Accurate: 67.2% → 97.4%

Time: 12.7s

Alternatives: 9

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. associate-/r* 69.2%

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

   2. cos-neg 69.2%

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

   3. distribute-rgt-neg-out 69.2%

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

   4. distribute-rgt-neg-out 69.2%

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

   5. *-commutative 69.2%

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

   6. distribute-rgt-neg-in 69.2%

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

   7. metadata-eval 69.2%

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

   8. *-commutative 69.2%

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

   9. associate-*l* 64.1%

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

   10. unpow2 64.1%

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

3. Simplified 64.1%

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

5. Taylor expanded in x around inf 64.6%

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

   1. associate-/r* 64.1%

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

   2. *-commutative 64.1%

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

   3. unpow2 64.1%

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

   4. unpow2 64.1%

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

   5. swap-sqr 79.5%

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

   6. unpow2 79.5%

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

   7. associate-/r* 80.0%

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

   8. *-commutative 80.0%

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

   9. unpow2 80.0%

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

   10. unpow2 80.0%

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

   11. swap-sqr 97.2%

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

   12. unpow2 97.2%

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

   13. *-commutative 97.2%

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

7. Simplified 97.2%

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

   1. unpow-prod-down 80.0%

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

   2. *-commutative 80.0%

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

   3. unpow-prod-down 64.6%

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

   4. /-rgt-identity 64.6%

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

   5. clear-num 64.6%

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

   6. div-inv 64.6%

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

   7. clear-num 64.6%

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

   8. div-inv 64.6%

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

   9. unpow2 64.6%

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

   10. associate-/l/ 64.6%

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

   11. unpow-prod-down 80.0%

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

   12. *-commutative 80.0%

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

   13. pow-flip 80.3%

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

   14. *-commutative 80.3%

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

   15. metadata-eval 80.3%

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

   16. sqr-pow 80.3%

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

   17. times-frac 97.1%

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

9. Applied egg-rr 97.6%

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

11. Applied egg-rr 97.5%

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

12. Final simplification 97.5%

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

Alternative 2: 97.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \frac{1}{t\_0} \cdot \frac{\cos \left(x \cdot -2\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x))) (* (/ 1.0 t_0) (/ (cos (* x -2.0)) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	return (1.0 / t_0) * (cos((x * -2.0)) / 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 = (c * s) * x
    code = (1.0d0 / t_0) * (cos((x * (-2.0d0))) / t_0)
end function

Java

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

Python

def code(x, c, s):
	t_0 = (c * s) * x
	return (1.0 / t_0) * (math.cos((x * -2.0)) / t_0)

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	return Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x * -2.0)) / t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = (1.0 / t_0) * (cos((x * -2.0)) / t_0);
end

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. associate-/r* 69.2%

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

   2. cos-neg 69.2%

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

   3. distribute-rgt-neg-out 69.2%

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

   4. distribute-rgt-neg-out 69.2%

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

   5. *-commutative 69.2%

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

   6. distribute-rgt-neg-in 69.2%

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

   7. metadata-eval 69.2%

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

   8. *-commutative 69.2%

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

   9. associate-*l* 64.1%

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

   10. unpow2 64.1%

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

3. Simplified 64.1%

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

5. Taylor expanded in x around inf 64.6%

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

   1. associate-/r* 64.1%

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

   2. *-commutative 64.1%

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

   3. unpow2 64.1%

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

   4. unpow2 64.1%

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

   5. swap-sqr 79.5%

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

   6. unpow2 79.5%

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

   7. associate-/r* 80.0%

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

   8. *-commutative 80.0%

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

   9. unpow2 80.0%

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

   10. unpow2 80.0%

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

   11. swap-sqr 97.2%

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

   12. unpow2 97.2%

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

   13. *-commutative 97.2%

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

7. Simplified 97.2%

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

   1. unpow-prod-down 80.0%

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

   2. *-commutative 80.0%

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

   3. unpow-prod-down 64.6%

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

   4. /-rgt-identity 64.6%

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

   5. clear-num 64.6%

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

   6. div-inv 64.6%

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

   7. clear-num 64.6%

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

   8. div-inv 64.6%

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

   9. unpow2 64.6%

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

   10. associate-/l/ 64.6%

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

   11. unpow-prod-down 80.0%

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

   12. *-commutative 80.0%

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

   13. pow-flip 80.3%

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

   14. *-commutative 80.3%

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

   15. metadata-eval 80.3%

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

   16. sqr-pow 80.3%

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

   17. times-frac 97.1%

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

9. Applied egg-rr 97.6%

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

    1. associate-/r* 97.5%

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

    2. div-inv 97.5%

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

    3. associate-/r/ 95.4%

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

    4. div-inv 95.4%

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

    5. clear-num 95.4%

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

    6. /-rgt-identity 95.4%

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

    7. associate-/r/ 97.5%

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

    8. div-inv 97.4%

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

    9. clear-num 97.4%

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

    10. /-rgt-identity 97.4%

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

11. Applied egg-rr 97.4%

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

12. Final simplification 97.4%

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

Alternative 3: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \frac{\cos \left(x \cdot -2\right)}{\frac{t\_0}{\frac{1}{t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x))) (/ (cos (* x -2.0)) (/ t_0 (/ 1.0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	return cos((x * -2.0)) / (t_0 / (1.0 / 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 = (c * s) * x
    code = cos((x * (-2.0d0))) / (t_0 / (1.0d0 / t_0))
end function

Java

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

Python

def code(x, c, s):
	t_0 = (c * s) * x
	return math.cos((x * -2.0)) / (t_0 / (1.0 / t_0))

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	return Float64(cos(Float64(x * -2.0)) / Float64(t_0 / Float64(1.0 / t_0)))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = cos((x * -2.0)) / (t_0 / (1.0 / t_0));
end

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. associate-/r* 69.2%

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

   2. cos-neg 69.2%

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

   3. distribute-rgt-neg-out 69.2%

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

   4. distribute-rgt-neg-out 69.2%

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

   5. *-commutative 69.2%

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

   6. distribute-rgt-neg-in 69.2%

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

   7. metadata-eval 69.2%

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

   8. *-commutative 69.2%

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

   9. associate-*l* 64.1%

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

   10. unpow2 64.1%

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

3. Simplified 64.1%

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

5. Taylor expanded in x around inf 64.6%

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

   1. associate-/r* 64.1%

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

   2. *-commutative 64.1%

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

   3. unpow2 64.1%

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

   4. unpow2 64.1%

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

   5. swap-sqr 79.5%

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

   6. unpow2 79.5%

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

   7. associate-/r* 80.0%

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

   8. *-commutative 80.0%

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

   9. unpow2 80.0%

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

   10. unpow2 80.0%

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

   11. swap-sqr 97.2%

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

   12. unpow2 97.2%

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

   13. *-commutative 97.2%

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

7. Simplified 97.2%

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

   1. unpow-prod-down 80.0%

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

   2. *-commutative 80.0%

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

   3. unpow-prod-down 64.6%

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

   4. /-rgt-identity 64.6%

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

   5. clear-num 64.6%

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

   6. div-inv 64.6%

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

   7. clear-num 64.6%

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

   8. div-inv 64.6%

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

   9. unpow2 64.6%

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

   10. associate-/l/ 64.6%

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

   11. unpow-prod-down 80.0%

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

   12. *-commutative 80.0%

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

   13. pow-flip 80.3%

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

   14. *-commutative 80.3%

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

   15. metadata-eval 80.3%

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

   16. sqr-pow 80.3%

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

   17. times-frac 97.1%

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

9. Applied egg-rr 97.6%

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

    1. clear-num 97.5%

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

    2. clear-num 97.5%

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

    3. div-inv 97.0%

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

    4. associate-/l/ 97.1%

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

    5. *-commutative 97.1%

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

    6. remove-double-div 97.1%

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

    7. inv-pow 97.1%

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

    8. metadata-eval 97.1%

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

    9. un-div-inv 97.2%

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

    10. associate-/r/ 95.1%

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

    11. div-inv 95.1%

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

    12. clear-num 95.1%

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

    13. /-rgt-identity 95.1%

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

    14. metadata-eval 95.1%

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

    15. inv-pow 95.1%

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

    16. associate-*r* 97.5%

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

11. Applied egg-rr 97.5%

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

12. Final simplification 97.5%

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

Alternative 4: 96.8% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* s x)))) (/ (cos (* x -2.0)) (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (s * x);
	return cos((x * -2.0)) / (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 = c * (s * x)
    code = cos((x * (-2.0d0))) / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. associate-/r* 69.2%

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

   2. cos-neg 69.2%

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

   3. distribute-rgt-neg-out 69.2%

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

   4. distribute-rgt-neg-out 69.2%

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

   5. *-commutative 69.2%

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

   6. distribute-rgt-neg-in 69.2%

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

   7. metadata-eval 69.2%

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

   8. *-commutative 69.2%

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

   9. associate-*l* 64.1%

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

   10. unpow2 64.1%

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

3. Simplified 64.1%

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

5. Taylor expanded in x around inf 64.6%

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

   1. associate-/r* 64.1%

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

   2. *-commutative 64.1%

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

   3. unpow2 64.1%

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

   4. unpow2 64.1%

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

   5. swap-sqr 79.5%

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

   6. unpow2 79.5%

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

   7. associate-/r* 80.0%

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

   8. *-commutative 80.0%

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

   9. unpow2 80.0%

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

   10. unpow2 80.0%

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

   11. swap-sqr 97.2%

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

   12. unpow2 97.2%

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

   13. *-commutative 97.2%

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

7. Simplified 97.2%

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

   1. unpow2 79.2%

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

9. Applied egg-rr 97.2%

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

10. Final simplification 97.2%

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

Alternative 5: 77.9% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in x around 0 57.9%

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

   1. associate-/r* 57.5%

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

   2. *-commutative 57.5%

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

   3. unpow2 57.5%

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

   4. unpow2 57.5%

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

   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* 68.5%

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

   8. unpow2 68.5%

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

   9. unpow2 68.5%

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

   10. swap-sqr 79.2%

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

   11. unpow2 79.2%

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

   12. *-commutative 79.2%

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

5. Simplified 79.2%

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

   1. unpow-prod-down 80.0%

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

   2. *-commutative 80.0%

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

   3. unpow-prod-down 64.6%

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

   4. /-rgt-identity 64.6%

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

   5. clear-num 64.6%

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

   6. div-inv 64.6%

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

   7. clear-num 64.6%

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

   8. div-inv 64.6%

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

   9. unpow2 64.6%

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

   10. associate-/l/ 64.6%

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

   11. unpow-prod-down 80.0%

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

   12. *-commutative 80.0%

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

   13. pow-flip 80.3%

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

   14. *-commutative 80.3%

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

   15. metadata-eval 80.3%

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

   16. sqr-pow 80.3%

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

   17. times-frac 97.1%

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

7. Applied egg-rr 79.3%

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

8. Final simplification 79.3%

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

Alternative 6: 78.1% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in x around 0 57.9%

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

   1. associate-/r* 57.5%

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

   2. *-commutative 57.5%

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

   3. unpow2 57.5%

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

   4. unpow2 57.5%

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

   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* 68.5%

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

   8. unpow2 68.5%

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

   9. unpow2 68.5%

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

   10. swap-sqr 79.2%

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

   11. unpow2 79.2%

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

   12. *-commutative 79.2%

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

5. Simplified 79.2%

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

   1. metadata-eval 79.2%

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

   2. *-commutative 79.2%

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

   3. unpow-prod-down 68.5%

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

   4. *-commutative 68.5%

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

   5. frac-times 68.0%

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

   6. unpow-prod-down 57.5%

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

   7. associate-/l/ 57.5%

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

   8. div-inv 57.9%

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

   9. add-sqr-sqrt 57.9%

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

7. Applied egg-rr 79.2%

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

8. Final simplification 79.2%

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

Alternative 7: 72.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in x around 0 57.9%

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

   1. associate-/r* 57.5%

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

   2. *-commutative 57.5%

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

   3. unpow2 57.5%

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

   4. unpow2 57.5%

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

   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* 68.5%

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

   8. unpow2 68.5%

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

   9. unpow2 68.5%

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

   10. swap-sqr 79.2%

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

   11. unpow2 79.2%

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

   12. *-commutative 79.2%

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

5. Simplified 79.2%

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

   1. /-rgt-identity 79.2%

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

   2. clear-num 79.2%

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

   3. pow-flip 79.2%

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

   4. metadata-eval 79.2%

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

7. Applied egg-rr 79.2%

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

   1. pow-flip 79.2%

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

   2. remove-double-div 79.1%

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

   3. *-commutative 79.1%

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

   4. associate-/l/ 79.1%

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

   5. div-inv 79.3%

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

   6. metadata-eval 79.3%

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

   7. pow2 79.3%

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

   8. associate-*l/ 76.1%

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

   9. *-un-lft-identity 76.1%

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

   10. times-frac 78.1%

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

   11. /-rgt-identity 78.1%

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

   12. associate-/r/ 76.9%

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

   13. div-inv 76.9%

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

   14. clear-num 76.9%

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

   15. /-rgt-identity 76.9%

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

   16. associate-/l/ 76.9%

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

   17. *-commutative 76.9%

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

9. Applied egg-rr 76.9%

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

    1. associate-/l* 73.3%

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

    2. *-commutative 73.3%

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

    3. associate-*l* 73.9%

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

    4. div-inv 73.9%

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

    5. remove-double-div 73.9%

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

11. Applied egg-rr 73.9%

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

12. Final simplification 73.9%

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

Alternative 8: 76.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in x around 0 57.9%

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

   1. associate-/r* 57.5%

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

   2. *-commutative 57.5%

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

   3. unpow2 57.5%

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

   4. unpow2 57.5%

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

   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* 68.5%

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

   8. unpow2 68.5%

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

   9. unpow2 68.5%

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

   10. swap-sqr 79.2%

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

   11. unpow2 79.2%

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

   12. *-commutative 79.2%

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

5. Simplified 79.2%

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

   1. /-rgt-identity 79.2%

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

   2. clear-num 79.2%

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

   3. pow-flip 79.2%

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

   4. metadata-eval 79.2%

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

7. Applied egg-rr 79.2%

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

   1. pow-flip 79.2%

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

   2. remove-double-div 79.1%

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

   3. *-commutative 79.1%

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

   4. associate-/l/ 79.1%

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

   5. div-inv 79.3%

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

   6. metadata-eval 79.3%

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

   7. pow2 79.3%

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

   8. associate-*l/ 76.1%

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

   9. *-un-lft-identity 76.1%

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

   10. times-frac 78.1%

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

   11. /-rgt-identity 78.1%

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

   12. associate-/r/ 76.9%

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

   13. div-inv 76.9%

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

   14. clear-num 76.9%

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

   15. /-rgt-identity 76.9%

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

   16. associate-/l/ 76.9%

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

   17. *-commutative 76.9%

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

9. Applied egg-rr 76.9%

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

    1. associate-/r/ 76.9%

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

    2. /-rgt-identity 76.9%

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

    3. *-commutative 76.9%

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

    4. associate-*r* 78.1%

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

11. Applied egg-rr 78.1%

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

12. Final simplification 78.1%

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

Alternative 9: 78.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in x around 0 57.9%

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

   1. associate-/r* 57.5%

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

   2. *-commutative 57.5%

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

   3. unpow2 57.5%

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

   4. unpow2 57.5%

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

   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* 68.5%

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

   8. unpow2 68.5%

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

   9. unpow2 68.5%

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

   10. swap-sqr 79.2%

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

   11. unpow2 79.2%

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

   12. *-commutative 79.2%

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

5. Simplified 79.2%

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

   1. unpow2 79.2%

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

7. Applied egg-rr 79.2%

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

8. Final simplification 79.2%

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

Reproduce

herbie shell --seed 2024075 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))