mixedcos

Percentage Accurate: 67.2% → 99.5%

Time: 12.0s

Alternatives: 12

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 7.3 \cdot 10^{-50}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 7.3e-50)
     (pow (* c (* x s)) -2.0)
     (/ (/ (cos (* x 2.0)) t_0) t_0))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 7.3e-50) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = (cos((x * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 7.3d-50) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = (cos((x * 2.0d0)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 7.3e-50) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = (Math.cos((x * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 7.3e-50:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = (math.cos((x * 2.0)) / t_0) / t_0
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 7.3e-50)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 7.3e-50)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = (cos((x * 2.0)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7.3e-50], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.30000000000000035e-50

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 62.2%

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

      4. *-commutative 62.2%

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

      5. unpow2 62.2%

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

      6. associate-*r* 66.1%

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

      7. associate-*r* 67.9%

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

      8. *-commutative 67.9%

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

      9. unpow2 67.9%

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

   3. Simplified 67.9%

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

      1. add-cube-cbrt 67.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

      2. times-frac 68.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

      3. associate-*r* 66.6%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

      4. swap-sqr 85.3%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      5. associate-*r* 94.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      6. *-commutative 94.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

      8. associate-*l* 95.9%

         \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

      9. add-cube-cbrt 96.1%

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

      10. associate-/r* 96.7%

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

   5. Applied egg-rr 96.6%

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

      1. un-div-inv 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

      2. associate-*r* 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*r* 59.1%

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

      5. associate-/r* 59.1%

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

      6. *-commutative 59.1%

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

      7. unpow2 59.1%

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

      8. unpow2 59.1%

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

      9. swap-sqr 68.8%

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

      10. unpow2 68.8%

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

      11. associate-/r* 68.8%

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

      12. unpow2 68.8%

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

      13. swap-sqr 82.7%

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

      14. associate-/r* 82.7%

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

      15. *-rgt-identity 82.7%

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

      16. associate-*r/ 82.7%

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

      17. unpow2 82.7%

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

   10. Simplified 83.9%

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

   if 7.30000000000000035e-50 < x

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-*l* 51.5%

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

      3. associate-*r* 53.0%

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

      4. *-commutative 53.0%

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

      5. unpow2 53.0%

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

      6. associate-*r* 59.2%

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

      7. associate-*r* 60.9%

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

      8. *-commutative 60.9%

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

      9. unpow2 60.9%

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

   3. Simplified 60.9%

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

      1. add-cube-cbrt 60.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

      2. times-frac 60.8%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

      3. associate-*r* 60.9%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

      4. swap-sqr 79.8%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      5. associate-*r* 89.7%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      6. *-commutative 89.7%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

      7. times-frac 89.7%

         \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

      8. associate-*l* 97.9%

         \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

      9. add-cube-cbrt 98.3%

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

      10. associate-/r* 98.2%

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

   5. Applied egg-rr 98.1%

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

      1. un-div-inv 98.2%

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

   7. Applied egg-rr 98.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.3 \cdot 10^{-50}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 2: 94.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+150}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= x 1.2e-8)
     (pow (* c (* x s)) -2.0)
     (if (<= x 1.45e+150)
       (/ t_0 (* x (* x (* (* c s) (* c s)))))
       (/ t_0 (* x (* (* c (* x c)) (* s s))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (x <= 1.2e-8) {
		tmp = pow((c * (x * s)), -2.0);
	} else if (x <= 1.45e+150) {
		tmp = t_0 / (x * (x * ((c * s) * (c * s))));
	} else {
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (x <= 1.2d-8) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else if (x <= 1.45d+150) then
        tmp = t_0 / (x * (x * ((c * s) * (c * s))))
    else
        tmp = t_0 / (x * ((c * (x * c)) * (s * s)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 1.2e-8) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else if (x <= 1.45e+150) {
		tmp = t_0 / (x * (x * ((c * s) * (c * s))));
	} else {
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 1.2e-8:
		tmp = math.pow((c * (x * s)), -2.0)
	elif x <= 1.45e+150:
		tmp = t_0 / (x * (x * ((c * s) * (c * s))))
	else:
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 1.2e-8)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	elseif (x <= 1.45e+150)
		tmp = Float64(t_0 / Float64(x * Float64(x * Float64(Float64(c * s) * Float64(c * s)))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(c * Float64(x * c)) * Float64(s * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 1.2e-8)
		tmp = (c * (x * s)) ^ -2.0;
	elseif (x <= 1.45e+150)
		tmp = t_0 / (x * (x * ((c * s) * (c * s))));
	else
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.2e-8], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x, 1.45e+150], N[(t$95$0 / N[(x * N[(x * N[(N[(c * s), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.19999999999999999e-8

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-*l* 62.3%

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

      3. associate-*r* 62.6%

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

      4. *-commutative 62.6%

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

      5. unpow2 62.6%

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

      6. associate-*r* 66.5%

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

      7. associate-*r* 68.3%

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

      8. *-commutative 68.3%

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

      9. unpow2 68.3%

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

   3. Simplified 68.3%

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

      1. add-cube-cbrt 68.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

      2. times-frac 68.3%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

      3. associate-*r* 67.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

      4. swap-sqr 85.2%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      5. associate-*r* 94.1%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      6. *-commutative 94.1%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

      7. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

      8. associate-*l* 96.0%

         \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

      9. add-cube-cbrt 96.2%

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

      10. associate-/r* 96.7%

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

   5. Applied egg-rr 96.7%

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

      1. un-div-inv 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in x around 0 59.8%

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

      1. unpow2 59.8%

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

      2. associate-*r* 59.9%

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

      3. *-commutative 59.9%

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

      4. associate-*r* 59.7%

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

      5. associate-/r* 59.6%

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

      6. *-commutative 59.6%

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

      7. unpow2 59.6%

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

      8. unpow2 59.6%

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

      9. swap-sqr 69.1%

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

      10. unpow2 69.1%

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

      11. associate-/r* 69.1%

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

      12. unpow2 69.1%

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

      13. swap-sqr 83.1%

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

      14. associate-/r* 83.1%

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

      15. *-rgt-identity 83.1%

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

      16. associate-*r/ 83.1%

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

      17. unpow2 83.1%

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

   10. Simplified 84.3%

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

   if 1.19999999999999999e-8 < x < 1.45000000000000005e150

   1. Initial program 48.0%

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

      1. associate-*r* 51.4%

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

      2. *-commutative 51.4%

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

      3. associate-*r* 51.4%

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

      4. unpow2 51.4%

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

      5. unpow2 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in c around 0 51.4%

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

      1. unpow2 51.4%

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

      2. associate-*l* 57.9%

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

   6. Simplified 57.9%

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

   7. Taylor expanded in c around 0 51.4%

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

      1. associate-*r* 51.4%

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

      2. *-commutative 51.4%

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

      3. *-commutative 51.4%

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

      4. *-commutative 51.4%

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

      5. unpow2 51.4%

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

      6. unpow2 51.4%

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

      7. unswap-sqr 90.1%

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

   9. Simplified 90.1%

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

   if 1.45000000000000005e150 < x

   1. Initial program 57.3%

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

      1. associate-*r* 60.2%

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

      2. *-commutative 60.2%

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

      3. associate-*r* 60.3%

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

      4. unpow2 60.3%

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

      5. unpow2 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in c around 0 60.3%

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

      1. unpow2 60.3%

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

      2. associate-*l* 69.6%

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

   6. Simplified 69.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+150}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 3: 82.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 4.95 \cdot 10^{+33}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= s 4.95e+33)
   (/ (cos (* x 2.0)) (* s (* (* x x) (* s (* c c)))))
   (pow (* c (* x s)) -2.0)))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (s <= 4.95e+33) {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 4.95d+33) then
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (s * (c * c))))
    else
        tmp = (c * (x * s)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (s <= 4.95e+33) {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = Math.pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if s <= 4.95e+33:
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))))
	else:
		tmp = math.pow((c * (x * s)), -2.0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (s <= 4.95e+33)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(s * Float64(c * c)))));
	else
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (s <= 4.95e+33)
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	else
		tmp = (c * (x * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[s, 4.95e+33], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(s * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 4.94999999999999993e33

   1. Initial program 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*l* 59.1%

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

      3. associate-*r* 59.6%

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

      4. *-commutative 59.6%

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

      5. unpow2 59.6%

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

      6. associate-*r* 65.3%

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

      7. associate-*r* 66.9%

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

      8. *-commutative 66.9%

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

      9. unpow2 66.9%

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

   3. Simplified 66.9%

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

   if 4.94999999999999993e33 < s

   1. Initial program 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*l* 59.0%

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

      3. associate-*r* 60.2%

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

      4. *-commutative 60.2%

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

      5. unpow2 60.2%

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

      6. associate-*r* 60.6%

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

      7. associate-*r* 63.2%

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

      8. *-commutative 63.2%

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

      9. unpow2 63.2%

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

   3. Simplified 63.2%

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

      1. add-cube-cbrt 63.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

      2. times-frac 63.3%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

      3. associate-*r* 61.6%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

      4. swap-sqr 84.5%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      5. associate-*r* 94.3%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      6. *-commutative 94.3%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

      7. times-frac 94.3%

         \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

      8. associate-*l* 96.0%

         \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

      9. add-cube-cbrt 96.2%

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

      10. associate-/r* 96.2%

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

   5. Applied egg-rr 96.0%

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

      1. un-div-inv 96.2%

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in x around 0 59.0%

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

      1. unpow2 59.0%

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

      2. associate-*r* 59.1%

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

      3. *-commutative 59.1%

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

      4. associate-*r* 58.2%

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

      5. associate-/r* 58.1%

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

      6. *-commutative 58.1%

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

      7. unpow2 58.1%

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

      8. unpow2 58.1%

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

      9. swap-sqr 68.6%

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

      10. unpow2 68.6%

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

      11. associate-/r* 68.6%

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

      12. unpow2 68.6%

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

      13. swap-sqr 82.4%

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

      14. associate-/r* 82.5%

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

      15. *-rgt-identity 82.5%

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

      16. associate-*r/ 82.4%

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

      17. unpow2 82.4%

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

   10. Simplified 85.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.95 \cdot 10^{+33}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 4: 93.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 4.2e-9)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* x (* x (* (* c s) (* c s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.2e-9) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (x * (x * ((c * s) * (c * s))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 4.2d-9) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (x * (x * ((c * s) * (c * s))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.2e-9) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * ((c * s) * (c * s))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 4.2e-9:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * ((c * s) * (c * s))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 4.2e-9)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(Float64(c * s) * Float64(c * s)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4.2e-9)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (x * (x * ((c * s) * (c * s))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 4.2e-9], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(N[(c * s), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000039e-9

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-*l* 62.3%

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

      3. associate-*r* 62.6%

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

      4. *-commutative 62.6%

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

      5. unpow2 62.6%

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

      6. associate-*r* 66.5%

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

      7. associate-*r* 68.3%

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

      8. *-commutative 68.3%

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

      9. unpow2 68.3%

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

   3. Simplified 68.3%

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

      1. add-cube-cbrt 68.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

      2. times-frac 68.3%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

      3. associate-*r* 67.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

      4. swap-sqr 85.2%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      5. associate-*r* 94.1%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      6. *-commutative 94.1%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

      7. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

      8. associate-*l* 96.0%

         \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

      9. add-cube-cbrt 96.2%

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

      10. associate-/r* 96.7%

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

   5. Applied egg-rr 96.7%

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

      1. un-div-inv 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in x around 0 59.8%

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

      1. unpow2 59.8%

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

      2. associate-*r* 59.9%

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

      3. *-commutative 59.9%

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

      4. associate-*r* 59.7%

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

      5. associate-/r* 59.6%

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

      6. *-commutative 59.6%

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

      7. unpow2 59.6%

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

      8. unpow2 59.6%

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

      9. swap-sqr 69.1%

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

      10. unpow2 69.1%

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

      11. associate-/r* 69.1%

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

      12. unpow2 69.1%

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

      13. swap-sqr 83.1%

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

      14. associate-/r* 83.1%

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

      15. *-rgt-identity 83.1%

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

      16. associate-*r/ 83.1%

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

      17. unpow2 83.1%

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

   10. Simplified 84.3%

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

   if 4.20000000000000039e-9 < x

   1. Initial program 52.8%

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

      1. associate-*r* 55.9%

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

      2. *-commutative 55.9%

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

      3. associate-*r* 56.0%

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

      4. unpow2 56.0%

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

      5. unpow2 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in c around 0 56.0%

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

      1. unpow2 56.0%

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

      2. associate-*l* 64.0%

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

   6. Simplified 64.0%

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

   7. Taylor expanded in c around 0 56.0%

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

      1. associate-*r* 54.4%

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

      2. *-commutative 54.4%

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

      3. *-commutative 54.4%

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

      4. *-commutative 54.4%

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

      5. unpow2 54.4%

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

      6. unpow2 54.4%

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

      7. unswap-sqr 82.7%

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

   9. Simplified 82.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 5: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 1.02 \cdot 10^{-50}:\\ \;\;\;\;{t_0}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= x 1.02e-50)
     (pow t_0 -2.0)
     (/ (cos (* x 2.0)) (* t_0 (* s (* x c)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1.02e-50) {
		tmp = pow(t_0, -2.0);
	} else {
		tmp = cos((x * 2.0)) / (t_0 * (s * (x * c)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 1.02d-50) then
        tmp = t_0 ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (t_0 * (s * (x * c)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1.02e-50) {
		tmp = Math.pow(t_0, -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * (s * (x * c)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 1.02e-50:
		tmp = math.pow(t_0, -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * (s * (x * c)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 1.02e-50)
		tmp = t_0 ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * Float64(s * Float64(x * c))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 1.02e-50)
		tmp = t_0 ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (t_0 * (s * (x * c)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.02e-50], N[Power[t$95$0, -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0199999999999999e-50

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 62.2%

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

      4. *-commutative 62.2%

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

      5. unpow2 62.2%

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

      6. associate-*r* 66.1%

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

      7. associate-*r* 67.9%

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

      8. *-commutative 67.9%

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

      9. unpow2 67.9%

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

   3. Simplified 67.9%

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

      1. add-cube-cbrt 67.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

      2. times-frac 68.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

      3. associate-*r* 66.6%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

      4. swap-sqr 85.3%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      5. associate-*r* 94.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      6. *-commutative 94.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

      8. associate-*l* 95.9%

         \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

      9. add-cube-cbrt 96.1%

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

      10. associate-/r* 96.7%

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

   5. Applied egg-rr 96.6%

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

      1. un-div-inv 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

      2. associate-*r* 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*r* 59.1%

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

      5. associate-/r* 59.1%

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

      6. *-commutative 59.1%

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

      7. unpow2 59.1%

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

      8. unpow2 59.1%

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

      9. swap-sqr 68.8%

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

      10. unpow2 68.8%

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

      11. associate-/r* 68.8%

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

      12. unpow2 68.8%

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

      13. swap-sqr 82.7%

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

      14. associate-/r* 82.7%

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

      15. *-rgt-identity 82.7%

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

      16. associate-*r/ 82.7%

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

      17. unpow2 82.7%

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

   10. Simplified 83.9%

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

   if 1.0199999999999999e-50 < x

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-*r* 51.5%

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

      3. associate-*r* 53.1%

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

      4. unpow2 53.1%

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

      5. unswap-sqr 66.6%

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

      6. unpow2 66.6%

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

      7. swap-sqr 98.3%

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

      8. *-commutative 98.3%

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

      9. *-commutative 98.3%

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

      10. *-commutative 98.3%

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

      11. *-commutative 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in s around 0 94.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{-50}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 6: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 1.1 \cdot 10^{-50}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 1.1e-50)
     (pow (* c (* x s)) -2.0)
     (/ (cos (* x 2.0)) (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 1.1e-50) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 1.1d-50) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 1.1e-50) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 1.1e-50:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 1.1e-50)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 1.1e-50)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.1e-50], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.0999999999999999e-50

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 62.2%

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

      4. *-commutative 62.2%

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

      5. unpow2 62.2%

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

      6. associate-*r* 66.1%

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

      7. associate-*r* 67.9%

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

      8. *-commutative 67.9%

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

      9. unpow2 67.9%

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

   3. Simplified 67.9%

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

      1. add-cube-cbrt 67.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

      2. times-frac 68.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

      3. associate-*r* 66.6%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

      4. swap-sqr 85.3%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      5. associate-*r* 94.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      6. *-commutative 94.0%

         \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

      8. associate-*l* 95.9%

         \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

      9. add-cube-cbrt 96.1%

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

      10. associate-/r* 96.7%

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

   5. Applied egg-rr 96.6%

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

      1. un-div-inv 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

      2. associate-*r* 59.3%

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

      3. *-commutative 59.3%

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

      4. associate-*r* 59.1%

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

      5. associate-/r* 59.1%

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

      6. *-commutative 59.1%

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

      7. unpow2 59.1%

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

      8. unpow2 59.1%

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

      9. swap-sqr 68.8%

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

      10. unpow2 68.8%

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

      11. associate-/r* 68.8%

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

      12. unpow2 68.8%

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

      13. swap-sqr 82.7%

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

      14. associate-/r* 82.7%

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

      15. *-rgt-identity 82.7%

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

      16. associate-*r/ 82.7%

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

      17. unpow2 82.7%

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

   10. Simplified 83.9%

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

   if 1.0999999999999999e-50 < x

   1. Initial program 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-*r* 51.5%

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

      3. associate-*r* 53.1%

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

      4. unpow2 53.1%

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

      5. unswap-sqr 66.6%

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

      6. unpow2 66.6%

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

      7. swap-sqr 98.3%

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

      8. *-commutative 98.3%

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

      9. *-commutative 98.3%

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

      10. *-commutative 98.3%

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

      11. *-commutative 98.3%

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

   3. Simplified 98.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-50}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 7: 79.8% accurate, 3.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (pow (* c (* x s)) -2.0))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return pow((c * (x * s)), -2.0);
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
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

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return math.pow((c * (x * s)), -2.0)

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(c * Float64(x * s)) ^ -2.0
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = (c * (x * s)) ^ -2.0;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 63.8%

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

   1. *-commutative 63.8%

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

   2. associate-*l* 59.1%

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

   3. associate-*r* 59.8%

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

   4. *-commutative 59.8%

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

   5. unpow2 59.8%

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

   6. associate-*r* 64.3%

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

   7. associate-*r* 66.1%

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

   8. *-commutative 66.1%

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

   9. unpow2 66.1%

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

3. Simplified 66.1%

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

   1. add-cube-cbrt 66.1%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

   2. times-frac 66.1%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

   3. associate-*r* 65.1%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

   4. swap-sqr 83.8%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

   5. associate-*r* 92.9%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

   6. *-commutative 92.9%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

   7. times-frac 92.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

   8. associate-*l* 96.4%

      \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

   9. add-cube-cbrt 96.7%

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

   10. associate-/r* 97.1%

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

5. Applied egg-rr 97.0%

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

   1. un-div-inv 97.1%

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

7. Applied egg-rr 97.1%

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

8. Taylor expanded in x around 0 56.1%

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

   1. unpow2 56.1%

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

   2. associate-*r* 56.2%

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

   3. *-commutative 56.2%

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

   4. associate-*r* 56.1%

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

   5. associate-/r* 56.0%

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

   6. *-commutative 56.0%

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

   7. unpow2 56.0%

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

   8. unpow2 56.0%

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

   9. swap-sqr 64.3%

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

   10. unpow2 64.3%

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

   11. associate-/r* 64.3%

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

   12. unpow2 64.3%

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

   13. swap-sqr 76.6%

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

   14. associate-/r* 76.7%

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

   15. *-rgt-identity 76.7%

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

   16. associate-*r/ 76.6%

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

   17. unpow2 76.6%

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

10. Simplified 77.4%

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

11. Final simplification 77.4%

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

Alternative 8: 78.6% accurate, 20.9× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x (* c s))))) (* t_0 t_0)))

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = 1.0d0 / (x * (c * s))
    code = t_0 * t_0
end function

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = 1.0 / (x * (c * s))
	return t_0 * t_0

Julia

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

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = 1.0 / (x * (c * s));
	tmp = t_0 * t_0;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(1.0 / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

   1. *-commutative 63.8%

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

   2. associate-*l* 59.1%

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

   3. associate-*r* 59.8%

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

   4. *-commutative 59.8%

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

   5. unpow2 59.8%

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

   6. associate-*r* 64.3%

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

   7. associate-*r* 66.1%

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

   8. *-commutative 66.1%

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

   9. unpow2 66.1%

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

3. Simplified 66.1%

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

4. Taylor expanded in x around 0 56.1%

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

   1. unpow2 56.1%

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

   2. associate-/r* 56.0%

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

   3. unpow2 56.0%

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

   4. *-commutative 56.0%

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

   5. unpow2 56.0%

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

   6. swap-sqr 64.3%

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

   7. unpow2 64.3%

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

   8. associate-/r* 64.3%

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

   9. unpow2 64.3%

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

   10. swap-sqr 76.6%

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

   11. unpow2 76.6%

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

   12. associate-*r* 77.3%

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

   13. *-commutative 77.3%

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

6. Simplified 77.3%

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

   1. associate-*r* 77.2%

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

   2. unpow-prod-down 68.0%

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

   3. pow2 68.0%

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

8. Applied egg-rr 68.0%

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

   1. add-sqr-sqrt 68.0%

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

   2. sqrt-div 68.0%

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

   3. metadata-eval 68.0%

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

   4. *-commutative 68.0%

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

   5. sqrt-prod 68.0%

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

   6. sqrt-prod 29.7%

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

   7. add-sqr-sqrt 48.4%

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

   8. unpow2 48.4%

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

   9. sqrt-prod 28.7%

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

   10. add-sqr-sqrt 53.3%

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

   11. *-commutative 53.3%

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

   12. sqrt-div 53.3%

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

   13. metadata-eval 53.3%

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

   14. *-commutative 53.3%

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

   15. sqrt-prod 53.2%

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

   16. sqrt-prod 27.6%

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

   17. add-sqr-sqrt 59.0%

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

   18. unpow2 59.0%

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

   19. sqrt-prod 43.1%

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

10. Applied egg-rr 77.2%

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

11. Final simplification 77.2%

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

Alternative 9: 79.8% accurate, 20.9× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ t_0 \cdot t_0 \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 c) (* x s)))) (* t_0 t_0)))

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = (1.0d0 / c) / (x * s)
    code = t_0 * t_0
end function

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = (1.0 / c) / (x * s)
	return t_0 * t_0

Julia

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

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = (1.0 / c) / (x * s);
	tmp = t_0 * t_0;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

   1. associate-/r* 63.9%

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

   2. unpow2 63.9%

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

   3. *-commutative 63.9%

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

   4. unpow2 63.9%

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

3. Simplified 63.9%

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

4. Taylor expanded in x around 0 59.0%

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

   1. unpow2 59.0%

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

6. Simplified 59.0%

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

7. Taylor expanded in x around 0 59.0%

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

   1. unpow2 59.0%

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

   2. associate-*l* 63.1%

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

9. Simplified 63.1%

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

    1. inv-pow 63.1%

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

    2. pow-prod-down 63.5%

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

    3. inv-pow 63.5%

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

    4. inv-pow 63.5%

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

11. Applied egg-rr 63.5%

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

    1. add-sqr-sqrt 63.4%

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

    2. sqrt-div 63.5%

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

    3. sqrt-prod 29.5%

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

    4. add-sqr-sqrt 47.9%

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

    5. associate-*r* 47.9%

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

    6. *-commutative 47.9%

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

    7. sqrt-prod 27.5%

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

    8. add-sqr-sqrt 50.2%

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

    9. *-commutative 50.2%

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

    10. sqrt-div 50.2%

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

    11. sqrt-prod 24.2%

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

    12. add-sqr-sqrt 50.1%

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

    13. associate-*r* 51.8%

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

    14. *-commutative 51.8%

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

    15. sqrt-prod 37.8%

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

    16. add-sqr-sqrt 77.4%

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

    17. *-commutative 77.4%

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

13. Applied egg-rr 77.4%

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

14. Final simplification 77.4%

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

Alternative 10: 77.7% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ 1.0 (* (* c (* x s)) (* s (* x c)))))

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
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)) * (s * (x * c)))
end function

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return 1.0 / ((c * (x * s)) * (s * (x * c)))

Julia

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

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = 1.0 / ((c * (x * s)) * (s * (x * c)));
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(1.0 / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

   1. *-commutative 63.8%

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

   2. associate-*l* 59.1%

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

   3. associate-*r* 59.8%

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

   4. *-commutative 59.8%

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

   5. unpow2 59.8%

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

   6. associate-*r* 64.3%

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

   7. associate-*r* 66.1%

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

   8. *-commutative 66.1%

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

   9. unpow2 66.1%

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

3. Simplified 66.1%

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

4. Taylor expanded in x around 0 56.1%

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

   1. unpow2 56.1%

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

   2. associate-/r* 56.0%

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

   3. unpow2 56.0%

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

   4. *-commutative 56.0%

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

   5. unpow2 56.0%

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

   6. swap-sqr 64.3%

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

   7. unpow2 64.3%

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

   8. associate-/r* 64.3%

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

   9. unpow2 64.3%

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

   10. swap-sqr 76.6%

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

   11. unpow2 76.6%

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

   12. associate-*r* 77.3%

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

   13. *-commutative 77.3%

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

6. Simplified 77.3%

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

   1. *-commutative 77.3%

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

   2. associate-*r* 76.6%

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

   3. pow2 76.6%

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

8. Applied egg-rr 76.6%

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

9. Taylor expanded in s around 0 75.5%

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

10. Final simplification 75.5%

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

Alternative 11: 77.9% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
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 = s * (x * c)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

   1. *-commutative 63.8%

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

   2. associate-*l* 59.1%

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

   3. associate-*r* 59.8%

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

   4. *-commutative 59.8%

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

   5. unpow2 59.8%

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

   6. associate-*r* 64.3%

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

   7. associate-*r* 66.1%

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

   8. *-commutative 66.1%

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

   9. unpow2 66.1%

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

3. Simplified 66.1%

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

4. Taylor expanded in x around 0 56.1%

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

   1. unpow2 56.1%

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

   2. associate-/r* 56.0%

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

   3. unpow2 56.0%

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

   4. *-commutative 56.0%

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

   5. unpow2 56.0%

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

   6. swap-sqr 64.3%

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

   7. unpow2 64.3%

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

   8. associate-/r* 64.3%

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

   9. unpow2 64.3%

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

   10. swap-sqr 76.6%

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

   11. unpow2 76.6%

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

   12. associate-*r* 77.3%

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

   13. *-commutative 77.3%

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

6. Simplified 77.3%

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

   1. *-commutative 77.3%

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

   2. associate-*r* 76.6%

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

   3. pow2 76.6%

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

8. Applied egg-rr 76.6%

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

9. Final simplification 76.6%

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

Alternative 12: 78.0% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c)))) (/ (/ 1.0 t_0) t_0)))

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
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 = s * (x * c)
    code = (1.0d0 / t_0) / t_0
end function

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	return (1.0 / t_0) / t_0

Julia

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

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = s * (x * c);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

   1. *-commutative 63.8%

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

   2. associate-*l* 59.1%

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

   3. associate-*r* 59.8%

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

   4. *-commutative 59.8%

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

   5. unpow2 59.8%

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

   6. associate-*r* 64.3%

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

   7. associate-*r* 66.1%

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

   8. *-commutative 66.1%

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

   9. unpow2 66.1%

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

3. Simplified 66.1%

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

   1. add-cube-cbrt 66.1%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{s \cdot \left(\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)\right)} \]

   2. times-frac 66.1%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

   3. associate-*r* 65.1%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(c \cdot c\right)\right) \cdot s}} \]

   4. swap-sqr 83.8%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

   5. associate-*r* 92.9%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

   6. *-commutative 92.9%

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}} \]

   7. times-frac 92.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

   8. associate-*l* 96.4%

      \[\leadsto \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]

   9. add-cube-cbrt 96.7%

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

   10. associate-/r* 97.1%

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

5. Applied egg-rr 97.0%

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

   1. un-div-inv 97.1%

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

7. Applied egg-rr 97.1%

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

8. Taylor expanded in x around 0 76.7%

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

9. Final simplification 76.7%

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

Reproduce

herbie shell --seed 2023252 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))