mixedcos

Percentage Accurate: 67.2% → 97.4%

Time: 17.0s

Alternatives: 12

Speedup: 24.1×

• 30.8% 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, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. associate-*r* 61.4%

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

   3. unpow2 61.4%

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

   4. *-commutative 61.4%

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

   5. associate-*r* 60.2%

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

   6. add-sqr-sqrt 60.2%

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

   7. unpow2 60.2%

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

   8. pow-prod-down 65.7%

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

   9. pow-prod-down 85.1%

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

   10. sqrt-pow1 98.2%

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

   11. metadata-eval 98.2%

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

   12. pow1 98.2%

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

3. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

   2. swap-sqr 78.0%

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

   3. swap-sqr 60.2%

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

   4. unpow2 60.2%

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

   5. unpow2 60.2%

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

   6. associate-/l/ 59.8%

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

   7. associate-/r* 60.3%

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

   8. div-inv 60.3%

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

   9. add-sqr-sqrt 60.2%

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

   10. times-frac 65.7%

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

   11. *-commutative 65.7%

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

   12. pow-prod-down 65.7%

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

   13. sqrt-pow1 50.9%

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

   14. metadata-eval 50.9%

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

   15. pow1 50.9%

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

5. Applied egg-rr 98.1%

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

   1. *-commutative 98.1%

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

   2. associate-/l/ 98.2%

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

   3. associate-*r* 97.0%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. associate-*r* 97.9%

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

   7. associate-*r/ 97.9%

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

   8. *-commutative 97.9%

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

   9. associate-*l* 97.2%

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

   10. *-commutative 97.2%

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

   11. *-commutative 97.2%

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

   12. associate-*l* 98.5%

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

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 92.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;s \leq 6 \cdot 10^{+160}:\\ \;\;\;\;\frac{t_1}{\left(x \cdot c\right) \cdot \left(s \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\left(x \cdot s\right) \cdot \left(c \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* x 2.0))))
   (if (<= s 6e+160)
     (/ t_1 (* (* x c) (* s t_0)))
     (/ t_1 (* (* x s) (* c t_0))))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (s <= 6e+160) {
		tmp = t_1 / ((x * c) * (s * t_0));
	} else {
		tmp = t_1 / ((x * s) * (c * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = cos((x * 2.0d0))
    if (s <= 6d+160) then
        tmp = t_1 / ((x * c) * (s * t_0))
    else
        tmp = t_1 / ((x * s) * (c * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (s <= 6e+160) {
		tmp = t_1 / ((x * c) * (s * t_0));
	} else {
		tmp = t_1 / ((x * s) * (c * t_0));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if s <= 6e+160:
		tmp = t_1 / ((x * c) * (s * t_0))
	else:
		tmp = t_1 / ((x * s) * (c * t_0))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (s <= 6e+160)
		tmp = Float64(t_1 / Float64(Float64(x * c) * Float64(s * t_0)));
	else
		tmp = Float64(t_1 / Float64(Float64(x * s) * Float64(c * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (s <= 6e+160)
		tmp = t_1 / ((x * c) * (s * t_0));
	else
		tmp = t_1 / ((x * s) * (c * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[s, 6e+160], N[(t$95$1 / N[(N[(x * c), $MachinePrecision] * N[(s * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(x * s), $MachinePrecision] * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if s < 5.9999999999999997e160

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*r* 62.8%

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

      3. unpow2 62.8%

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

      4. *-commutative 62.8%

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

      5. associate-*r* 61.5%

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

      6. add-sqr-sqrt 61.5%

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

      7. unpow2 61.5%

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

      8. pow-prod-down 66.3%

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

      9. pow-prod-down 85.1%

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

      10. sqrt-pow1 98.1%

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

      11. metadata-eval 98.1%

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

      12. pow1 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. unpow2 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*r* 95.3%

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

      4. associate-*l* 93.9%

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

   5. Applied egg-rr 93.9%

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

   if 5.9999999999999997e160 < s

   1. Initial program 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-*r* 48.3%

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

      3. unpow2 48.3%

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

      4. *-commutative 48.3%

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

      5. associate-*r* 48.3%

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

      6. add-sqr-sqrt 48.3%

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

      7. unpow2 48.3%

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

      8. pow-prod-down 60.8%

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

      9. pow-prod-down 84.5%

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

      10. sqrt-pow1 99.6%

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

      11. metadata-eval 99.6%

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

      12. pow1 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow2 99.6%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.8%

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

      5. associate-*l* 99.8%

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

      6. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 6 \cdot 10^{+160}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 3: 86.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2.3e-16)
   (pow (* c (* x s)) -2.0)
   (/ (/ (cos (* x 2.0)) s) (* (* x (* s c)) (* x c)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.3e-16) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = (cos((x * 2.0)) / s) / ((x * (s * c)) * (x * c));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 2.3d-16) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = (cos((x * 2.0d0)) / s) / ((x * (s * c)) * (x * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.3e-16) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = (Math.cos((x * 2.0)) / s) / ((x * (s * c)) * (x * c));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2.3e-16:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = (math.cos((x * 2.0)) / s) / ((x * (s * c)) * (x * c))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2.3e-16)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / s) / Float64(Float64(x * Float64(s * c)) * Float64(x * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.3e-16)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = (cos((x * 2.0)) / s) / ((x * (s * c)) * (x * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2.3e-16], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s), $MachinePrecision] / N[(N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999999e-16

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

      2. cos-neg 68.4%

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

      3. *-commutative 68.4%

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

      4. distribute-rgt-neg-in 68.4%

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

      5. metadata-eval 68.4%

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

      6. *-commutative 68.4%

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

      7. *-commutative 68.4%

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

      8. associate-*r* 59.6%

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

      9. unpow2 59.6%

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

      10. *-commutative 59.6%

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

      11. unpow2 59.6%

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

      12. associate-*r* 65.1%

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

      13. associate-*r* 59.6%

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

      14. unpow2 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in x around 0 54.7%

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

      1. associate-/r* 54.6%

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

      2. *-commutative 54.6%

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

      3. unpow2 54.6%

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

      4. unpow2 54.6%

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

      5. swap-sqr 72.1%

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

      6. unpow2 72.1%

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

      7. associate-/r* 72.2%

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

      8. unpow2 72.2%

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

      9. unpow2 72.2%

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

      10. swap-sqr 84.8%

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

      11. unpow2 84.8%

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

      12. *-commutative 84.8%

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

   6. Simplified 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*l* 86.1%

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

      3. pow-flip 86.1%

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

      4. associate-*l* 84.8%

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

      5. metadata-eval 84.8%

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

   8. Applied egg-rr 84.8%

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

   if 2.2999999999999999e-16 < x

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*r* 66.2%

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

      3. unpow2 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*r* 66.2%

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

      6. add-sqr-sqrt 66.1%

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

      7. unpow2 66.1%

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

      8. pow-prod-down 72.1%

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

      9. pow-prod-down 91.8%

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

      10. sqrt-pow1 97.3%

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

      11. metadata-eval 97.3%

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

      12. pow1 97.3%

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

   3. Applied egg-rr 97.3%

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

      1. unpow2 97.3%

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

      2. swap-sqr 83.2%

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

      3. swap-sqr 66.2%

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

      4. unpow2 66.2%

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

      5. unpow2 66.2%

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

      6. associate-/l/ 64.7%

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

      7. associate-/r* 65.6%

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

      8. div-inv 65.6%

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

      9. add-sqr-sqrt 65.5%

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

      10. times-frac 72.0%

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

      11. *-commutative 72.0%

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

      12. pow-prod-down 72.0%

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

      13. sqrt-pow1 59.5%

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

      14. metadata-eval 59.5%

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

      15. pow1 59.5%

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

   5. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-/l/ 97.2%

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

      3. associate-*r* 95.9%

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

      4. frac-times 94.6%

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

      5. *-un-lft-identity 94.6%

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

      6. *-commutative 94.6%

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

      7. associate-*l* 94.6%

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

   7. Applied egg-rr 94.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 4: 97.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $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 68.9%

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

   1. *-commutative 68.9%

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

   2. cos-neg 68.9%

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

   3. *-commutative 68.9%

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

   4. distribute-rgt-neg-in 68.9%

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

   5. metadata-eval 68.9%

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

   6. *-commutative 68.9%

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

   7. *-commutative 68.9%

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

   8. associate-*r* 61.4%

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

   9. unpow2 61.4%

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

   10. *-commutative 61.4%

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

   11. unpow2 61.4%

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

   12. associate-*r* 66.6%

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

   13. associate-*r* 61.4%

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

   14. unpow2 61.4%

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

3. Simplified 61.4%

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

   1. *-un-lft-identity 61.4%

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

   2. metadata-eval 61.4%

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

   3. add-sqr-sqrt 61.4%

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

   4. times-frac 61.4%

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

   5. metadata-eval 61.4%

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

   6. sqrt-prod 61.3%

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

   7. unpow2 61.3%

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

   8. sqrt-prod 28.3%

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

   9. add-sqr-sqrt 43.0%

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

   10. pow-prod-down 43.1%

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

   11. sqrt-pow1 44.7%

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

   12. metadata-eval 44.7%

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

   13. pow1 44.7%

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

   14. *-commutative 44.7%

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

5. Applied egg-rr 97.9%

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

6. Final simplification 97.9%

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

Alternative 5: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / s), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. associate-*r* 61.4%

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

   3. unpow2 61.4%

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

   4. *-commutative 61.4%

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

   5. associate-*r* 60.2%

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

   6. add-sqr-sqrt 60.2%

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

   7. unpow2 60.2%

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

   8. pow-prod-down 65.7%

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

   9. pow-prod-down 85.1%

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

   10. sqrt-pow1 98.2%

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

   11. metadata-eval 98.2%

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

   12. pow1 98.2%

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

3. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

   2. swap-sqr 78.0%

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

   3. swap-sqr 60.2%

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

   4. unpow2 60.2%

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

   5. unpow2 60.2%

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

   6. associate-/l/ 59.8%

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

   7. associate-/r* 60.3%

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

   8. div-inv 60.3%

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

   9. add-sqr-sqrt 60.2%

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

   10. times-frac 65.7%

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

   11. *-commutative 65.7%

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

   12. pow-prod-down 65.7%

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

   13. sqrt-pow1 50.9%

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

   14. metadata-eval 50.9%

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

   15. pow1 50.9%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 6: 93.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. associate-*r* 61.4%

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

   3. unpow2 61.4%

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

   4. *-commutative 61.4%

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

   5. associate-*r* 60.2%

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

   6. add-sqr-sqrt 60.2%

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

   7. unpow2 60.2%

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

   8. pow-prod-down 65.7%

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

   9. pow-prod-down 85.1%

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

   10. sqrt-pow1 98.2%

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

   11. metadata-eval 98.2%

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

   12. pow1 98.2%

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

3. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

   2. associate-*l* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 94.5%

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

   5. associate-*l* 94.8%

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

   6. *-commutative 94.8%

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

5. Applied egg-rr 94.8%

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

6. Final simplification 94.8%

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

Alternative 7: 95.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. associate-*r* 61.4%

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

   3. unpow2 61.4%

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

   4. *-commutative 61.4%

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

   5. associate-*r* 60.2%

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

   6. add-sqr-sqrt 60.2%

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

   7. unpow2 60.2%

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

   8. pow-prod-down 65.7%

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

   9. pow-prod-down 85.1%

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

   10. sqrt-pow1 98.2%

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

   11. metadata-eval 98.2%

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

   12. pow1 98.2%

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

3. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

   2. swap-sqr 78.0%

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

   3. swap-sqr 60.2%

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

   4. unpow2 60.2%

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

   5. unpow2 60.2%

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

   6. associate-/l/ 59.8%

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

   7. associate-/r* 60.3%

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

   8. div-inv 60.3%

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

   9. add-sqr-sqrt 60.2%

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

   10. times-frac 65.7%

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

   11. *-commutative 65.7%

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

   12. pow-prod-down 65.7%

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

   13. sqrt-pow1 50.9%

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

   14. metadata-eval 50.9%

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

   15. pow1 50.9%

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

5. Applied egg-rr 98.1%

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

   1. associate-/l/ 98.2%

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

   2. associate-*r* 97.0%

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

   3. un-div-inv 97.0%

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

   4. *-commutative 97.0%

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

   5. associate-*l* 97.5%

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

7. Applied egg-rr 97.5%

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

8. Final simplification 97.5%

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

Alternative 8: 79.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return pow((c * (x * s)), -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 * (x * s)) ** (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. cos-neg 68.9%

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

   3. *-commutative 68.9%

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

   4. distribute-rgt-neg-in 68.9%

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

   5. metadata-eval 68.9%

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

   6. *-commutative 68.9%

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

   7. *-commutative 68.9%

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

   8. associate-*r* 61.4%

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

   9. unpow2 61.4%

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

   10. *-commutative 61.4%

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

   11. unpow2 61.4%

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

   12. associate-*r* 66.6%

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

   13. associate-*r* 61.4%

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

   14. unpow2 61.4%

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

3. Simplified 61.4%

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

4. Taylor expanded in x around 0 53.2%

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

   1. associate-/r* 53.2%

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

   2. *-commutative 53.2%

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

   3. unpow2 53.2%

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

   4. unpow2 53.2%

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

   5. swap-sqr 67.2%

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

   6. unpow2 67.2%

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

   7. associate-/r* 67.2%

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

   8. unpow2 67.2%

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

   9. unpow2 67.2%

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

   10. swap-sqr 78.3%

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

   11. unpow2 78.3%

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

   12. *-commutative 78.3%

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

6. Simplified 78.3%

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

   1. *-commutative 78.3%

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

   2. associate-*l* 79.2%

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

   3. pow-flip 79.2%

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

   4. associate-*l* 78.4%

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

   5. metadata-eval 78.4%

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

8. Applied egg-rr 78.4%

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

9. Final simplification 78.4%

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

Alternative 9: 76.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. associate-*r* 61.4%

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

   3. unpow2 61.4%

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

   4. *-commutative 61.4%

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

   5. associate-*r* 60.2%

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

   6. add-sqr-sqrt 60.2%

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

   7. unpow2 60.2%

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

   8. pow-prod-down 65.7%

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

   9. pow-prod-down 85.1%

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

   10. sqrt-pow1 98.2%

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

   11. metadata-eval 98.2%

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

   12. pow1 98.2%

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

3. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

   2. associate-*l* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-*r* 97.5%

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

   5. associate-*r* 92.8%

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

   6. associate-*l* 92.6%

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

5. Applied egg-rr 92.6%

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

6. Taylor expanded in x around 0 75.0%

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

7. Final simplification 75.0%

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

Alternative 10: 76.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. cos-neg 68.9%

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

   3. *-commutative 68.9%

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

   4. distribute-rgt-neg-in 68.9%

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

   5. metadata-eval 68.9%

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

   6. *-commutative 68.9%

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

   7. *-commutative 68.9%

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

   8. associate-*r* 61.4%

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

   9. unpow2 61.4%

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

   10. *-commutative 61.4%

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

   11. unpow2 61.4%

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

   12. associate-*r* 66.6%

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

   13. associate-*r* 61.4%

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

   14. unpow2 61.4%

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

3. Simplified 61.4%

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

4. Taylor expanded in x around 0 53.2%

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

   1. associate-/r* 53.2%

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

   2. *-commutative 53.2%

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

   3. unpow2 53.2%

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

   4. unpow2 53.2%

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

   5. swap-sqr 67.2%

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

   6. unpow2 67.2%

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

   7. associate-/r* 67.2%

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

   8. unpow2 67.2%

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

   9. unpow2 67.2%

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

   10. swap-sqr 78.3%

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

   11. unpow2 78.3%

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

   12. *-commutative 78.3%

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

6. Simplified 78.3%

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

   1. *-commutative 78.3%

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

   2. associate-*l* 79.2%

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

   3. add-sqr-sqrt 79.1%

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

   4. sqrt-div 79.2%

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

   5. metadata-eval 79.2%

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

   6. unpow2 79.2%

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

   7. sqrt-prod 45.0%

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

   8. add-sqr-sqrt 58.5%

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

   9. associate-*l* 57.7%

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

   10. sqrt-div 57.7%

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

   11. metadata-eval 57.7%

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

   12. unpow2 57.7%

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

   13. sqrt-prod 37.4%

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

   14. add-sqr-sqrt 78.3%

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

   15. associate-*l* 78.3%

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

8. Applied egg-rr 78.3%

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

   1. associate-/r* 78.4%

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

   2. frac-times 77.2%

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

   3. div-inv 77.2%

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

   4. *-commutative 77.2%

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

   5. associate-*l* 77.2%

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

   6. *-commutative 77.2%

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

10. Applied egg-rr 77.2%

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

11. Final simplification 77.2%

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

Alternative 11: 78.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. cos-neg 68.9%

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

   3. *-commutative 68.9%

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

   4. distribute-rgt-neg-in 68.9%

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

   5. metadata-eval 68.9%

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

   6. *-commutative 68.9%

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

   7. *-commutative 68.9%

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

   8. associate-*r* 61.4%

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

   9. unpow2 61.4%

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

   10. *-commutative 61.4%

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

   11. unpow2 61.4%

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

   12. associate-*r* 66.6%

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

   13. associate-*r* 61.4%

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

   14. unpow2 61.4%

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

3. Simplified 61.4%

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

4. Taylor expanded in x around 0 53.2%

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

   1. associate-/r* 53.2%

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

   2. *-commutative 53.2%

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

   3. unpow2 53.2%

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

   4. unpow2 53.2%

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

   5. swap-sqr 67.2%

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

   6. unpow2 67.2%

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

   7. associate-/r* 67.2%

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

   8. unpow2 67.2%

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

   9. unpow2 67.2%

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

   10. swap-sqr 78.3%

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

   11. unpow2 78.3%

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

   12. *-commutative 78.3%

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

6. Simplified 78.3%

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

   1. *-commutative 78.3%

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

   2. associate-*l* 79.2%

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

   3. add-sqr-sqrt 79.1%

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

   4. sqrt-div 79.2%

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

   5. metadata-eval 79.2%

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

   6. unpow2 79.2%

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

   7. sqrt-prod 45.0%

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

   8. add-sqr-sqrt 58.5%

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

   9. associate-*l* 57.7%

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

   10. sqrt-div 57.7%

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

   11. metadata-eval 57.7%

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

   12. unpow2 57.7%

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

   13. sqrt-prod 37.4%

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

   14. add-sqr-sqrt 78.3%

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

   15. associate-*l* 78.3%

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

8. Applied egg-rr 78.3%

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

   1. un-div-inv 78.3%

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

   2. *-commutative 78.3%

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

   3. associate-*l* 78.3%

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

   4. *-commutative 78.3%

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

   5. associate-*l* 79.2%

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

10. Applied egg-rr 79.2%

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

11. Final simplification 79.2%

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

Alternative 12: 78.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. cos-neg 68.9%

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

   3. *-commutative 68.9%

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

   4. distribute-rgt-neg-in 68.9%

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

   5. metadata-eval 68.9%

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

   6. *-commutative 68.9%

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

   7. *-commutative 68.9%

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

   8. associate-*r* 61.4%

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

   9. unpow2 61.4%

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

   10. *-commutative 61.4%

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

   11. unpow2 61.4%

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

   12. associate-*r* 66.6%

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

   13. associate-*r* 61.4%

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

   14. unpow2 61.4%

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

3. Simplified 61.4%

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

4. Taylor expanded in x around 0 53.2%

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

   1. associate-/r* 53.2%

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

   2. *-commutative 53.2%

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

   3. unpow2 53.2%

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

   4. unpow2 53.2%

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

   5. swap-sqr 67.2%

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

   6. unpow2 67.2%

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

   7. associate-/r* 67.2%

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

   8. unpow2 67.2%

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

   9. unpow2 67.2%

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

   10. swap-sqr 78.3%

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

   11. unpow2 78.3%

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

   12. *-commutative 78.3%

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

6. Simplified 78.3%

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

   1. *-commutative 78.3%

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

   2. associate-*l* 79.2%

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

   3. add-sqr-sqrt 79.1%

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

   4. sqrt-div 79.2%

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

   5. metadata-eval 79.2%

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

   6. unpow2 79.2%

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

   7. sqrt-prod 45.0%

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

   8. add-sqr-sqrt 58.5%

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

   9. associate-*l* 57.7%

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

   10. sqrt-div 57.7%

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

   11. metadata-eval 57.7%

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

   12. unpow2 57.7%

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

   13. sqrt-prod 37.4%

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

   14. add-sqr-sqrt 78.3%

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

   15. associate-*l* 78.3%

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

8. Applied egg-rr 78.3%

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

   1. frac-2neg 78.3%

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

   2. metadata-eval 78.3%

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

   3. metadata-eval 78.3%

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

   4. frac-2neg 78.3%

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

   5. metadata-eval 78.3%

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

   6. metadata-eval 78.3%

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

   7. frac-times 78.3%

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

   8. metadata-eval 78.3%

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

   9. metadata-eval 78.3%

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

   10. metadata-eval 78.3%

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

   11. distribute-rgt-neg-in 78.3%

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

   12. *-commutative 78.3%

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

   13. distribute-rgt-neg-in 78.3%

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

   14. *-commutative 78.3%

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

10. Applied egg-rr 78.3%

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

11. Final simplification 78.3%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))