mixedcos

Percentage Accurate: 66.6% → 99.3%

Time: 13.5s

Alternatives: 14

Speedup: 24.1×

• 31.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 7.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{t_0}}{t_0}\\ \end{array} \end{array} \]

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 7.8e-54)
     (* (/ 1.0 c) (/ (/ 1.0 (* c (* x s))) (* x s)))
     (/ (/ (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 <= 7.8e-54) {
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s));
	} 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 <= 7.8d-54) then
        tmp = (1.0d0 / c) * ((1.0d0 / (c * (x * s))) / (x * s))
    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 <= 7.8e-54) {
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s));
	} 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 <= 7.8e-54:
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s))
	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 <= 7.8e-54)
		tmp = Float64(Float64(1.0 / c) * Float64(Float64(1.0 / Float64(c * Float64(x * s))) / Float64(x * s)));
	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 <= 7.8e-54)
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s));
	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, 7.8e-54], N[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]), $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 < 7.8e-54

   1. Initial program 62.1%

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

      1. associate-/r* 62.1%

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

      2. unpow2 62.1%

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

      3. *-commutative 62.1%

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

      4. unpow2 62.1%

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

   3. Simplified 62.1%

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

      1. associate-*r* 57.2%

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

      2. pow2 57.2%

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

      3. pow2 57.2%

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

      4. pow-prod-down 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. div-inv 76.8%

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

      2. *-commutative 76.8%

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

      3. frac-times 77.5%

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

      4. pow2 77.5%

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

      5. *-commutative 77.5%

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

      6. unpow-prod-down 97.1%

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

      7. *-commutative 97.1%

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

      8. *-commutative 97.1%

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

      9. associate-*r* 96.1%

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

      10. pow2 96.1%

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

      11. frac-times 96.5%

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

      12. *-commutative 96.5%

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

      13. associate-*l/ 96.5%

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

      14. associate-*r* 94.4%

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

      15. *-commutative 94.4%

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

      16. *-commutative 94.4%

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

   7. Applied egg-rr 89.6%

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

   8. Taylor expanded in x around 0 77.3%

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

   if 7.8e-54 < x

   1. Initial program 70.9%

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

      1. associate-*r* 69.8%

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

      2. *-commutative 69.8%

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

      3. *-commutative 69.8%

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

      4. associate-*r* 65.7%

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

      5. *-commutative 65.7%

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

      6. unpow2 65.7%

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

      7. unpow2 65.7%

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

   3. Simplified 65.7%

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

      1. *-un-lft-identity 65.7%

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

      2. associate-*r* 69.8%

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

      3. *-commutative 69.8%

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

      4. add-sqr-sqrt 69.8%

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

      5. times-frac 69.7%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-un-lft-identity 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*r* 96.9%

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

   10. Simplified 96.9%

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

   11. Taylor expanded in x around 0 95.4%

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

       1. associate-*r* 99.5%

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

       2. *-commutative 99.5%

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

       3. associate-*r* 96.9%

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

   13. Simplified 96.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 2: 88.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 9 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 3.58 \cdot 10^{+152}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+213}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

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 (cos (* x 2.0))))
   (if (<= x 9e-9)
     (pow (* c (* x s)) -2.0)
     (if (<= x 3.58e+152)
       (/ t_0 (* s (* (* x x) (* c (* c s)))))
       (if (<= x 4.9e+213)
         (/ t_0 (* x (* x (* (* c c) (* s s)))))
         (/ (/ 1.0 s) (* (* x c) (* 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 = cos((x * 2.0));
	double tmp;
	if (x <= 9e-9) {
		tmp = pow((c * (x * s)), -2.0);
	} else if (x <= 3.58e+152) {
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	} else if (x <= 4.9e+213) {
		tmp = t_0 / (x * (x * ((c * c) * (s * s))));
	} else {
		tmp = (1.0 / s) / ((x * c) * (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 = cos((x * 2.0d0))
    if (x <= 9d-9) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else if (x <= 3.58d+152) then
        tmp = t_0 / (s * ((x * x) * (c * (c * s))))
    else if (x <= 4.9d+213) then
        tmp = t_0 / (x * (x * ((c * c) * (s * s))))
    else
        tmp = (1.0d0 / s) / ((x * c) * (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 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 9e-9) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else if (x <= 3.58e+152) {
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	} else if (x <= 4.9e+213) {
		tmp = t_0 / (x * (x * ((c * c) * (s * s))));
	} else {
		tmp = (1.0 / s) / ((x * c) * (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 = math.cos((x * 2.0))
	tmp = 0
	if x <= 9e-9:
		tmp = math.pow((c * (x * s)), -2.0)
	elif x <= 3.58e+152:
		tmp = t_0 / (s * ((x * x) * (c * (c * s))))
	elif x <= 4.9e+213:
		tmp = t_0 / (x * (x * ((c * c) * (s * s))))
	else:
		tmp = (1.0 / s) / ((x * c) * (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 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 9e-9)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	elseif (x <= 3.58e+152)
		tmp = Float64(t_0 / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(c * s)))));
	elseif (x <= 4.9e+213)
		tmp = Float64(t_0 / Float64(x * Float64(x * Float64(Float64(c * c) * Float64(s * s)))));
	else
		tmp = Float64(Float64(1.0 / s) / Float64(Float64(x * c) * 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 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 9e-9)
		tmp = (c * (x * s)) ^ -2.0;
	elseif (x <= 3.58e+152)
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	elseif (x <= 4.9e+213)
		tmp = t_0 / (x * (x * ((c * c) * (s * s))));
	else
		tmp = (1.0 / s) / ((x * c) * (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[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 9e-9], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x, 3.58e+152], N[(t$95$0 / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9e+213], N[(t$95$0 / N[(x * N[(x * N[(N[(c * c), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / s), $MachinePrecision] / N[(N[(x * c), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.99999999999999953e-9

   1. Initial program 62.0%

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

      1. associate-*r* 62.7%

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

      2. *-commutative 62.7%

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

      3. *-commutative 62.7%

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

      4. associate-*r* 61.3%

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

      5. *-commutative 61.3%

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

      6. unpow2 61.3%

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

      7. unpow2 61.3%

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

   3. Simplified 61.3%

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

      1. *-un-lft-identity 61.3%

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

      2. associate-*r* 62.7%

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

      3. *-commutative 62.7%

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

      4. add-sqr-sqrt 62.6%

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

      5. times-frac 62.6%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in x around 0 51.7%

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

   8. Simplified 81.3%

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

   if 8.99999999999999953e-9 < x < 3.58000000000000006e152

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-*l* 71.2%

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

      3. associate-*r* 71.9%

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

      4. *-commutative 71.9%

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

      5. unpow2 71.9%

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

      6. associate-*r* 73.7%

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

      7. associate-*r* 77.3%

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

      8. *-commutative 77.3%

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

      9. unpow2 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in c around 0 77.3%

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

      1. *-commutative 77.3%

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

      2. unpow2 77.3%

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

      3. associate-*r* 89.1%

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

   6. Simplified 89.1%

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

   if 3.58000000000000006e152 < x < 4.89999999999999997e213

   1. Initial program 75.1%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-*r* 69.0%

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

      5. *-commutative 69.0%

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

      6. unpow2 69.0%

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

      7. unpow2 69.0%

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

   3. Simplified 69.0%

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

   if 4.89999999999999997e213 < x

   1. Initial program 77.7%

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

      1. associate-/r* 70.0%

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

      2. unpow2 70.0%

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

      3. *-commutative 70.0%

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

      4. unpow2 70.0%

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

   3. Simplified 70.0%

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

      1. associate-*r* 50.3%

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

      2. pow2 50.3%

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

      3. pow2 50.3%

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

      4. pow-prod-down 70.4%

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

   5. Applied egg-rr 70.4%

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

      1. div-inv 70.2%

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

      2. *-commutative 70.2%

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

      3. frac-times 77.8%

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

      4. pow2 77.8%

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

      5. *-commutative 77.8%

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

      6. unpow-prod-down 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r* 99.8%

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

      10. pow2 99.8%

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

      11. frac-times 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-*l/ 99.4%

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

      14. associate-*r* 99.7%

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

      15. *-commutative 99.7%

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

      16. *-commutative 99.7%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 64.6%

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

      1. frac-times 64.6%

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

      2. *-un-lft-identity 64.6%

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

      3. associate-*r* 64.6%

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

      4. *-commutative 64.6%

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

      5. associate-*r* 64.6%

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

      6. inv-pow 64.6%

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

      7. *-commutative 64.6%

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

      8. associate-*r* 64.6%

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

      9. *-commutative 64.6%

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

      10. associate-*r* 64.6%

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

      11. pow1 64.6%

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

      12. pow-div 64.6%

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

      13. metadata-eval 64.6%

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

      14. metadata-eval 64.6%

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

      15. pow-prod-up 64.6%

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

      16. inv-pow 64.6%

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

      17. inv-pow 64.6%

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

      18. associate-/r* 64.6%

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

      19. frac-times 64.6%

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

      20. *-un-lft-identity 64.6%

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

   10. Applied egg-rr 64.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(c \cdot x\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 3.58 \cdot 10^{+152}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+213}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 3: 87.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 9.6 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 3.58 \cdot 10^{+152}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

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 9.6e-9)
     (pow (* c (* x s)) -2.0)
     (if (<= x 3.58e+152)
       (/ t_0 (* s (* (* x x) (* c (* c s)))))
       (/ t_0 (* x (* (* x (* c 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 <= 9.6e-9) {
		tmp = pow((c * (x * s)), -2.0);
	} else if (x <= 3.58e+152) {
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	} else {
		tmp = t_0 / (x * ((x * (c * 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 <= 9.6d-9) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else if (x <= 3.58d+152) then
        tmp = t_0 / (s * ((x * x) * (c * (c * s))))
    else
        tmp = t_0 / (x * ((x * (c * 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 <= 9.6e-9) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else if (x <= 3.58e+152) {
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	} else {
		tmp = t_0 / (x * ((x * (c * 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 <= 9.6e-9:
		tmp = math.pow((c * (x * s)), -2.0)
	elif x <= 3.58e+152:
		tmp = t_0 / (s * ((x * x) * (c * (c * s))))
	else:
		tmp = t_0 / (x * ((x * (c * 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 <= 9.6e-9)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	elseif (x <= 3.58e+152)
		tmp = Float64(t_0 / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(c * s)))));
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(x * Float64(c * 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 <= 9.6e-9)
		tmp = (c * (x * s)) ^ -2.0;
	elseif (x <= 3.58e+152)
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	else
		tmp = t_0 / (x * ((x * (c * 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, 9.6e-9], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x, 3.58e+152], N[(t$95$0 / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(N[(x * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.5999999999999999e-9

   1. Initial program 62.0%

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

      1. associate-*r* 62.7%

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

      2. *-commutative 62.7%

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

      3. *-commutative 62.7%

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

      4. associate-*r* 61.3%

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

      5. *-commutative 61.3%

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

      6. unpow2 61.3%

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

      7. unpow2 61.3%

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

   3. Simplified 61.3%

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

      1. *-un-lft-identity 61.3%

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

      2. associate-*r* 62.7%

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

      3. *-commutative 62.7%

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

      4. add-sqr-sqrt 62.6%

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

      5. times-frac 62.6%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in x around 0 51.7%

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

   8. Simplified 81.3%

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

   if 9.5999999999999999e-9 < x < 3.58000000000000006e152

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-*l* 71.2%

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

      3. associate-*r* 71.9%

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

      4. *-commutative 71.9%

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

      5. unpow2 71.9%

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

      6. associate-*r* 73.7%

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

      7. associate-*r* 77.3%

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

      8. *-commutative 77.3%

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

      9. unpow2 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in c around 0 77.3%

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

      1. *-commutative 77.3%

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

      2. unpow2 77.3%

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

      3. associate-*r* 89.1%

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

   6. Simplified 89.1%

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

   if 3.58000000000000006e152 < x

   1. Initial program 76.1%

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

      1. associate-*r* 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r* 75.8%

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

      4. unpow2 75.8%

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

      5. unpow2 75.8%

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

   3. Simplified 75.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.6 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 3.58 \cdot 10^{+152}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 4: 86.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

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 9.5e-9)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* s (* (* x x) (* c (* 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 <= 9.5e-9) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (c * (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 <= 9.5d-9) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (c * (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 <= 9.5e-9) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (c * (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 <= 9.5e-9:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (c * (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 <= 9.5e-9)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(c * 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 <= 9.5e-9)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (s * ((x * x) * (c * (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, 9.5e-9], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000007e-9

   1. Initial program 62.0%

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

      1. associate-*r* 62.7%

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

      2. *-commutative 62.7%

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

      3. *-commutative 62.7%

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

      4. associate-*r* 61.3%

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

      5. *-commutative 61.3%

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

      6. unpow2 61.3%

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

      7. unpow2 61.3%

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

   3. Simplified 61.3%

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

      1. *-un-lft-identity 61.3%

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

      2. associate-*r* 62.7%

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

      3. *-commutative 62.7%

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

      4. add-sqr-sqrt 62.6%

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

      5. times-frac 62.6%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in x around 0 51.7%

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

   8. Simplified 81.3%

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

   if 9.5000000000000007e-9 < x

   1. Initial program 73.7%

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

      1. *-commutative 73.7%

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

      2. associate-*l* 62.7%

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

      3. associate-*r* 62.8%

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

      4. *-commutative 62.8%

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

      5. unpow2 62.8%

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

      6. associate-*r* 64.2%

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

      7. associate-*r* 66.0%

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

      8. *-commutative 66.0%

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

      9. unpow2 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in c around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. unpow2 66.0%

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

      3. associate-*r* 75.7%

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

   6. Simplified 75.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 5: 96.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 2.55 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{t_0}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

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 2.55e-89)
     (* (/ 1.0 c) (/ (/ 1.0 t_0) (* x s)))
     (/ (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 <= 2.55e-89) {
		tmp = (1.0 / c) * ((1.0 / t_0) / (x * s));
	} 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 <= 2.55d-89) then
        tmp = (1.0d0 / c) * ((1.0d0 / t_0) / (x * s))
    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 <= 2.55e-89) {
		tmp = (1.0 / c) * ((1.0 / t_0) / (x * s));
	} 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 <= 2.55e-89:
		tmp = (1.0 / c) * ((1.0 / t_0) / (x * s))
	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 <= 2.55e-89)
		tmp = Float64(Float64(1.0 / c) * Float64(Float64(1.0 / t_0) / Float64(x * s)));
	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 <= 2.55e-89)
		tmp = (1.0 / c) * ((1.0 / t_0) / (x * s));
	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, 2.55e-89], N[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]), $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 < 2.55000000000000002e-89

   1. Initial program 61.0%

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

      1. associate-/r* 61.0%

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

      2. unpow2 61.0%

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

      3. *-commutative 61.0%

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

      4. unpow2 61.0%

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

   3. Simplified 61.0%

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

      1. associate-*r* 55.9%

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

      2. pow2 55.9%

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

      3. pow2 55.9%

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

      4. pow-prod-down 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. div-inv 75.8%

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

      2. *-commutative 75.8%

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

      3. frac-times 76.5%

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

      4. pow2 76.5%

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

      5. *-commutative 76.5%

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

      6. unpow-prod-down 97.0%

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

      7. *-commutative 97.0%

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

      8. *-commutative 97.0%

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

      9. associate-*r* 95.9%

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

      10. pow2 95.9%

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

      11. frac-times 96.4%

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

      12. *-commutative 96.4%

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

      13. associate-*l/ 96.4%

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

      14. associate-*r* 94.1%

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

      15. *-commutative 94.1%

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

      16. *-commutative 94.1%

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

   7. Applied egg-rr 89.1%

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

   8. Taylor expanded in x around 0 76.4%

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

   if 2.55000000000000002e-89 < x

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*r* 64.9%

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

      3. associate-*r* 65.0%

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

      4. unpow2 65.0%

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

      5. unswap-sqr 77.3%

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

      6. unpow2 77.3%

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

      7. swap-sqr 95.2%

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

      8. *-commutative 95.2%

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

      9. *-commutative 95.2%

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

      10. *-commutative 95.2%

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

      11. *-commutative 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in s around 0 93.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 6: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 2.45 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

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 2.45e-82)
     (* (/ 1.0 c) (/ (/ 1.0 (* c (* x s))) (* x s)))
     (/ (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 <= 2.45e-82) {
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s));
	} 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 <= 2.45d-82) then
        tmp = (1.0d0 / c) * ((1.0d0 / (c * (x * s))) / (x * s))
    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 <= 2.45e-82) {
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s));
	} 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 <= 2.45e-82:
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s))
	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 <= 2.45e-82)
		tmp = Float64(Float64(1.0 / c) * Float64(Float64(1.0 / Float64(c * Float64(x * s))) / Float64(x * s)));
	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 <= 2.45e-82)
		tmp = (1.0 / c) * ((1.0 / (c * (x * s))) / (x * s));
	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, 2.45e-82], N[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]), $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 < 2.4500000000000001e-82

   1. Initial program 61.6%

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

      1. associate-/r* 61.6%

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

      2. unpow2 61.6%

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

      3. *-commutative 61.6%

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

      4. unpow2 61.6%

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

   3. Simplified 61.6%

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

      1. associate-*r* 56.6%

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

      2. pow2 56.6%

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

      3. pow2 56.6%

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

      4. pow-prod-down 76.7%

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

   5. Applied egg-rr 76.7%

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

      1. div-inv 76.2%

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

      2. *-commutative 76.2%

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

      3. frac-times 76.9%

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

      4. pow2 76.9%

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

      5. *-commutative 76.9%

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

      6. unpow-prod-down 97.0%

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

      7. *-commutative 97.0%

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

      8. *-commutative 97.0%

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

      9. associate-*r* 96.0%

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

      10. pow2 96.0%

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

      11. frac-times 96.4%

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

      12. *-commutative 96.4%

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

      13. associate-*l/ 96.4%

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

      14. associate-*r* 94.2%

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

      15. *-commutative 94.2%

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

      16. *-commutative 94.2%

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

   7. Applied egg-rr 89.3%

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

   8. Taylor expanded in x around 0 76.7%

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

   if 2.4500000000000001e-82 < x

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-*r* 63.4%

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

      3. associate-*r* 63.5%

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

      4. unpow2 63.5%

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

      5. unswap-sqr 76.3%

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

      6. unpow2 76.3%

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

      7. swap-sqr 95.0%

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

      8. *-commutative 95.0%

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

      9. *-commutative 95.0%

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

      10. *-commutative 95.0%

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

      11. *-commutative 95.0%

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

   3. Simplified 95.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 7: 79.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 2.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{0.5 - x \cdot x}{\frac{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

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 2.7e-78)
   (/ (- 0.5 (* x x)) (/ (pow (* s (* x c)) 2.0) 2.0))
   (/ 1.0 (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 <= 2.7e-78) {
		tmp = (0.5 - (x * x)) / (pow((s * (x * c)), 2.0) / 2.0);
	} else {
		tmp = 1.0 / 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 <= 2.7d-78) then
        tmp = (0.5d0 - (x * x)) / (((s * (x * c)) ** 2.0d0) / 2.0d0)
    else
        tmp = 1.0d0 / ((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 <= 2.7e-78) {
		tmp = (0.5 - (x * x)) / (Math.pow((s * (x * c)), 2.0) / 2.0);
	} else {
		tmp = 1.0 / 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 <= 2.7e-78:
		tmp = (0.5 - (x * x)) / (math.pow((s * (x * c)), 2.0) / 2.0)
	else:
		tmp = 1.0 / 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 <= 2.7e-78)
		tmp = Float64(Float64(0.5 - Float64(x * x)) / Float64((Float64(s * Float64(x * c)) ^ 2.0) / 2.0));
	else
		tmp = Float64(1.0 / (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 <= 2.7e-78)
		tmp = (0.5 - (x * x)) / (((s * (x * c)) ^ 2.0) / 2.0);
	else
		tmp = 1.0 / ((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, 2.7e-78], N[(N[(0.5 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 2.69999999999999994e-78

   1. Initial program 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-*r* 58.4%

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

      3. associate-*r* 58.4%

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

      4. unpow2 58.4%

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

      5. unswap-sqr 74.2%

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

      6. unpow2 74.2%

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

      7. swap-sqr 96.9%

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

      8. *-commutative 96.9%

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

      9. *-commutative 96.9%

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

      10. *-commutative 96.9%

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

      11. *-commutative 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around 0 29.1%

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

      1. unpow2 29.1%

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

      2. unpow2 29.1%

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

      3. associate-*r* 27.8%

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

      4. *-commutative 27.8%

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

      5. unpow2 27.8%

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

      6. associate-*r/ 27.8%

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

      7. metadata-eval 27.8%

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

      8. unpow2 27.8%

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

      9. unpow2 27.8%

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

   6. Simplified 27.8%

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

      1. associate-/r* 27.8%

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

      2. clear-num 27.8%

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

      3. frac-sub 12.2%

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

      4. unswap-sqr 12.2%

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

      5. *-commutative 12.2%

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

      6. *-commutative 12.2%

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

      7. pow2 12.2%

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

      8. unswap-sqr 15.5%

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

      9. *-commutative 15.5%

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

      10. *-commutative 15.5%

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

      11. pow2 15.5%

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

   8. Applied egg-rr 29.2%

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

      1. associate-*r/ 29.2%

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

      2. lft-mult-inverse 55.4%

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

      3. metadata-eval 55.4%

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

      4. *-rgt-identity 55.4%

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

      5. associate-*r/ 55.4%

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

      6. unpow2 55.4%

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

      7. swap-sqr 66.3%

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

      8. unpow2 66.3%

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

      9. associate-*r* 66.3%

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

      10. *-commutative 66.3%

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

      11. associate-*r* 67.3%

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

   10. Simplified 67.3%

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

   if 2.69999999999999994e-78 < s

   1. Initial program 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*r* 58.3%

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

      3. associate-*r* 59.6%

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

      4. unpow2 59.6%

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

      5. unswap-sqr 73.9%

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

      6. unpow2 73.9%

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

      7. swap-sqr 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. *-commutative 98.4%

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

      11. *-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. unpow2 55.3%

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

      2. unpow2 55.3%

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

      3. *-commutative 55.3%

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

      4. associate-*r* 55.5%

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

      5. *-commutative 55.5%

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

      6. unpow2 55.5%

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

      7. swap-sqr 64.0%

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

      8. unpow2 64.0%

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

      9. associate-*r* 68.9%

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

      10. *-commutative 68.9%

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

      11. *-commutative 68.9%

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

      12. associate-*r* 64.0%

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

      13. unpow2 64.0%

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

      14. swap-sqr 81.9%

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

      15. unpow2 81.9%

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

      16. associate-*r* 80.5%

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

      17. *-commutative 80.5%

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

      18. *-commutative 80.5%

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

      19. *-commutative 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{0.5 - x \cdot x}{\frac{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 8: 79.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}} \end{array} \]

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 (/ 1.0 (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 1.0 / 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 = 1.0d0 / ((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 1.0 / 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 1.0 / 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(1.0 / (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 = 1.0 / ((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[(1.0 / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

   1. *-commutative 64.3%

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

   2. associate-*r* 58.4%

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

   3. associate-*r* 58.7%

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

   4. unpow2 58.7%

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

   5. unswap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. swap-sqr 97.3%

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

   8. *-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. *-commutative 97.3%

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

   11. *-commutative 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 53.1%

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

   1. unpow2 53.1%

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

   2. unpow2 53.1%

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

   3. *-commutative 53.1%

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

   4. associate-*r* 53.5%

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

   5. *-commutative 53.5%

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

   6. unpow2 53.5%

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

   7. swap-sqr 64.8%

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

   8. unpow2 64.8%

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

   9. associate-*r* 72.2%

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

   10. *-commutative 72.2%

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

   11. *-commutative 72.2%

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

   12. associate-*r* 64.8%

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

   13. unpow2 64.8%

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

   14. swap-sqr 79.6%

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

   15. unpow2 79.6%

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

   16. associate-*r* 78.4%

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

   17. *-commutative 78.4%

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

   18. *-commutative 78.4%

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

   19. *-commutative 78.4%

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

6. Simplified 78.4%

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

7. Final simplification 78.4%

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

Alternative 9: 79.9% 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 64.3%

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

   1. associate-*r* 64.9%

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

   2. *-commutative 64.9%

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

   3. *-commutative 64.9%

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

   4. associate-*r* 62.8%

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

   5. *-commutative 62.8%

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

   6. unpow2 62.8%

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

   7. unpow2 62.8%

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

3. Simplified 62.8%

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

   1. *-un-lft-identity 62.8%

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

   2. associate-*r* 64.9%

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

   3. *-commutative 64.9%

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

   4. add-sqr-sqrt 64.8%

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

   5. times-frac 64.8%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 53.5%

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

8. Simplified 78.4%

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

9. Final simplification 78.4%

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

Alternative 10: 79.7% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;c \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= c 1.7e-27)
   (/ 1.0 (* (* c (* x s)) (* x (* c s))))
   (/ 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) {
	double tmp;
	if (c <= 1.7e-27) {
		tmp = 1.0 / ((c * (x * s)) * (x * (c * s)));
	} else {
		tmp = 1.0 / (c * ((x * s) * (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) :: tmp
    if (c <= 1.7d-27) then
        tmp = 1.0d0 / ((c * (x * s)) * (x * (c * s)))
    else
        tmp = 1.0d0 / (c * ((x * s) * (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 tmp;
	if (c <= 1.7e-27) {
		tmp = 1.0 / ((c * (x * s)) * (x * (c * s)));
	} else {
		tmp = 1.0 / (c * ((x * s) * (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):
	tmp = 0
	if c <= 1.7e-27:
		tmp = 1.0 / ((c * (x * s)) * (x * (c * s)))
	else:
		tmp = 1.0 / (c * ((x * s) * (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)
	tmp = 0.0
	if (c <= 1.7e-27)
		tmp = Float64(1.0 / Float64(Float64(c * Float64(x * s)) * Float64(x * Float64(c * s))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s) * 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)
	tmp = 0.0;
	if (c <= 1.7e-27)
		tmp = 1.0 / ((c * (x * s)) * (x * (c * s)));
	else
		tmp = 1.0 / (c * ((x * s) * (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_] := If[LessEqual[c, 1.7e-27], N[(1.0 / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(N[(x * s), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.69999999999999985e-27

   1. Initial program 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-*r* 56.8%

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

      3. associate-*r* 57.4%

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

      4. unpow2 57.4%

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

      5. unswap-sqr 73.7%

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

      6. unpow2 73.7%

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

      7. swap-sqr 96.9%

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

      8. *-commutative 96.9%

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

      9. *-commutative 96.9%

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

      10. *-commutative 96.9%

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

      11. *-commutative 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around 0 49.8%

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

      1. unpow2 49.8%

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

      2. unpow2 49.8%

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

      3. *-commutative 49.8%

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

      4. associate-*r* 50.4%

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

      5. *-commutative 50.4%

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

      6. unpow2 50.4%

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

      7. swap-sqr 61.0%

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

      8. unpow2 61.0%

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

      9. associate-*r* 67.0%

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

      10. *-commutative 67.0%

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

      11. *-commutative 67.0%

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

      12. associate-*r* 61.0%

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

      13. unpow2 61.0%

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

      14. swap-sqr 76.4%

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

      15. unpow2 76.4%

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

      16. associate-*r* 76.1%

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

      17. *-commutative 76.1%

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

      18. *-commutative 76.1%

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

      19. *-commutative 76.1%

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

   6. Simplified 76.1%

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

      1. *-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*r* 75.2%

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

      4. pow2 75.2%

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

   8. Applied egg-rr 75.2%

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

   9. Taylor expanded in x around 0 74.7%

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

   if 1.69999999999999985e-27 < c

   1. Initial program 66.9%

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

      1. associate-/r* 66.9%

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

      2. unpow2 66.9%

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

      3. *-commutative 66.9%

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

      4. unpow2 66.9%

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

   3. Simplified 66.9%

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

      1. associate-*r* 62.1%

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

      2. pow2 62.1%

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

      3. pow2 62.1%

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

      4. pow-prod-down 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. div-inv 78.9%

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

      2. *-commutative 78.9%

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

      3. frac-times 79.9%

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

      4. pow2 79.9%

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

      5. *-commutative 79.9%

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

      6. unpow-prod-down 95.6%

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

      7. *-commutative 95.6%

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

      8. *-commutative 95.6%

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

      9. associate-*r* 96.7%

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

      10. pow2 96.7%

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

      11. frac-times 97.0%

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

      12. *-commutative 97.0%

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

      13. associate-*l/ 97.1%

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

      14. associate-*r* 93.4%

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

      15. *-commutative 93.4%

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

      16. *-commutative 93.4%

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

   7. Applied egg-rr 89.7%

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

   8. Taylor expanded in x around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-/l/ 79.9%

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

      3. frac-times 80.0%

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

      4. metadata-eval 80.0%

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

      5. *-commutative 80.0%

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

      6. associate-*r* 77.6%

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

      7. *-commutative 77.6%

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

      8. associate-*r* 80.0%

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

   10. Applied egg-rr 80.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 11: 79.3% accurate, 20.9× speedup

Math

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

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) (/ (/ 1.0 (* c (* x s))) (* x s))))

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) * ((1.0 / (c * (x * s))) / (x * s));
}

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) * ((1.0d0 / (c * (x * s))) / (x * s))
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) * ((1.0 / (c * (x * s))) / (x * s));
}

Python

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

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	return Float64(Float64(1.0 / c) * Float64(Float64(1.0 / Float64(c * Float64(x * s))) / Float64(x * s)))
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) * ((1.0 / (c * (x * s))) / (x * s));
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[(N[(1.0 / c), $MachinePrecision] * N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

   1. associate-/r* 64.2%

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

   2. unpow2 64.2%

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

   3. *-commutative 64.2%

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

   4. unpow2 64.2%

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

3. Simplified 64.2%

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

   1. associate-*r* 58.6%

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

   2. pow2 58.6%

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

   3. pow2 58.6%

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

   4. pow-prod-down 76.5%

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

5. Applied egg-rr 76.5%

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

   1. div-inv 76.1%

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

   2. *-commutative 76.1%

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

   3. frac-times 76.7%

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

   4. pow2 76.7%

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

   5. *-commutative 76.7%

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

   6. unpow-prod-down 96.8%

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

   7. *-commutative 96.8%

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

   8. *-commutative 96.8%

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

   9. associate-*r* 96.4%

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

   10. pow2 96.4%

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

   11. frac-times 97.2%

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

   12. *-commutative 97.2%

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

   13. associate-*l/ 97.3%

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

   14. associate-*r* 95.3%

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

   15. *-commutative 95.3%

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

   16. *-commutative 95.3%

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

7. Applied egg-rr 90.5%

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

8. Taylor expanded in x around 0 76.0%

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

9. Final simplification 76.0%

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

Alternative 12: 55.0% accurate, 24.1× speedup

Math

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

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 (* (* s s) (* (* x x) (* c 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 / ((s * s) * ((x * x) * (c * 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 / ((s * s) * ((x * x) * (c * 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 / ((s * s) * ((x * x) * (c * c)));
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return 1.0 / ((s * s) * ((x * x) * (c * 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(s * s) * Float64(Float64(x * x) * Float64(c * 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 / ((s * s) * ((x * x) * (c * 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[(s * s), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

   1. associate-*r* 64.9%

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

   2. *-commutative 64.9%

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

   3. *-commutative 64.9%

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

   4. associate-*r* 62.8%

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

   5. *-commutative 62.8%

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

   6. unpow2 62.8%

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

   7. unpow2 62.8%

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

3. Simplified 62.8%

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

   1. *-un-lft-identity 62.8%

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

   2. associate-*r* 64.9%

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

   3. *-commutative 64.9%

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

   4. add-sqr-sqrt 64.8%

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

   5. times-frac 64.8%

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

5. Applied egg-rr 97.2%

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

   1. associate-*l/ 97.3%

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

   2. *-un-lft-identity 97.3%

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

7. Applied egg-rr 97.3%

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

8. Taylor expanded in x around 0 95.3%

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

   1. associate-*r* 97.3%

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

   2. *-commutative 97.3%

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

   3. associate-*r* 96.0%

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

10. Simplified 96.0%

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

11. Taylor expanded in x around 0 53.5%

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

    1. unpow2 53.5%

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

    2. unpow2 53.5%

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

    3. unpow2 53.5%

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

13. Simplified 53.5%

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

14. Final simplification 53.5%

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

Alternative 13: 78.6% accurate, 24.1× speedup

Math

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

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)) (* x (* c s)))))

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)) * (x * (c * s)));
}

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)) * (x * (c * s)))
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)) * (x * (c * s)));
}

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)) * (x * (c * s)))

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(x * Float64(c * s))))
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)) * (x * (c * s)));
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[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

   1. *-commutative 64.3%

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

   2. associate-*r* 58.4%

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

   3. associate-*r* 58.7%

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

   4. unpow2 58.7%

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

   5. unswap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. swap-sqr 97.3%

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

   8. *-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. *-commutative 97.3%

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

   11. *-commutative 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 53.1%

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

   1. unpow2 53.1%

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

   2. unpow2 53.1%

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

   3. *-commutative 53.1%

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

   4. associate-*r* 53.5%

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

   5. *-commutative 53.5%

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

   6. unpow2 53.5%

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

   7. swap-sqr 64.8%

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

   8. unpow2 64.8%

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

   9. associate-*r* 72.2%

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

   10. *-commutative 72.2%

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

   11. *-commutative 72.2%

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

   12. associate-*r* 64.8%

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

   13. unpow2 64.8%

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

   14. swap-sqr 79.6%

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

   15. unpow2 79.6%

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

   16. associate-*r* 78.4%

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

   17. *-commutative 78.4%

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

   18. *-commutative 78.4%

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

   19. *-commutative 78.4%

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

6. Simplified 78.4%

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

   1. *-commutative 78.4%

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

   2. *-commutative 78.4%

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

   3. associate-*r* 78.1%

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

   4. pow2 78.1%

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

8. Applied egg-rr 78.1%

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

9. Taylor expanded in x around 0 76.6%

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

10. Final simplification 76.6%

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

Alternative 14: 28.5% accurate, 34.8× speedup

Math

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

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 (/ -2.0 (* (* c c) (* s s))))

C

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

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 = (-2.0d0) / ((c * c) * (s * s))
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 -2.0 / ((c * c) * (s * s));
}

Python

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

Julia

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

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / ((c * c) * (s * s));
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[(-2.0 / N[(N[(c * c), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

   1. *-commutative 64.3%

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

   2. associate-*r* 58.4%

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

   3. associate-*r* 58.7%

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

   4. unpow2 58.7%

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

   5. unswap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. swap-sqr 97.3%

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

   8. *-commutative 97.3%

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

   9. *-commutative 97.3%

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

   10. *-commutative 97.3%

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

   11. *-commutative 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in x around 0 32.2%

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

   1. unpow2 32.2%

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

   2. unpow2 32.2%

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

   3. associate-*r* 31.3%

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

   4. *-commutative 31.3%

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

   5. unpow2 31.3%

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

   6. associate-*r/ 31.3%

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

   7. metadata-eval 31.3%

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

   8. unpow2 31.3%

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

   9. unpow2 31.3%

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

6. Simplified 31.3%

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

7. Taylor expanded in x around inf 25.0%

   \[\leadsto \color{blue}{\frac{-2}{{s}^{2} \cdot {c}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 25.0%

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

   2. *-commutative 25.0%

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

   3. unpow2 25.0%

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

9. Simplified 25.0%

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

10. Final simplification 25.0%

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

Reproduce

herbie shell --seed 2023240 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))