mixedcos

Percentage Accurate: 67.5% → 99.5%

Time: 12.7s

Alternatives: 15

Speedup: 24.1×

• 32.0% 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: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 3.15 \cdot 10^{-65}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 3.15e-65)
     (pow (* c (* x s)) -2.0)
     (/ (/ (cos (* x 2.0)) t_0) t_0))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 3.15e-65) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = (cos((x * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 3.15d-65) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = (cos((x * 2.0d0)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 3.15e-65) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = (Math.cos((x * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 3.15e-65:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = (math.cos((x * 2.0)) / t_0) / t_0
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 3.15e-65)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 3.15e-65)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = (cos((x * 2.0)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.15e-65], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1499999999999998e-65

   1. Initial program 68.0%

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

      1. associate-/r* 68.0%

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

      2. unpow2 68.0%

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

      3. *-commutative 68.0%

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

      4. unpow2 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 61.5%

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

      1. unpow2 61.5%

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

   6. Simplified 61.5%

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

   7. Taylor expanded in x around 0 61.5%

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

      1. unpow2 61.5%

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

      2. associate-*l* 68.5%

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

   9. Simplified 68.5%

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

       1. add-sqr-sqrt 68.5%

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

       2. pow2 68.5%

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

       3. sqrt-div 68.4%

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

       4. sqrt-div 68.4%

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

       5. metadata-eval 68.4%

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

       6. sqrt-prod 36.0%

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

       7. add-sqr-sqrt 74.4%

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

       8. associate-*r* 77.2%

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

       9. *-commutative 77.2%

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

       10. sqrt-prod 54.0%

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

       11. add-sqr-sqrt 85.0%

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

       12. associate-/r* 84.9%

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

       13. pow2 84.9%

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

       14. clear-num 84.9%

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

       15. *-commutative 84.9%

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

       16. associate-*r* 82.2%

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

       17. clear-num 82.2%

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

   11. Applied egg-rr 84.9%

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

   if 3.1499999999999998e-65 < x

   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* 66.9%

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

      3. associate-*r* 67.1%

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

      4. *-commutative 67.1%

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

      5. unpow2 67.1%

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

      6. associate-*r* 70.0%

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

      7. associate-*r* 70.1%

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

      8. *-commutative 70.1%

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

      9. unpow2 70.1%

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

   3. Simplified 70.1%

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

      1. add-cube-cbrt 70.0%

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

      2. times-frac 70.1%

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

      3. associate-*r* 70.1%

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

      4. swap-sqr 87.0%

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

      5. associate-*r* 92.0%

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

      6. *-commutative 92.0%

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

      7. times-frac 92.5%

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

      8. associate-*l* 95.8%

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

      9. add-cube-cbrt 96.0%

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

      10. associate-/r* 96.1%

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

   5. Applied egg-rr 96.1%

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

      1. un-div-inv 96.1%

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

   7. Applied egg-rr 96.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.15 \cdot 10^{-65}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 2: 87.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 2 \cdot 10^{+207}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= (pow s 2.0) 2e+207)
   (/ (cos (* x 2.0)) (* x (* (* c c) (* s (* x s)))))
   (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 (pow(s, 2.0) <= 2e+207) {
		tmp = cos((x * 2.0)) / (x * ((c * c) * (s * (x * s))));
	} else {
		tmp = pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((s ** 2.0d0) <= 2d+207) then
        tmp = cos((x * 2.0d0)) / (x * ((c * c) * (s * (x * s))))
    else
        tmp = (c * (x * s)) ** (-2.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (Math.pow(s, 2.0) <= 2e+207) {
		tmp = Math.cos((x * 2.0)) / (x * ((c * c) * (s * (x * s))));
	} else {
		tmp = Math.pow((c * (x * s)), -2.0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if math.pow(s, 2.0) <= 2e+207:
		tmp = math.cos((x * 2.0)) / (x * ((c * c) * (s * (x * s))))
	else:
		tmp = math.pow((c * (x * s)), -2.0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if ((s ^ 2.0) <= 2e+207)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(c * c) * Float64(s * Float64(x * s)))));
	else
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((s ^ 2.0) <= 2e+207)
		tmp = cos((x * 2.0)) / (x * ((c * c) * (s * (x * s))));
	else
		tmp = (c * (x * s)) ^ -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[N[Power[s, 2.0], $MachinePrecision], 2e+207], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(c * c), $MachinePrecision] * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s 2) < 2.0000000000000001e207

   1. Initial program 74.1%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

      3. associate-*r* 74.2%

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

      4. unpow2 74.2%

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

      5. unpow2 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in c around 0 74.2%

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

      1. unpow2 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*r* 74.9%

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

      4. unpow2 74.9%

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

      5. *-commutative 74.9%

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

      6. associate-*l* 78.6%

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

   6. Simplified 78.6%

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

   if 2.0000000000000001e207 < (pow.f64 s 2)

   1. Initial program 60.3%

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

      1. associate-/r* 60.3%

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

      2. unpow2 60.3%

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

      3. *-commutative 60.3%

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

      4. unpow2 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in x around 0 59.4%

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

      1. unpow2 59.4%

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

   6. Simplified 59.4%

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

   7. Taylor expanded in x around 0 59.4%

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

      1. unpow2 59.4%

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

      2. associate-*l* 71.4%

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

   9. Simplified 71.4%

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

       1. add-sqr-sqrt 71.4%

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

       2. pow2 71.4%

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

       3. sqrt-div 71.4%

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

       4. sqrt-div 71.4%

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

       5. metadata-eval 71.4%

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

       6. sqrt-prod 38.6%

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

       7. add-sqr-sqrt 77.1%

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

       8. associate-*r* 82.1%

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

       9. *-commutative 82.1%

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

       10. sqrt-prod 46.9%

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

       11. add-sqr-sqrt 88.7%

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

       12. associate-/r* 88.6%

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

       13. pow2 88.6%

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

       14. clear-num 88.6%

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

       15. *-commutative 88.6%

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

       16. associate-*r* 83.3%

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

       17. clear-num 83.3%

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

   11. Applied egg-rr 88.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 2 \cdot 10^{+207}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 3: 85.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \frac{1}{s \cdot \left(x \cdot c\right)}\\ \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 3.95 \cdot 10^{+182}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* s (* x c)))))
   (if (<= x 0.112)
     (pow (* c (* x s)) -2.0)
     (if (<= x 3.95e+182)
       (/ (cos (* x 2.0)) (* s (* (* c c) (* x (* x s)))))
       (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = 1.0 / (s * (x * c));
	double tmp;
	if (x <= 0.112) {
		tmp = pow((c * (x * s)), -2.0);
	} else if (x <= 3.95e+182) {
		tmp = cos((x * 2.0)) / (s * ((c * c) * (x * (x * s))));
	} else {
		tmp = 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 = 1.0d0 / (s * (x * c))
    if (x <= 0.112d0) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else if (x <= 3.95d+182) then
        tmp = cos((x * 2.0d0)) / (s * ((c * c) * (x * (x * s))))
    else
        tmp = 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 = 1.0 / (s * (x * c));
	double tmp;
	if (x <= 0.112) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else if (x <= 3.95e+182) {
		tmp = Math.cos((x * 2.0)) / (s * ((c * c) * (x * (x * s))));
	} else {
		tmp = 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 = 1.0 / (s * (x * c))
	tmp = 0
	if x <= 0.112:
		tmp = math.pow((c * (x * s)), -2.0)
	elif x <= 3.95e+182:
		tmp = math.cos((x * 2.0)) / (s * ((c * c) * (x * (x * s))))
	else:
		tmp = 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(1.0 / Float64(s * Float64(x * c)))
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	elseif (x <= 3.95e+182)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(c * c) * Float64(x * Float64(x * s)))));
	else
		tmp = 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 = 1.0 / (s * (x * c));
	tmp = 0.0;
	if (x <= 0.112)
		tmp = (c * (x * s)) ^ -2.0;
	elseif (x <= 3.95e+182)
		tmp = cos((x * 2.0)) / (s * ((c * c) * (x * (x * s))));
	else
		tmp = 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[(1.0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.112], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x, 3.95e+182], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(c * c), $MachinePrecision] * N[(x * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.112000000000000002

   1. Initial program 68.6%

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

      1. associate-/r* 68.6%

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

      2. unpow2 68.6%

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

      3. *-commutative 68.6%

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

      4. unpow2 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 62.5%

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

      1. unpow2 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in x around 0 62.5%

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

      1. unpow2 62.5%

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

      2. associate-*l* 69.1%

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

   9. Simplified 69.1%

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

       1. add-sqr-sqrt 69.0%

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

       2. pow2 69.0%

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

       3. sqrt-div 69.0%

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

       4. sqrt-div 69.0%

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

       5. metadata-eval 69.0%

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

       6. sqrt-prod 35.7%

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

       7. add-sqr-sqrt 74.6%

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

       8. associate-*r* 77.2%

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

       9. *-commutative 77.2%

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

       10. sqrt-prod 54.5%

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

       11. add-sqr-sqrt 85.6%

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

       12. associate-/r* 85.5%

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

       13. pow2 85.5%

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

       14. clear-num 85.5%

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

       15. *-commutative 85.5%

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

       16. associate-*r* 83.0%

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

       17. clear-num 83.0%

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

   11. Applied egg-rr 85.6%

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

   if 0.112000000000000002 < x < 3.9500000000000001e182

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l* 70.2%

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

      3. associate-*r* 73.3%

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

      4. *-commutative 73.3%

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

      5. unpow2 73.3%

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

      6. associate-*r* 76.3%

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

      7. associate-*r* 73.4%

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

      8. *-commutative 73.4%

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

      9. unpow2 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in x around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. unpow2 73.4%

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

      3. associate-*r* 70.5%

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

      4. unpow2 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*r* 76.4%

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

   6. Simplified 76.4%

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

   if 3.9500000000000001e182 < x

   1. Initial program 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*l* 58.6%

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

      3. associate-*r* 58.6%

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

      4. *-commutative 58.6%

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

      5. unpow2 58.6%

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

      6. associate-*r* 62.6%

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

      7. associate-*r* 63.0%

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

      8. *-commutative 63.0%

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

      9. unpow2 63.0%

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

   3. Simplified 63.0%

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

      1. add-cube-cbrt 63.0%

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

      2. times-frac 63.0%

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

      3. associate-*r* 63.0%

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

      4. swap-sqr 90.2%

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

      5. associate-*r* 93.5%

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

      6. *-commutative 93.5%

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

      7. times-frac 93.6%

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

      8. associate-*l* 93.6%

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

      9. add-cube-cbrt 93.7%

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

      10. associate-/r* 93.7%

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

   5. Applied egg-rr 93.7%

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

   6. Taylor expanded in x around 0 72.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 3.95 \cdot 10^{+182}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 4: 86.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \frac{1}{s \cdot \left(x \cdot c\right)}\\ \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* s (* x c)))))
   (if (<= x 0.112)
     (pow (* c (* x s)) -2.0)
     (if (<= x 1.26e+154)
       (/ (cos (* x 2.0)) (* s (* (* x x) (* s (* c c)))))
       (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = 1.0 / (s * (x * c));
	double tmp;
	if (x <= 0.112) {
		tmp = pow((c * (x * s)), -2.0);
	} else if (x <= 1.26e+154) {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = 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 = 1.0d0 / (s * (x * c))
    if (x <= 0.112d0) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else if (x <= 1.26d+154) then
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (s * (c * c))))
    else
        tmp = 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 = 1.0 / (s * (x * c));
	double tmp;
	if (x <= 0.112) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else if (x <= 1.26e+154) {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	} else {
		tmp = 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 = 1.0 / (s * (x * c))
	tmp = 0
	if x <= 0.112:
		tmp = math.pow((c * (x * s)), -2.0)
	elif x <= 1.26e+154:
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))))
	else:
		tmp = 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(1.0 / Float64(s * Float64(x * c)))
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	elseif (x <= 1.26e+154)
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(s * Float64(c * c)))));
	else
		tmp = 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 = 1.0 / (s * (x * c));
	tmp = 0.0;
	if (x <= 0.112)
		tmp = (c * (x * s)) ^ -2.0;
	elseif (x <= 1.26e+154)
		tmp = cos((x * 2.0)) / (s * ((x * x) * (s * (c * c))));
	else
		tmp = 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[(1.0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.112], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], If[LessEqual[x, 1.26e+154], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(s * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.112000000000000002

   1. Initial program 68.6%

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

      1. associate-/r* 68.6%

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

      2. unpow2 68.6%

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

      3. *-commutative 68.6%

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

      4. unpow2 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 62.5%

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

      1. unpow2 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in x around 0 62.5%

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

      1. unpow2 62.5%

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

      2. associate-*l* 69.1%

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

   9. Simplified 69.1%

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

       1. add-sqr-sqrt 69.0%

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

       2. pow2 69.0%

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

       3. sqrt-div 69.0%

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

       4. sqrt-div 69.0%

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

       5. metadata-eval 69.0%

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

       6. sqrt-prod 35.7%

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

       7. add-sqr-sqrt 74.6%

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

       8. associate-*r* 77.2%

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

       9. *-commutative 77.2%

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

       10. sqrt-prod 54.5%

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

       11. add-sqr-sqrt 85.6%

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

       12. associate-/r* 85.5%

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

       13. pow2 85.5%

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

       14. clear-num 85.5%

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

       15. *-commutative 85.5%

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

       16. associate-*r* 83.0%

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

       17. clear-num 83.0%

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

   11. Applied egg-rr 85.6%

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

   if 0.112000000000000002 < x < 1.26e154

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*l* 78.1%

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

      3. associate-*r* 81.8%

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

      4. *-commutative 81.8%

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

      5. unpow2 81.8%

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

      6. associate-*r* 85.5%

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

      7. associate-*r* 81.9%

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

      8. *-commutative 81.9%

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

      9. unpow2 81.9%

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

   3. Simplified 81.9%

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

   if 1.26e154 < x

   1. Initial program 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-*l* 54.6%

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

      3. associate-*r* 54.6%

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

      4. *-commutative 54.6%

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

      5. unpow2 54.6%

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

      6. associate-*r* 57.9%

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

      7. associate-*r* 58.2%

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

      8. *-commutative 58.2%

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

      9. unpow2 58.2%

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

   3. Simplified 58.2%

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

      1. add-cube-cbrt 58.2%

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

      2. times-frac 58.2%

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

      3. associate-*r* 58.2%

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

      4. swap-sqr 89.1%

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

      5. associate-*r* 94.5%

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

      6. *-commutative 94.5%

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

      7. times-frac 94.5%

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

      8. associate-*l* 94.5%

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

      9. add-cube-cbrt 94.7%

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

      10. associate-/r* 94.7%

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

   5. Applied egg-rr 94.7%

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

   6. Taylor expanded in x around 0 66.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+154}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 5: 91.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.295:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

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 0.295)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* x (* (* c (* x c)) (* s s))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.295) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.295d0) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (x * ((c * (x * c)) * (s * s)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.295) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.295:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.295)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(c * Float64(x * c)) * Float64(s * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.295)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.295], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.294999999999999984

   1. Initial program 68.6%

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

      1. associate-/r* 68.6%

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

      2. unpow2 68.6%

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

      3. *-commutative 68.6%

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

      4. unpow2 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in x around 0 62.5%

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

      1. unpow2 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in x around 0 62.5%

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

      1. unpow2 62.5%

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

      2. associate-*l* 69.1%

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

   9. Simplified 69.1%

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

       1. add-sqr-sqrt 69.0%

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

       2. pow2 69.0%

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

       3. sqrt-div 69.0%

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

       4. sqrt-div 69.0%

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

       5. metadata-eval 69.0%

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

       6. sqrt-prod 35.7%

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

       7. add-sqr-sqrt 74.6%

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

       8. associate-*r* 77.2%

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

       9. *-commutative 77.2%

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

       10. sqrt-prod 54.5%

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

       11. add-sqr-sqrt 85.6%

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

       12. associate-/r* 85.5%

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

       13. pow2 85.5%

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

       14. clear-num 85.5%

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

       15. *-commutative 85.5%

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

       16. associate-*r* 83.0%

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

       17. clear-num 83.0%

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

   11. Applied egg-rr 85.6%

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

   if 0.294999999999999984 < x

   1. Initial program 70.0%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

      3. associate-*r* 71.6%

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

      4. unpow2 71.6%

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

      5. unpow2 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in c around 0 71.6%

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

      1. unpow2 71.6%

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

      2. associate-*l* 81.1%

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

   6. Simplified 81.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.295:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 6: 95.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{+203}:\\ \;\;\;\;\frac{t_0}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (* c (* x s))))
   (if (<= x 1.9e+203)
     (/ t_0 (* t_1 t_1))
     (/ t_0 (* x (* (* c (* x c)) (* s s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 1.9e+203) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    t_1 = c * (x * s)
    if (x <= 1.9d+203) then
        tmp = t_0 / (t_1 * t_1)
    else
        tmp = t_0 / (x * ((c * (x * c)) * (s * s)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 1.9e+203) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	t_1 = c * (x * s)
	tmp = 0
	if x <= 1.9e+203:
		tmp = t_0 / (t_1 * t_1)
	else:
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 1.9e+203)
		tmp = Float64(t_0 / Float64(t_1 * t_1));
	else
		tmp = Float64(t_0 / Float64(x * Float64(Float64(c * Float64(x * c)) * Float64(s * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 1.9e+203)
		tmp = t_0 / (t_1 * t_1);
	else
		tmp = t_0 / (x * ((c * (x * c)) * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9e+203], N[(t$95$0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x * N[(N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.90000000000000012e203

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*r* 66.0%

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

      3. associate-*r* 64.4%

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

      4. unpow2 64.4%

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

      5. unswap-sqr 79.5%

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

      6. unpow2 79.5%

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

      7. swap-sqr 96.3%

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

      8. *-commutative 96.3%

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

      9. *-commutative 96.3%

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

      10. *-commutative 96.3%

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

      11. *-commutative 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in s around 0 94.0%

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

   5. Taylor expanded in s around 0 97.0%

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

   if 1.90000000000000012e203 < x

   1. Initial program 59.7%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

      3. associate-*r* 60.0%

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

      4. unpow2 60.0%

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

      5. unpow2 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in c around 0 60.0%

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

      1. unpow2 60.0%

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

      2. associate-*l* 81.7%

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

   6. Simplified 81.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+203}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 7: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{-64}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 2.4e-64)
     (pow (* c (* x s)) -2.0)
     (/ (cos (* x 2.0)) (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 2.4e-64) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 2.4d-64) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 2.4e-64) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 2.4e-64:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 2.4e-64)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 2.4e-64)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.4e-64], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999998e-64

   1. Initial program 68.0%

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

      1. associate-/r* 68.0%

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

      2. unpow2 68.0%

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

      3. *-commutative 68.0%

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

      4. unpow2 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 61.5%

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

      1. unpow2 61.5%

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

   6. Simplified 61.5%

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

   7. Taylor expanded in x around 0 61.5%

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

      1. unpow2 61.5%

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

      2. associate-*l* 68.5%

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

   9. Simplified 68.5%

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

       1. add-sqr-sqrt 68.5%

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

       2. pow2 68.5%

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

       3. sqrt-div 68.4%

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

       4. sqrt-div 68.4%

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

       5. metadata-eval 68.4%

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

       6. sqrt-prod 36.0%

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

       7. add-sqr-sqrt 74.4%

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

       8. associate-*r* 77.2%

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

       9. *-commutative 77.2%

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

       10. sqrt-prod 54.0%

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

       11. add-sqr-sqrt 85.0%

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

       12. associate-/r* 84.9%

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

       13. pow2 84.9%

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

       14. clear-num 84.9%

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

       15. *-commutative 84.9%

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

       16. associate-*r* 82.2%

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

       17. clear-num 82.2%

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

   11. Applied egg-rr 84.9%

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

   if 2.39999999999999998e-64 < x

   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%

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

      2. associate-*r* 66.9%

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

      3. associate-*r* 67.0%

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

      4. unpow2 67.0%

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

      5. unswap-sqr 82.8%

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

      6. unpow2 82.8%

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

      7. swap-sqr 96.0%

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

      8. *-commutative 96.0%

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

      9. *-commutative 96.0%

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

      10. *-commutative 96.0%

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

      11. *-commutative 96.0%

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

   3. Simplified 96.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-64}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 8: 80.7% 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 68.9%

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

   1. associate-/r* 68.9%

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

   2. unpow2 68.9%

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

   3. *-commutative 68.9%

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

   4. unpow2 68.9%

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

3. Simplified 68.9%

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

4. Taylor expanded in x around 0 60.6%

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

   1. unpow2 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in x around 0 60.6%

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

   1. unpow2 60.6%

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

   2. associate-*l* 66.0%

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

9. Simplified 66.0%

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

    1. add-sqr-sqrt 66.0%

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

    2. pow2 66.0%

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

    3. sqrt-div 66.0%

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

    4. sqrt-div 66.0%

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

    5. metadata-eval 66.0%

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

    6. sqrt-prod 35.5%

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

    7. add-sqr-sqrt 71.2%

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

    8. associate-*r* 73.1%

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

    9. *-commutative 73.1%

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

    10. sqrt-prod 49.1%

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

    11. add-sqr-sqrt 79.9%

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

    12. associate-/r* 79.8%

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

    13. pow2 79.8%

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

    14. clear-num 79.8%

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

    15. *-commutative 79.8%

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

    16. associate-*r* 77.8%

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

    17. clear-num 77.8%

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

11. Applied egg-rr 79.9%

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

12. Final simplification 79.9%

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

Alternative 9: 58.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 5.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\\ \end{array} \end{array} \]

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 (<= s 5.6e+157)
   (/ 1.0 (* (* c c) (* (* s s) (* x x))))
   (/ -2.0 (* (* s s) (* c 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 (s <= 5.6e+157) {
		tmp = 1.0 / ((c * c) * ((s * s) * (x * x)));
	} else {
		tmp = -2.0 / ((s * s) * (c * 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 (s <= 5.6d+157) then
        tmp = 1.0d0 / ((c * c) * ((s * s) * (x * x)))
    else
        tmp = (-2.0d0) / ((s * s) * (c * 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 (s <= 5.6e+157) {
		tmp = 1.0 / ((c * c) * ((s * s) * (x * x)));
	} else {
		tmp = -2.0 / ((s * s) * (c * 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 s <= 5.6e+157:
		tmp = 1.0 / ((c * c) * ((s * s) * (x * x)))
	else:
		tmp = -2.0 / ((s * s) * (c * 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 (s <= 5.6e+157)
		tmp = Float64(1.0 / Float64(Float64(c * c) * Float64(Float64(s * s) * Float64(x * x))));
	else
		tmp = Float64(-2.0 / Float64(Float64(s * s) * Float64(c * 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 (s <= 5.6e+157)
		tmp = 1.0 / ((c * c) * ((s * s) * (x * x)));
	else
		tmp = -2.0 / ((s * s) * (c * 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[s, 5.6e+157], N[(1.0 / N[(N[(c * c), $MachinePrecision] * N[(N[(s * s), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 5.6000000000000005e157

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-*r* 65.0%

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

      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 80.5%

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

      6. unpow2 80.5%

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

      7. swap-sqr 96.1%

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

      8. *-commutative 96.1%

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

      9. *-commutative 96.1%

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

      10. *-commutative 96.1%

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

      11. *-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in s around 0 93.4%

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

   5. Taylor expanded in s around 0 96.5%

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

   6. Taylor expanded in x around 0 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*r* 56.9%

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

      3. *-commutative 56.9%

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

      4. unpow2 56.9%

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

      5. unpow2 56.9%

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

      6. unpow2 56.9%

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

   8. Simplified 56.9%

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

   if 5.6000000000000005e157 < s

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*l* 63.4%

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

      3. associate-*r* 63.4%

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

      4. *-commutative 63.4%

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

      5. unpow2 63.4%

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

      6. associate-*r* 70.5%

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

      7. associate-*r* 70.5%

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

      8. *-commutative 70.5%

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

      9. unpow2 70.5%

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

   3. Simplified 70.5%

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

      1. add-cube-cbrt 70.5%

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

      2. times-frac 70.5%

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

      3. associate-*r* 67.5%

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

      4. swap-sqr 87.0%

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

      5. associate-*r* 93.4%

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

      6. *-commutative 93.4%

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

      7. times-frac 93.5%

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

      8. associate-*l* 94.8%

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

      9. add-cube-cbrt 95.0%

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

      10. associate-/r* 95.0%

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

   5. Applied egg-rr 95.0%

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

   6. Taylor expanded in x around 0 85.4%

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

   7. Taylor expanded in x around inf 70.5%

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

      1. unpow2 70.5%

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

      2. unpow2 70.5%

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

   9. Simplified 70.5%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\\ \end{array} \]

Alternative 10: 76.1% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{1}{c \cdot c}}{x \cdot \left(s \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

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 2.8e-202)
   (/ (/ 1.0 (* c c)) (* x (* s (* x s))))
   (/ 1.0 (* (* x c) (* s (* c (* x 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 <= 2.8e-202) {
		tmp = (1.0 / (c * c)) / (x * (s * (x * s)));
	} else {
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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 <= 2.8d-202) then
        tmp = (1.0d0 / (c * c)) / (x * (s * (x * s)))
    else
        tmp = 1.0d0 / ((x * c) * (s * (c * (x * 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 <= 2.8e-202) {
		tmp = (1.0 / (c * c)) / (x * (s * (x * s)));
	} else {
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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 <= 2.8e-202:
		tmp = (1.0 / (c * c)) / (x * (s * (x * s)))
	else:
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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 <= 2.8e-202)
		tmp = Float64(Float64(1.0 / Float64(c * c)) / Float64(x * Float64(s * Float64(x * s))));
	else
		tmp = Float64(1.0 / Float64(Float64(x * c) * Float64(s * Float64(c * Float64(x * 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 <= 2.8e-202)
		tmp = (1.0 / (c * c)) / (x * (s * (x * s)));
	else
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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, 2.8e-202], N[(N[(1.0 / N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(x * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * c), $MachinePrecision] * N[(s * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8000000000000001e-202

   1. Initial program 66.4%

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

      1. associate-/r* 66.5%

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

      2. unpow2 66.5%

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

      3. *-commutative 66.5%

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

      4. unpow2 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in x around 0 58.9%

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

      1. unpow2 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in x around 0 58.9%

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

      1. unpow2 58.9%

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

      2. associate-*l* 65.8%

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

   9. Simplified 65.8%

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

   if 2.8000000000000001e-202 < 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%

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

      2. associate-*l* 65.6%

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

      3. associate-*r* 65.8%

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

      4. *-commutative 65.8%

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

      5. unpow2 65.8%

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

      6. associate-*r* 70.0%

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

      7. associate-*r* 69.2%

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

      8. *-commutative 69.2%

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

      9. unpow2 69.2%

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

   3. Simplified 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. times-frac 69.1%

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

      3. associate-*r* 68.2%

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

      4. swap-sqr 88.5%

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

      5. associate-*r* 93.2%

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

      6. *-commutative 93.2%

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

      7. times-frac 93.0%

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

      8. associate-*l* 96.3%

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

      9. associate-*r* 95.4%

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

      10. times-frac 96.0%

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

      11. *-commutative 96.0%

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

   5. Applied egg-rr 96.0%

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

   6. Taylor expanded in x around 0 75.3%

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

      1. frac-times 74.8%

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

      2. metadata-eval 74.8%

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

      3. associate-*r* 74.4%

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

      4. *-commutative 74.4%

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

      5. *-commutative 74.4%

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

      6. *-commutative 74.4%

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

      7. *-commutative 74.4%

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

   8. Applied egg-rr 74.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{1}{c \cdot c}}{x \cdot \left(s \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 11: 77.6% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{1}{c \cdot c}}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

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 6e-123)
   (/ (/ 1.0 (* c c)) (* (* x s) (* x s)))
   (/ 1.0 (* (* x c) (* s (* c (* x 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 <= 6e-123) {
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s));
	} else {
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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 <= 6d-123) then
        tmp = (1.0d0 / (c * c)) / ((x * s) * (x * s))
    else
        tmp = 1.0d0 / ((x * c) * (s * (c * (x * 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 <= 6e-123) {
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s));
	} else {
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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 <= 6e-123:
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s))
	else:
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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 <= 6e-123)
		tmp = Float64(Float64(1.0 / Float64(c * c)) / Float64(Float64(x * s) * Float64(x * s)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * c) * Float64(s * Float64(c * Float64(x * 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 <= 6e-123)
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s));
	else
		tmp = 1.0 / ((x * c) * (s * (c * (x * 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, 6e-123], N[(N[(1.0 / N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * c), $MachinePrecision] * N[(s * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999968e-123

   1. Initial program 66.7%

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

      1. associate-/r* 66.7%

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

      2. unpow2 66.7%

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

      3. *-commutative 66.7%

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

      4. unpow2 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x around 0 59.8%

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

      1. unpow2 59.8%

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

   6. Simplified 59.8%

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

      1. add-sqr-sqrt 59.8%

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

      2. sqrt-unprod 58.1%

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

      3. associate-*r* 55.1%

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

      4. associate-*r* 55.1%

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

      5. swap-sqr 45.4%

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

      6. pow2 45.4%

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

      7. pow2 45.4%

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

      8. pow-prod-up 45.4%

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

      9. metadata-eval 45.4%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{\color{blue}{4}} \cdot \left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right)}} \]

      10. pow2 45.4%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot \left(\color{blue}{{s}^{2}} \cdot \left(s \cdot s\right)\right)}} \]

      11. pow2 45.4%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot \left({s}^{2} \cdot \color{blue}{{s}^{2}}\right)}} \]

      12. pow-prod-up 45.4%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot \color{blue}{{s}^{\left(2 + 2\right)}}}} \]

      13. metadata-eval 45.4%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot {s}^{\color{blue}{4}}}} \]

   8. Applied egg-rr 45.4%

      \[\leadsto \frac{\frac{1}{c \cdot c}}{\color{blue}{\sqrt{{x}^{4} \cdot {s}^{4}}}} \]
   9. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\color{blue}{{s}^{4} \cdot {x}^{4}}}} \]

      2. metadata-eval 45.4%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{s}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {x}^{4}}} \]

      3. pow-sqr 45.4%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\color{blue}{\left({s}^{2} \cdot {s}^{2}\right)} \cdot {x}^{4}}} \]

      4. unpow2 45.4%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\left(\color{blue}{\left(s \cdot s\right)} \cdot {s}^{2}\right) \cdot {x}^{4}}} \]

      5. unpow2 45.4%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\left(\left(s \cdot s\right) \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot {x}^{4}}} \]

      6. metadata-eval 45.4%

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

      7. pow-sqr 45.4%

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

      8. unpow2 45.4%

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

      9. unpow2 45.4%

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

      10. unswap-sqr 55.1%

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

      11. *-commutative 55.1%

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

      12. swap-sqr 55.1%

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

      13. unpow2 55.1%

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

      14. *-commutative 55.1%

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

      15. swap-sqr 65.6%

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

      16. unpow2 65.6%

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

      17. pow-sqr 65.6%

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

      18. *-commutative 65.6%

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

      19. metadata-eval 65.6%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{\left(s \cdot x\right)}^{\color{blue}{4}}}} \]

   10. Simplified 65.6%

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

       1. sqrt-pow1 69.1%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\color{blue}{{\left(s \cdot x\right)}^{\left(\frac{4}{2}\right)}}} \]

       2. metadata-eval 69.1%

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

       3. pow2 69.1%

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

       4. *-commutative 69.1%

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

       5. *-commutative 69.1%

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

   12. Applied egg-rr 69.1%

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

   if 5.99999999999999968e-123 < x

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*l* 69.6%

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

      3. associate-*r* 69.7%

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

      4. *-commutative 69.7%

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

      5. unpow2 69.7%

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

      6. associate-*r* 72.4%

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

      7. associate-*r* 72.5%

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

      8. *-commutative 72.5%

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

      9. unpow2 72.5%

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

   3. Simplified 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. times-frac 72.5%

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

      3. associate-*r* 71.4%

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

      4. swap-sqr 87.6%

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

      5. associate-*r* 93.2%

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

      6. *-commutative 93.2%

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

      7. times-frac 93.6%

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

      8. associate-*l* 96.4%

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

      9. associate-*r* 95.4%

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

      10. times-frac 95.3%

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

      11. *-commutative 95.3%

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

   5. Applied egg-rr 95.3%

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

   6. Taylor expanded in x around 0 71.1%

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

      1. frac-times 71.0%

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

      2. metadata-eval 71.0%

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

      3. associate-*r* 70.6%

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

      4. *-commutative 70.6%

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

      5. *-commutative 70.6%

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

      6. *-commutative 70.6%

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

      7. *-commutative 70.6%

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

   8. Applied egg-rr 70.6%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{1}{c \cdot c}}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 12: 79.1% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{c \cdot c}}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 2.6e-171)
     (/ (/ 1.0 (* c c)) (* (* x s) (* x s)))
     (/ (/ 1.0 t_0) t_0))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 2.6e-171) {
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s));
	} else {
		tmp = (1.0 / 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 <= 2.6d-171) then
        tmp = (1.0d0 / (c * c)) / ((x * s) * (x * s))
    else
        tmp = (1.0d0 / 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 <= 2.6e-171) {
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s));
	} else {
		tmp = (1.0 / 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 <= 2.6e-171:
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s))
	else:
		tmp = (1.0 / 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 <= 2.6e-171)
		tmp = Float64(Float64(1.0 / Float64(c * c)) / Float64(Float64(x * s) * Float64(x * s)));
	else
		tmp = Float64(Float64(1.0 / 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 <= 2.6e-171)
		tmp = (1.0 / (c * c)) / ((x * s) * (x * s));
	else
		tmp = (1.0 / 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, 2.6e-171], N[(N[(1.0 / N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000005e-171

   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. Taylor expanded in x around 0 59.5%

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

      1. unpow2 59.5%

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

   6. Simplified 59.5%

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

      1. add-sqr-sqrt 59.5%

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

      2. sqrt-unprod 58.4%

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

      3. associate-*r* 55.2%

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

      4. associate-*r* 55.1%

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

      5. swap-sqr 45.8%

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

      6. pow2 45.8%

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

      7. pow2 45.8%

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

      8. pow-prod-up 45.8%

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

      9. metadata-eval 45.8%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{\color{blue}{4}} \cdot \left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right)}} \]

      10. pow2 45.8%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot \left(\color{blue}{{s}^{2}} \cdot \left(s \cdot s\right)\right)}} \]

      11. pow2 45.8%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot \left({s}^{2} \cdot \color{blue}{{s}^{2}}\right)}} \]

      12. pow-prod-up 45.8%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot \color{blue}{{s}^{\left(2 + 2\right)}}}} \]

      13. metadata-eval 45.8%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{x}^{4} \cdot {s}^{\color{blue}{4}}}} \]

   8. Applied egg-rr 45.8%

      \[\leadsto \frac{\frac{1}{c \cdot c}}{\color{blue}{\sqrt{{x}^{4} \cdot {s}^{4}}}} \]
   9. Step-by-step derivation

      1. *-commutative 45.8%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\color{blue}{{s}^{4} \cdot {x}^{4}}}} \]

      2. metadata-eval 45.8%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{s}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {x}^{4}}} \]

      3. pow-sqr 45.8%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\color{blue}{\left({s}^{2} \cdot {s}^{2}\right)} \cdot {x}^{4}}} \]

      4. unpow2 45.8%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\left(\color{blue}{\left(s \cdot s\right)} \cdot {s}^{2}\right) \cdot {x}^{4}}} \]

      5. unpow2 45.8%

         \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{\left(\left(s \cdot s\right) \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot {x}^{4}}} \]

      6. metadata-eval 45.8%

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

      7. pow-sqr 45.8%

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

      8. unpow2 45.8%

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

      9. unpow2 45.8%

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

      10. unswap-sqr 55.1%

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

      11. *-commutative 55.1%

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

      12. swap-sqr 55.1%

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

      13. unpow2 55.1%

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

      14. *-commutative 55.1%

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

      15. swap-sqr 65.6%

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

      16. unpow2 65.6%

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

      17. pow-sqr 65.6%

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

      18. *-commutative 65.6%

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

      19. metadata-eval 65.6%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\sqrt{{\left(s \cdot x\right)}^{\color{blue}{4}}}} \]

   10. Simplified 65.6%

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

       1. sqrt-pow1 68.7%

          \[\leadsto \frac{\frac{1}{c \cdot c}}{\color{blue}{{\left(s \cdot x\right)}^{\left(\frac{4}{2}\right)}}} \]

       2. metadata-eval 68.7%

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

       3. pow2 68.7%

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

       4. *-commutative 68.7%

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

       5. *-commutative 68.7%

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

   12. Applied egg-rr 68.7%

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

   if 2.60000000000000005e-171 < x

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*l* 68.0%

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

      3. associate-*r* 68.2%

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

      4. *-commutative 68.2%

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

      5. unpow2 68.2%

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

      6. associate-*r* 71.6%

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

      7. associate-*r* 70.7%

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

      8. *-commutative 70.7%

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

      9. unpow2 70.7%

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

   3. Simplified 70.7%

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

      1. add-cube-cbrt 70.6%

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

      2. times-frac 70.6%

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

      3. associate-*r* 69.7%

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

      4. swap-sqr 87.7%

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

      5. associate-*r* 92.6%

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

      6. *-commutative 92.6%

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

      7. times-frac 92.4%

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

      8. associate-*l* 96.0%

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

      9. add-cube-cbrt 96.1%

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

      10. associate-/r* 96.9%

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

   5. Applied egg-rr 96.8%

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

      1. un-div-inv 96.9%

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

   7. Applied egg-rr 96.9%

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

   8. Taylor expanded in x around 0 74.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{c \cdot c}}{\left(x \cdot s\right) \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 13: 80.6% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. associate-/r* 68.9%

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

   2. unpow2 68.9%

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

   3. *-commutative 68.9%

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

   4. unpow2 68.9%

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

3. Simplified 68.9%

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

4. Taylor expanded in x around 0 60.6%

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

   1. unpow2 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in x around 0 60.6%

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

   1. unpow2 60.6%

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

   2. associate-*l* 66.0%

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

9. Simplified 66.0%

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

    1. add-sqr-sqrt 66.0%

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

    2. sqrt-div 66.0%

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

    3. sqrt-div 66.0%

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

    4. metadata-eval 66.0%

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

    5. sqrt-prod 33.0%

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

    6. add-sqr-sqrt 50.3%

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

    7. associate-*r* 50.3%

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

    8. *-commutative 50.3%

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

    9. sqrt-prod 30.0%

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

    10. add-sqr-sqrt 47.8%

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

    11. associate-/r* 47.8%

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

    12. *-commutative 47.8%

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

    13. sqrt-div 47.8%

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

    14. sqrt-div 47.8%

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

    15. metadata-eval 47.8%

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

    16. sqrt-prod 25.3%

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

    17. add-sqr-sqrt 52.2%

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

    18. associate-*r* 52.9%

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

    19. *-commutative 52.9%

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

11. Applied egg-rr 79.8%

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

12. Final simplification 79.8%

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

Alternative 14: 75.6% accurate, 24.1× speedup

Math

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

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 (* (* x c) (* s (* c (* 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 / ((x * c) * (s * (c * (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 / ((x * c) * (s * (c * (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 / ((x * c) * (s * (c * (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 / ((x * c) * (s * (c * (x * 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(x * c) * Float64(s * Float64(c * 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 / ((x * c) * (s * (c * (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[(1.0 / N[(N[(x * c), $MachinePrecision] * N[(s * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. associate-*l* 64.9%

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

   3. associate-*r* 64.2%

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

   4. *-commutative 64.2%

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

   5. unpow2 64.2%

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

   6. associate-*r* 69.9%

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

   7. associate-*r* 70.3%

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

   8. *-commutative 70.3%

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

   9. unpow2 70.3%

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

3. Simplified 70.3%

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

   1. *-un-lft-identity 70.3%

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

   2. times-frac 70.4%

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

   3. associate-*r* 68.9%

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

   4. swap-sqr 87.2%

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

   5. associate-*r* 93.2%

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

   6. *-commutative 93.2%

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

   7. times-frac 93.1%

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

   8. associate-*l* 95.9%

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

   9. associate-*r* 92.7%

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

   10. times-frac 92.7%

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

   11. *-commutative 92.7%

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

5. Applied egg-rr 92.7%

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

6. Taylor expanded in x around 0 77.1%

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

   1. frac-times 77.0%

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

   2. metadata-eval 77.0%

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

   3. associate-*r* 76.0%

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

   4. *-commutative 76.0%

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

   5. *-commutative 76.0%

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

   6. *-commutative 76.0%

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

   7. *-commutative 76.0%

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

8. Applied egg-rr 76.0%

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

9. Final simplification 76.0%

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

Alternative 15: 28.7% accurate, 34.8× speedup

Math

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

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

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return -2.0 / ((s * s) * (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 = (-2.0d0) / ((s * s) * (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 -2.0 / ((s * s) * (c * c));
}

Python

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

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(s * s) * 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 = -2.0 / ((s * s) * (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[(-2.0 / N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.9%

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

   1. *-commutative 68.9%

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

   2. associate-*l* 64.9%

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

   3. associate-*r* 64.2%

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

   4. *-commutative 64.2%

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

   5. unpow2 64.2%

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

   6. associate-*r* 69.9%

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

   7. associate-*r* 70.3%

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

   8. *-commutative 70.3%

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

   9. unpow2 70.3%

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

3. Simplified 70.3%

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

   1. add-cube-cbrt 70.3%

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

   2. times-frac 70.3%

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

   3. associate-*r* 68.9%

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

   4. swap-sqr 87.1%

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

   5. associate-*r* 93.0%

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

   6. *-commutative 93.0%

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

   7. times-frac 92.9%

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

   8. associate-*l* 95.8%

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

   9. add-cube-cbrt 95.9%

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

   10. associate-/r* 96.2%

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

5. Applied egg-rr 96.2%

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

6. Taylor expanded in x around 0 64.8%

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

7. Taylor expanded in x around inf 31.3%

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

   1. unpow2 31.3%

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

   2. unpow2 31.3%

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

9. Simplified 31.3%

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

10. Final simplification 31.3%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))