mixedcos

Percentage Accurate: 66.1% → 99.1%

Time: 11.0s

Alternatives: 10

Speedup: 24.1×

• 30.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% 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: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-60}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 1.2e-60)
     (pow (* c (* x s)) -2.0)
     (/ (cos (* x 2.0)) (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 1.2e-60) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 1.2d-60) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 1.2e-60) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 1.2e-60:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 1.2e-60)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 1.2e-60)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.2e-60], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.20000000000000005e-60

   1. Initial program 65.4%

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

      1. associate-*r* 67.5%

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

      2. *-commutative 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-*r* 65.4%

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

      5. *-commutative 65.4%

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

      6. unpow2 65.4%

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

      7. unpow2 65.4%

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

   3. Simplified 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. associate-*r* 66.9%

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

      3. *-commutative 66.9%

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

      4. add-sqr-sqrt 66.9%

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

      5. times-frac 66.9%

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

   5. Applied egg-rr 97.2%

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

      1. associate-*l/ 97.3%

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

      2. *-un-lft-identity 97.3%

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

   7. Applied egg-rr 97.3%

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

   8. Taylor expanded in c around 0 94.9%

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

   9. Taylor expanded in x around 0 53.8%

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

       1. associate-/r* 53.7%

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

       2. unpow2 53.7%

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

       3. unpow2 53.7%

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

       4. swap-sqr 66.3%

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

       5. *-commutative 66.3%

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

       6. associate-*r* 64.6%

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

       7. associate-/r* 64.7%

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

       8. unpow2 64.7%

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

       9. associate-*r* 71.7%

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

       10. associate-/r* 71.7%

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

       11. *-rgt-identity 71.7%

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

       12. associate-*r* 73.8%

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

       13. *-commutative 73.8%

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

       14. associate-*r* 80.8%

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

       15. *-commutative 80.8%

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

       16. times-frac 83.7%

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

       17. associate-/r* 83.7%

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

       18. unpow-1 83.7%

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

       19. unpow-1 83.7%

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

       20. pow-sqr 83.8%

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

   11. Simplified 83.8%

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

   if 1.20000000000000005e-60 < x

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*r* 63.0%

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

      3. associate-*r* 62.9%

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

      4. unpow2 62.9%

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

      5. unswap-sqr 77.3%

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

      6. unpow2 77.3%

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

      7. swap-sqr 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%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-60}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 2: 93.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-17}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 8.8e-17)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x 2.0)) (* s (* s (* x (* c (* x c))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 8.8e-17) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 8.8d-17) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (s * (s * (x * (c * (x * c)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 8.8e-17) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 8.8e-17:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 8.8e-17)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(s * Float64(x * Float64(c * Float64(x * c))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 8.8e-17)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 8.8e-17], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(s * N[(x * N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8e-17

   1. Initial program 66.8%

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

      1. associate-*r* 68.8%

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

      2. *-commutative 68.8%

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

      3. *-commutative 68.8%

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

      4. associate-*r* 66.9%

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

      5. *-commutative 66.9%

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

      6. unpow2 66.9%

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

      7. unpow2 66.9%

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

   3. Simplified 66.9%

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

      1. *-un-lft-identity 66.9%

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

      2. associate-*r* 68.2%

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

      3. *-commutative 68.2%

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

      4. add-sqr-sqrt 68.2%

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

      5. times-frac 68.2%

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

   5. Applied egg-rr 97.4%

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

      1. associate-*l/ 97.4%

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

      2. *-un-lft-identity 97.4%

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

   7. Applied egg-rr 97.4%

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

   8. Taylor expanded in c around 0 95.2%

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

   9. Taylor expanded in x around 0 56.0%

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

       1. associate-/r* 56.0%

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

       2. unpow2 56.0%

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

       3. unpow2 56.0%

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

       4. swap-sqr 67.7%

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

       5. *-commutative 67.7%

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

       6. associate-*r* 66.1%

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

       7. associate-/r* 66.1%

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

       8. unpow2 66.1%

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

       9. associate-*r* 73.1%

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

       10. associate-/r* 73.1%

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

       11. *-rgt-identity 73.1%

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

       12. associate-*r* 75.1%

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

       13. *-commutative 75.1%

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

       14. associate-*r* 82.2%

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

       15. *-commutative 82.2%

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

       16. times-frac 84.8%

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

       17. associate-/r* 84.8%

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

       18. unpow-1 84.8%

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

       19. unpow-1 84.8%

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

       20. pow-sqr 84.9%

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

   11. Simplified 84.9%

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

   if 8.8e-17 < x

   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. *-commutative 66.4%

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

      2. associate-*l* 57.2%

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

      3. associate-*r* 55.4%

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

      4. *-commutative 55.4%

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

      5. unpow2 55.4%

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

      6. associate-*r* 65.3%

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

      7. associate-*r* 69.0%

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

      8. *-commutative 69.0%

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

      9. unpow2 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in x around 0 67.3%

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

      1. unpow2 55.8%

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

      2. *-commutative 55.8%

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

      3. associate-*r* 58.0%

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

      4. unpow2 58.0%

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

      5. associate-*r* 60.4%

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

      6. *-commutative 60.4%

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

   6. Simplified 84.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-17}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 3: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 5.2 \cdot 10^{-45}:\\ \;\;\;\;{t_0}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= x 5.2e-45)
     (pow t_0 -2.0)
     (/ (cos (* x 2.0)) (* t_0 (* s (* x c)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 5.2e-45) {
		tmp = pow(t_0, -2.0);
	} else {
		tmp = cos((x * 2.0)) / (t_0 * (s * (x * c)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 5.2d-45) then
        tmp = t_0 ** (-2.0d0)
    else
        tmp = cos((x * 2.0d0)) / (t_0 * (s * (x * c)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 5.2e-45) {
		tmp = Math.pow(t_0, -2.0);
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * (s * (x * c)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 5.2e-45:
		tmp = math.pow(t_0, -2.0)
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * (s * (x * c)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 5.2e-45)
		tmp = t_0 ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * Float64(s * Float64(x * c))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 5.2e-45)
		tmp = t_0 ^ -2.0;
	else
		tmp = cos((x * 2.0)) / (t_0 * (s * (x * c)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.2e-45], N[Power[t$95$0, -2.0], $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999973e-45

   1. Initial program 66.1%

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

      1. associate-*r* 68.2%

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

      2. *-commutative 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r* 66.2%

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

      5. *-commutative 66.2%

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

      6. unpow2 66.2%

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

      7. unpow2 66.2%

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

   3. Simplified 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. associate-*r* 67.6%

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

      3. *-commutative 67.6%

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

      4. add-sqr-sqrt 67.6%

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

      5. times-frac 67.6%

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

   5. Applied egg-rr 97.3%

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

      1. associate-*l/ 97.3%

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

      2. *-un-lft-identity 97.3%

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

   7. Applied egg-rr 97.3%

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

   8. Taylor expanded in c around 0 95.0%

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

   9. Taylor expanded in x around 0 54.7%

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

       1. associate-/r* 54.7%

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

       2. unpow2 54.7%

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

       3. unpow2 54.7%

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

       4. swap-sqr 67.0%

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

       5. *-commutative 67.0%

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

       6. associate-*r* 65.4%

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

       7. associate-/r* 65.4%

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

       8. unpow2 65.4%

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

       9. associate-*r* 72.2%

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

       10. associate-/r* 72.2%

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

       11. *-rgt-identity 72.2%

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

       12. associate-*r* 74.3%

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

       13. *-commutative 74.3%

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

       14. associate-*r* 81.2%

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

       15. *-commutative 81.2%

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

       16. times-frac 84.0%

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

       17. associate-/r* 84.0%

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

       18. unpow-1 84.0%

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

       19. unpow-1 84.0%

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

       20. pow-sqr 84.1%

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

   11. Simplified 84.1%

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

   if 5.19999999999999973e-45 < x

   1. Initial program 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-*r* 60.8%

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

      3. associate-*r* 60.7%

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

      4. unpow2 60.7%

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

      5. unswap-sqr 75.9%

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

      6. unpow2 75.9%

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

      7. swap-sqr 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 91.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-45}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 4: 79.2% 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 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* 68.6%

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

   2. *-commutative 68.6%

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

   3. *-commutative 68.6%

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

   4. associate-*r* 66.0%

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

   5. *-commutative 66.0%

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

   6. unpow2 66.0%

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

   7. unpow2 66.0%

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

3. Simplified 66.0%

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

   1. *-un-lft-identity 66.0%

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

   2. associate-*r* 68.2%

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

   3. *-commutative 68.2%

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

   4. add-sqr-sqrt 68.2%

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

   5. times-frac 68.2%

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

5. Applied egg-rr 97.2%

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

   1. associate-*l/ 97.2%

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

   2. *-un-lft-identity 97.2%

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

7. Applied egg-rr 97.2%

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

8. Taylor expanded in c around 0 94.8%

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

9. Taylor expanded in x around 0 54.3%

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

    1. associate-/r* 54.2%

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

    2. unpow2 54.2%

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

    3. unpow2 54.2%

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

    4. swap-sqr 64.3%

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

    5. *-commutative 64.3%

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

    6. associate-*r* 63.1%

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

    7. associate-/r* 63.2%

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

    8. unpow2 63.2%

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

    9. associate-*r* 70.4%

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

    10. associate-/r* 70.4%

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

    11. *-rgt-identity 70.4%

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

    12. associate-*r* 72.0%

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

    13. *-commutative 72.0%

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

    14. associate-*r* 77.7%

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

    15. *-commutative 77.7%

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

    16. times-frac 79.8%

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

    17. associate-/r* 79.8%

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

    18. unpow-1 79.8%

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

    19. unpow-1 79.8%

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

    20. pow-sqr 79.8%

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

11. Simplified 80.0%

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

12. Final simplification 80.0%

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

Alternative 5: 67.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-201}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 2.3e-201)
   (/ 1.0 (* s (* s (* x (* c (* x c))))))
   (/ 1.0 (* s (* (* x x) (* c (* c s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.3e-201) {
		tmp = 1.0 / (s * (s * (x * (c * (x * c)))));
	} else {
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 2.3d-201) then
        tmp = 1.0d0 / (s * (s * (x * (c * (x * c)))))
    else
        tmp = 1.0d0 / (s * ((x * x) * (c * (c * s))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.3e-201) {
		tmp = 1.0 / (s * (s * (x * (c * (x * c)))));
	} else {
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 2.3e-201:
		tmp = 1.0 / (s * (s * (x * (c * (x * c)))))
	else:
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 2.3e-201)
		tmp = Float64(1.0 / Float64(s * Float64(s * Float64(x * Float64(c * Float64(x * c))))));
	else
		tmp = Float64(1.0 / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(c * s)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.3e-201)
		tmp = 1.0 / (s * (s * (x * (c * (x * c)))));
	else
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 2.3e-201], N[(1.0 / N[(s * N[(s * N[(x * N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.29999999999999986e-201

   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. *-commutative 66.7%

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

      2. associate-*l* 61.0%

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

      3. associate-*r* 59.6%

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

      4. *-commutative 59.6%

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

      5. unpow2 59.6%

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

      6. associate-*r* 64.8%

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

      7. associate-*r* 68.0%

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

      8. *-commutative 68.0%

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

      9. unpow2 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in x around 0 60.5%

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

   5. Taylor expanded in x around 0 60.0%

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

      1. unpow2 60.0%

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

      2. *-commutative 60.0%

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

      3. associate-*r* 65.2%

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

      4. unpow2 65.2%

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

      5. associate-*r* 71.9%

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

      6. *-commutative 71.9%

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

   7. Simplified 71.9%

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

   if 2.29999999999999986e-201 < x

   1. Initial program 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-*l* 61.1%

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

      3. associate-*r* 61.0%

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

      4. *-commutative 61.0%

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

      5. unpow2 61.0%

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

      6. associate-*r* 70.3%

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

      7. associate-*r* 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. Taylor expanded in x around 0 64.6%

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

      1. expm1-log1p-u 42.6%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c \cdot c\right) \cdot s\right)\right)}\right)} \]

      2. expm1-udef 26.3%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(c \cdot c\right) \cdot s\right)} - 1\right)}\right)} \]

      3. associate-*l* 26.3%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \left(c \cdot s\right)}\right)} - 1\right)\right)} \]

      4. add-sqr-sqrt 14.6%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\left(\sqrt{c \cdot s} \cdot \sqrt{c \cdot s}\right)}\right)} - 1\right)\right)} \]

      5. sqrt-prod 26.8%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\sqrt{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}}\right)} - 1\right)\right)} \]

      6. unswap-sqr 24.6%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \sqrt{\color{blue}{\left(c \cdot c\right) \cdot \left(s \cdot s\right)}}\right)} - 1\right)\right)} \]

      7. *-commutative 24.6%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \sqrt{\color{blue}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}}\right)} - 1\right)\right)} \]

      8. sqrt-prod 24.6%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\left(\sqrt{s \cdot s} \cdot \sqrt{c \cdot c}\right)}\right)} - 1\right)\right)} \]

      9. sqrt-prod 15.7%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(\color{blue}{\left(\sqrt{s} \cdot \sqrt{s}\right)} \cdot \sqrt{c \cdot c}\right)\right)} - 1\right)\right)} \]

      10. add-sqr-sqrt 25.8%

          \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(\color{blue}{s} \cdot \sqrt{c \cdot c}\right)\right)} - 1\right)\right)} \]

      11. sqrt-prod 13.5%

          \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(s \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)}\right)\right)} - 1\right)\right)} \]

      12. add-sqr-sqrt 26.3%

          \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(s \cdot \color{blue}{c}\right)\right)} - 1\right)\right)} \]

   6. Applied egg-rr 26.3%

      \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot \left(s \cdot c\right)\right)} - 1\right)}\right)} \]
   7. Step-by-step derivation

      1. expm1-def 43.8%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(s \cdot c\right)\right)\right)}\right)} \]

      2. expm1-log1p 66.9%

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

      3. *-commutative 66.9%

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

   8. Simplified 66.9%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-201}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 6: 68.0% accurate, 20.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;c \leq 2.06 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= c 2.06e-17)
   (/ 1.0 (* s (* (* x x) (* c (* c s)))))
   (/ 1.0 (* x (* (* x (* c c)) (* s s))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (c <= 2.06e-17) {
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))));
	} else {
		tmp = 1.0 / (x * ((x * (c * c)) * (s * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (c <= 2.06d-17) then
        tmp = 1.0d0 / (s * ((x * x) * (c * (c * s))))
    else
        tmp = 1.0d0 / (x * ((x * (c * c)) * (s * s)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 2.06e-17) {
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))));
	} else {
		tmp = 1.0 / (x * ((x * (c * c)) * (s * s)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if c <= 2.06e-17:
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))))
	else:
		tmp = 1.0 / (x * ((x * (c * c)) * (s * s)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (c <= 2.06e-17)
		tmp = Float64(1.0 / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(c * s)))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x * Float64(c * c)) * Float64(s * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 2.06e-17)
		tmp = 1.0 / (s * ((x * x) * (c * (c * s))));
	else
		tmp = 1.0 / (x * ((x * (c * c)) * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[c, 2.06e-17], N[(1.0 / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(N[(x * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.06000000000000013e-17

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-*l* 61.9%

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

      3. associate-*r* 60.2%

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

      4. *-commutative 60.2%

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

      5. unpow2 60.2%

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

      6. associate-*r* 66.3%

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

      7. associate-*r* 69.5%

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

      8. *-commutative 69.5%

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

      9. unpow2 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around 0 61.1%

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

      1. expm1-log1p-u 46.2%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c \cdot c\right) \cdot s\right)\right)}\right)} \]

      2. expm1-udef 27.7%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(c \cdot c\right) \cdot s\right)} - 1\right)}\right)} \]

      3. associate-*l* 29.3%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \left(c \cdot s\right)}\right)} - 1\right)\right)} \]

      4. add-sqr-sqrt 11.8%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\left(\sqrt{c \cdot s} \cdot \sqrt{c \cdot s}\right)}\right)} - 1\right)\right)} \]

      5. sqrt-prod 20.2%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\sqrt{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}}\right)} - 1\right)\right)} \]

      6. unswap-sqr 18.4%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \sqrt{\color{blue}{\left(c \cdot c\right) \cdot \left(s \cdot s\right)}}\right)} - 1\right)\right)} \]

      7. *-commutative 18.4%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \sqrt{\color{blue}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}}\right)} - 1\right)\right)} \]

      8. sqrt-prod 18.4%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\left(\sqrt{s \cdot s} \cdot \sqrt{c \cdot c}\right)}\right)} - 1\right)\right)} \]

      9. sqrt-prod 17.3%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(\color{blue}{\left(\sqrt{s} \cdot \sqrt{s}\right)} \cdot \sqrt{c \cdot c}\right)\right)} - 1\right)\right)} \]

      10. add-sqr-sqrt 30.7%

          \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(\color{blue}{s} \cdot \sqrt{c \cdot c}\right)\right)} - 1\right)\right)} \]

      11. sqrt-prod 8.5%

          \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(s \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)}\right)\right)} - 1\right)\right)} \]

      12. add-sqr-sqrt 29.3%

          \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(e^{\mathsf{log1p}\left(c \cdot \left(s \cdot \color{blue}{c}\right)\right)} - 1\right)\right)} \]

   6. Applied egg-rr 29.3%

      \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot \left(s \cdot c\right)\right)} - 1\right)}\right)} \]
   7. Step-by-step derivation

      1. expm1-def 52.3%

         \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(s \cdot c\right)\right)\right)}\right)} \]

      2. expm1-log1p 67.8%

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

      3. *-commutative 67.8%

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

   8. Simplified 67.8%

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

   if 2.06000000000000013e-17 < c

   1. Initial program 63.4%

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

      1. associate-*r* 67.3%

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

      2. *-commutative 67.3%

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

      3. associate-*r* 65.7%

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

      4. unpow2 65.7%

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

      5. unpow2 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in x around 0 63.2%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.06 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 7: 67.4% accurate, 24.1× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 66.7%

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

   2. associate-*l* 61.0%

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

   3. associate-*r* 60.1%

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

   4. *-commutative 60.1%

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

   5. unpow2 60.1%

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

   6. associate-*r* 66.8%

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

   7. associate-*r* 69.6%

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

   8. *-commutative 69.6%

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

   9. unpow2 69.6%

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

3. Simplified 69.6%

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

4. Taylor expanded in x around 0 62.0%

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

5. Taylor expanded in x around 0 60.6%

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

   1. unpow2 60.6%

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

   2. *-commutative 60.6%

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

   3. associate-*r* 64.5%

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

   4. unpow2 64.5%

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

   5. associate-*r* 70.4%

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

   6. *-commutative 70.4%

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

7. Simplified 70.4%

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

8. Final simplification 70.4%

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

Alternative 8: 76.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. *-commutative 66.7%

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

   2. associate-*r* 61.0%

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

   3. associate-*r* 60.4%

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

   4. unpow2 60.4%

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

   5. unswap-sqr 74.9%

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

   6. unpow2 74.9%

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

   7. swap-sqr 96.9%

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

   8. *-commutative 96.9%

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

   9. *-commutative 96.9%

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

   10. *-commutative 96.9%

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

   11. *-commutative 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in x around 0 79.9%

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

5. Final simplification 79.9%

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

Alternative 9: 77.8% accurate, 24.1× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 68.6%

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

   2. *-commutative 68.6%

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

   3. *-commutative 68.6%

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

   4. associate-*r* 66.0%

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

   5. *-commutative 66.0%

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

   6. unpow2 66.0%

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

   7. unpow2 66.0%

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

3. Simplified 66.0%

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

   1. *-un-lft-identity 66.0%

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

   2. associate-*r* 68.2%

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

   3. *-commutative 68.2%

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

   4. add-sqr-sqrt 68.2%

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

   5. times-frac 68.2%

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

5. Applied egg-rr 97.2%

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

6. Taylor expanded in x around 0 54.3%

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

   1. unpow2 54.3%

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

   2. unpow2 54.3%

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

   3. unpow2 54.3%

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

   4. swap-sqr 64.4%

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

   5. swap-sqr 79.9%

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

   6. associate-*r* 78.7%

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

   7. associate-*r* 80.4%

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

   8. *-commutative 80.4%

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

   9. *-commutative 80.4%

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

   10. unpow2 80.4%

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

   11. *-commutative 80.4%

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

8. Simplified 80.4%

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

   1. unpow2 80.4%

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

   2. *-commutative 80.4%

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

   3. *-commutative 80.4%

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

10. Applied egg-rr 80.4%

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

11. Final simplification 80.4%

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

Alternative 10: 28.2% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. *-commutative 66.7%

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

   2. associate-*r* 61.0%

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

   3. associate-*r* 60.4%

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

   4. unpow2 60.4%

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

   5. unswap-sqr 74.9%

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

   6. unpow2 74.9%

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

   7. swap-sqr 96.9%

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

   8. *-commutative 96.9%

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

   9. *-commutative 96.9%

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

   10. *-commutative 96.9%

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

   11. *-commutative 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in x around 0 29.8%

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

   1. unpow2 29.8%

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

   2. unpow2 29.8%

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

   3. associate-*r* 30.2%

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

   4. *-commutative 30.2%

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

   5. associate-*r* 30.0%

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

   6. unpow2 30.0%

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

   7. associate-*r/ 30.0%

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

   8. metadata-eval 30.0%

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

   9. unpow2 30.0%

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

   10. unpow2 30.0%

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

6. Simplified 30.0%

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

7. Taylor expanded in x around inf 26.9%

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

   1. unpow2 26.9%

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

   2. unpow2 26.9%

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

9. Simplified 26.9%

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

10. Final simplification 26.9%

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

Reproduce

herbie shell --seed 2023257 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))