mixedcos

Percentage Accurate: 66.5% → 99.0%

Time: 11.7s

Alternatives: 13

Speedup: 24.1×

• 31.7% 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: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 11600000000000:\\ \;\;\;\;\frac{\frac{\frac{t_0}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{t_0}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= x 11600000000000.0)
     (/ (/ (/ t_0 c) (* x s)) (* c (* x s)))
     (* (/ 1.0 s) (/ t_0 (* (* x c) (* s (* x c))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (x <= 11600000000000.0) {
		tmp = ((t_0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (x <= 11600000000000.0d0) then
        tmp = ((t_0 / c) / (x * s)) / (c * (x * s))
    else
        tmp = (1.0d0 / s) * (t_0 / ((x * c) * (s * (x * c))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 11600000000000.0) {
		tmp = ((t_0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 11600000000000.0:
		tmp = ((t_0 / c) / (x * s)) / (c * (x * s))
	else:
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 11600000000000.0)
		tmp = Float64(Float64(Float64(t_0 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(Float64(1.0 / s) * Float64(t_0 / Float64(Float64(x * c) * Float64(s * Float64(x * c)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 11600000000000.0)
		tmp = ((t_0 / c) / (x * s)) / (c * (x * s));
	else
		tmp = (1.0 / s) * (t_0 / ((x * c) * (s * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 11600000000000.0], N[(N[(N[(t$95$0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / s), $MachinePrecision] * N[(t$95$0 / N[(N[(x * c), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.16e13

   1. Initial program 62.2%

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

      1. unpow2 62.2%

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

      2. *-commutative 62.2%

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

      3. unpow2 62.2%

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

   3. Simplified 62.2%

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

      1. *-un-lft-identity 62.2%

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

      2. add-sqr-sqrt 62.2%

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

      3. times-frac 62.2%

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

      4. sqrt-prod 62.2%

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

      5. sqrt-prod 26.5%

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

      6. add-sqr-sqrt 38.1%

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

      7. associate-*r* 35.4%

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

      8. sqrt-prod 35.4%

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

      9. sqrt-prod 11.0%

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

      10. add-sqr-sqrt 38.9%

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

      11. sqrt-prod 19.6%

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

      12. add-sqr-sqrt 41.4%

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

      13. *-commutative 41.4%

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

   5. Applied egg-rr 95.9%

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

      1. associate-*l/ 95.9%

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

      2. *-un-lft-identity 95.9%

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

   7. Applied egg-rr 95.9%

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

   8. Taylor expanded in x around inf 95.9%

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

      1. associate-/r* 95.9%

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

      2. *-commutative 95.9%

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

   10. Simplified 95.9%

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

   if 1.16e13 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*r* 65.8%

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

      3. associate-*r* 64.1%

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

      4. unpow2 64.1%

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

      5. unswap-sqr 82.5%

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

      6. unpow2 82.5%

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

      7. swap-sqr 99.2%

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

      8. *-commutative 99.2%

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

      9. *-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. *-commutative 99.2%

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

   3. Simplified 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. associate-*l* 99.3%

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

      3. times-frac 99.3%

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

      4. *-commutative 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 11600000000000:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 2: 86.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 3.35e-18)
   (/ (/ (/ 1.0 (* x s)) c) (* c (* x s)))
   (/ (cos (* x 2.0)) (* s (* (* x x) (* c (* c s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.35e-18) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = cos((x * 2.0)) / (s * ((x * x) * (c * (c * s))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 3.35d-18) then
        tmp = ((1.0d0 / (x * s)) / c) / (c * (x * s))
    else
        tmp = cos((x * 2.0d0)) / (s * ((x * x) * (c * (c * s))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.35e-18) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = Math.cos((x * 2.0)) / (s * ((x * x) * (c * (c * s))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 3.35e-18:
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s))
	else:
		tmp = math.cos((x * 2.0)) / (s * ((x * x) * (c * (c * s))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 3.35e-18)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) / c) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(c * s)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 3.35e-18)
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	else
		tmp = cos((x * 2.0)) / (s * ((x * x) * (c * (c * s))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 3.35e-18], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3499999999999999e-18

   1. Initial program 62.5%

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

      1. unpow2 62.5%

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

      2. *-commutative 62.5%

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

      3. unpow2 62.5%

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

   3. Simplified 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. add-sqr-sqrt 62.5%

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

      3. times-frac 62.5%

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

      4. sqrt-prod 62.5%

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

      5. sqrt-prod 27.2%

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

      6. add-sqr-sqrt 38.8%

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

      7. associate-*r* 36.0%

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

      8. sqrt-prod 36.0%

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

      9. sqrt-prod 10.4%

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

      10. add-sqr-sqrt 39.6%

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

      11. sqrt-prod 20.5%

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

      12. add-sqr-sqrt 42.3%

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

      13. *-commutative 42.3%

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

   5. Applied egg-rr 95.7%

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

      1. associate-*l/ 95.7%

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

      2. *-un-lft-identity 95.7%

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

   7. Applied egg-rr 95.7%

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

   8. Taylor expanded in x around inf 95.7%

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

      1. associate-/r* 95.8%

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

      2. *-commutative 95.8%

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

   10. Simplified 95.8%

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

   11. Taylor expanded in x around 0 81.2%

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

       1. associate-/r* 81.2%

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

       2. associate-/l/ 83.2%

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

       3. associate-/l/ 83.2%

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

       4. *-commutative 83.2%

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

   13. Simplified 83.2%

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

   if 3.3499999999999999e-18 < x

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-*l* 64.5%

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

      3. associate-*r* 62.8%

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

      4. *-commutative 62.8%

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

      5. unpow2 62.8%

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

      6. associate-*r* 75.1%

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

      7. associate-*r* 79.5%

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

      8. *-commutative 79.5%

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

      9. unpow2 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in c around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. unpow2 79.5%

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

      3. associate-*l* 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 3: 90.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 2.8e-5)
   (/ (/ (/ 1.0 (* x s)) c) (* c (* x s)))
   (/ (cos (* x 2.0)) (* x (* c (* c (* s (* x s))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.8e-5) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 2.8d-5) then
        tmp = ((1.0d0 / (x * s)) / c) / (c * (x * s))
    else
        tmp = cos((x * 2.0d0)) / (x * (c * (c * (s * (x * s)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.8e-5) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 2.8e-5:
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s))
	else:
		tmp = math.cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 2.8e-5)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) / c) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(c * Float64(c * Float64(s * Float64(x * s))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.8e-5)
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	else
		tmp = cos((x * 2.0)) / (x * (c * (c * (s * (x * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 2.8e-5], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(c * N[(c * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.79999999999999996e-5

   1. Initial program 63.1%

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

      1. unpow2 63.1%

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

      2. *-commutative 63.1%

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

      3. unpow2 63.1%

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

   3. Simplified 63.1%

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

      1. *-un-lft-identity 63.1%

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

      2. add-sqr-sqrt 63.1%

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

      3. times-frac 63.1%

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

      4. sqrt-prod 63.1%

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

      5. sqrt-prod 27.2%

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

      6. add-sqr-sqrt 38.7%

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

      7. associate-*r* 36.0%

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

      8. sqrt-prod 35.9%

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

      9. sqrt-prod 10.8%

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

      10. add-sqr-sqrt 39.5%

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

      11. sqrt-prod 20.2%

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

      12. add-sqr-sqrt 41.7%

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

      13. *-commutative 41.7%

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

   5. Applied egg-rr 95.8%

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

      1. associate-*l/ 95.8%

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

      2. *-un-lft-identity 95.8%

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

   7. Applied egg-rr 95.8%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. associate-/r* 95.8%

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

      2. *-commutative 95.8%

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

   10. Simplified 95.8%

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

   11. Taylor expanded in x around 0 81.5%

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

       1. associate-/r* 81.5%

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

       2. associate-/l/ 83.5%

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

       3. associate-/l/ 83.4%

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

       4. *-commutative 83.4%

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

   13. Simplified 83.4%

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

   if 2.79999999999999996e-5 < x

   1. Initial program 69.5%

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

      1. associate-*r* 69.0%

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

      2. *-commutative 69.0%

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

      3. associate-*r* 68.9%

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

      4. unpow2 68.9%

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

      5. unpow2 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in c around 0 68.9%

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

      1. unpow2 68.9%

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

      2. associate-*r* 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-*r* 69.0%

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

      5. *-commutative 69.0%

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

      6. unpow2 69.0%

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

      7. associate-*l* 73.6%

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

      8. *-commutative 73.6%

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

      9. associate-*l* 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4: 90.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0195:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.0195)
   (/ (/ (/ 1.0 (* x s)) c) (* c (* x s)))
   (/ (cos (* x 2.0)) (* x (* (* c (* x c)) (* s s))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0195) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.0195d0) then
        tmp = ((1.0d0 / (x * s)) / c) / (c * (x * s))
    else
        tmp = cos((x * 2.0d0)) / (x * ((c * (x * c)) * (s * s)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0195) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = Math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.0195:
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s))
	else:
		tmp = math.cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.0195)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) / c) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(c * Float64(x * c)) * Float64(s * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.0195)
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	else
		tmp = cos((x * 2.0)) / (x * ((c * (x * c)) * (s * s)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.0195], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0195

   1. Initial program 63.0%

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

      1. unpow2 63.0%

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

      2. *-commutative 63.0%

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

      3. unpow2 63.0%

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

   3. Simplified 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. add-sqr-sqrt 62.9%

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

      3. times-frac 62.9%

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

      4. sqrt-prod 63.0%

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

      5. sqrt-prod 27.0%

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

      6. add-sqr-sqrt 38.8%

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

      7. associate-*r* 36.1%

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

      8. sqrt-prod 36.1%

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

      9. sqrt-prod 11.2%

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

      10. add-sqr-sqrt 39.6%

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

      11. sqrt-prod 20.0%

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

      12. add-sqr-sqrt 41.8%

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

      13. *-commutative 41.8%

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

   5. Applied egg-rr 95.8%

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

      1. associate-*l/ 95.8%

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

      2. *-un-lft-identity 95.8%

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

   7. Applied egg-rr 95.8%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. associate-/r* 95.9%

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

      2. *-commutative 95.9%

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

   10. Simplified 95.9%

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

   11. Taylor expanded in x around 0 81.5%

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

       1. associate-/r* 81.5%

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

       2. associate-/l/ 83.4%

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

       3. associate-/l/ 83.4%

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

       4. *-commutative 83.4%

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

   13. Simplified 83.4%

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

   if 0.0195 < x

   1. Initial program 70.2%

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

      1. associate-*r* 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-*r* 69.6%

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

      4. unpow2 69.6%

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

      5. unpow2 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in c around 0 69.6%

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

      1. *-commutative 69.6%

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

      2. unpow2 69.6%

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

      3. associate-*r* 74.2%

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

      4. *-commutative 74.2%

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

   6. Simplified 74.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0195:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]

Alternative 5: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

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 1e-13)
     (/ (/ (/ 1.0 (* x s)) c) t_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 <= 1e-13) {
		tmp = ((1.0 / (x * s)) / c) / t_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 <= 1d-13) then
        tmp = ((1.0d0 / (x * s)) / c) / t_0
    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 <= 1e-13) {
		tmp = ((1.0 / (x * s)) / c) / t_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 <= 1e-13:
		tmp = ((1.0 / (x * s)) / c) / t_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 <= 1e-13)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) / c) / t_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 <= 1e-13)
		tmp = ((1.0 / (x * s)) / c) / t_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, 1e-13], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / t$95$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 < 1e-13

   1. Initial program 62.9%

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

      1. unpow2 62.9%

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

      2. *-commutative 62.9%

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

      3. unpow2 62.9%

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

   3. Simplified 62.9%

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

      1. *-un-lft-identity 62.9%

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

      2. add-sqr-sqrt 62.9%

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

      3. times-frac 62.9%

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

      4. sqrt-prod 62.9%

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

      5. sqrt-prod 27.4%

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

      6. add-sqr-sqrt 38.9%

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

      7. associate-*r* 36.2%

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

      8. sqrt-prod 36.1%

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

      9. sqrt-prod 10.9%

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

      10. add-sqr-sqrt 39.7%

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

      11. sqrt-prod 20.3%

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

      12. add-sqr-sqrt 41.9%

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

      13. *-commutative 41.9%

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

   5. Applied egg-rr 95.8%

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

      1. associate-*l/ 95.8%

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

      2. *-un-lft-identity 95.8%

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

   7. Applied egg-rr 95.8%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. associate-/r* 95.8%

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

      2. *-commutative 95.8%

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

   10. Simplified 95.8%

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

   11. Taylor expanded in x around 0 81.4%

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

       1. associate-/r* 81.4%

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

       2. associate-/l/ 83.4%

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

       3. associate-/l/ 83.3%

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

       4. *-commutative 83.3%

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

   13. Simplified 83.3%

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

   if 1e-13 < x

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*r* 63.5%

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

      3. associate-*r* 61.9%

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

      4. unpow2 61.9%

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

      5. unswap-sqr 78.5%

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

      6. unpow2 78.5%

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

      7. swap-sqr 99.2%

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

      8. *-commutative 99.2%

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

      9. *-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around 0 97.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 6: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 2.25 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= x 2.25e-13)
     (/ (/ (/ 1.0 (* x s)) c) (* c (* x s)))
     (/ (cos (* x 2.0)) (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 2.25e-13) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 2.25d-13) then
        tmp = ((1.0d0 / (x * s)) / c) / (c * (x * s))
    else
        tmp = cos((x * 2.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 2.25e-13) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 2.25e-13:
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s))
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 2.25e-13)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) / c) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 2.25e-13)
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	else
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.25e-13], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25e-13

   1. Initial program 62.9%

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

      1. unpow2 62.9%

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

      2. *-commutative 62.9%

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

      3. unpow2 62.9%

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

   3. Simplified 62.9%

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

      1. *-un-lft-identity 62.9%

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

      2. add-sqr-sqrt 62.9%

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

      3. times-frac 62.9%

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

      4. sqrt-prod 62.9%

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

      5. sqrt-prod 27.4%

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

      6. add-sqr-sqrt 38.9%

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

      7. associate-*r* 36.2%

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

      8. sqrt-prod 36.1%

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

      9. sqrt-prod 10.9%

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

      10. add-sqr-sqrt 39.7%

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

      11. sqrt-prod 20.3%

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

      12. add-sqr-sqrt 41.9%

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

      13. *-commutative 41.9%

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

   5. Applied egg-rr 95.8%

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

      1. associate-*l/ 95.8%

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

      2. *-un-lft-identity 95.8%

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

   7. Applied egg-rr 95.8%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. associate-/r* 95.8%

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

      2. *-commutative 95.8%

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

   10. Simplified 95.8%

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

   11. Taylor expanded in x around 0 81.4%

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

       1. associate-/r* 81.4%

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

       2. associate-/l/ 83.4%

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

       3. associate-/l/ 83.3%

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

       4. *-commutative 83.3%

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

   13. Simplified 83.3%

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

   if 2.25e-13 < x

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*r* 63.5%

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

      3. associate-*r* 61.9%

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

      4. unpow2 61.9%

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

      5. unswap-sqr 78.5%

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

      6. unpow2 78.5%

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

      7. swap-sqr 99.2%

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

      8. *-commutative 99.2%

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

      9. *-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. *-commutative 99.2%

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

   3. Simplified 99.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 7: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ t_2 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 9 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{t_1}{t_2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

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))) (t_1 (cos (* x 2.0))) (t_2 (* c (* x s))))
   (if (<= x 9e+119) (/ (/ t_1 t_2) t_2) (/ t_1 (* 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 t_1 = cos((x * 2.0));
	double t_2 = c * (x * s);
	double tmp;
	if (x <= 9e+119) {
		tmp = (t_1 / t_2) / t_2;
	} else {
		tmp = t_1 / (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = cos((x * 2.0d0))
    t_2 = c * (x * s)
    if (x <= 9d+119) then
        tmp = (t_1 / t_2) / t_2
    else
        tmp = t_1 / (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 t_1 = Math.cos((x * 2.0));
	double t_2 = c * (x * s);
	double tmp;
	if (x <= 9e+119) {
		tmp = (t_1 / t_2) / t_2;
	} else {
		tmp = t_1 / (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)
	t_1 = math.cos((x * 2.0))
	t_2 = c * (x * s)
	tmp = 0
	if x <= 9e+119:
		tmp = (t_1 / t_2) / t_2
	else:
		tmp = t_1 / (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))
	t_1 = cos(Float64(x * 2.0))
	t_2 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 9e+119)
		tmp = Float64(Float64(t_1 / t_2) / t_2);
	else
		tmp = Float64(t_1 / 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);
	t_1 = cos((x * 2.0));
	t_2 = c * (x * s);
	tmp = 0.0;
	if (x <= 9e+119)
		tmp = (t_1 / t_2) / t_2;
	else
		tmp = t_1 / (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]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9e+119], N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.00000000000000039e119

   1. Initial program 64.0%

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

      1. unpow2 64.0%

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

      2. *-commutative 64.0%

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

      3. unpow2 64.0%

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

   3. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. add-sqr-sqrt 63.9%

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

      3. times-frac 64.0%

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

      4. sqrt-prod 64.0%

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

      5. sqrt-prod 27.3%

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

      6. add-sqr-sqrt 39.9%

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

      7. associate-*r* 37.5%

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

      8. sqrt-prod 37.5%

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

      9. sqrt-prod 15.9%

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

      10. add-sqr-sqrt 40.6%

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

      11. sqrt-prod 20.7%

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

      12. add-sqr-sqrt 44.6%

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

      13. *-commutative 44.6%

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

   5. Applied egg-rr 96.3%

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

      1. associate-*l/ 96.3%

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

      2. *-un-lft-identity 96.3%

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

   7. Applied egg-rr 96.3%

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

   if 9.00000000000000039e119 < x

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*r* 57.2%

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

      3. associate-*r* 57.0%

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

      4. unpow2 57.0%

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

      5. unswap-sqr 87.8%

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

      6. unpow2 87.8%

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

      7. swap-sqr 99.1%

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

      8. *-commutative 99.1%

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

      9. *-commutative 99.1%

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

      10. *-commutative 99.1%

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

      11. *-commutative 99.1%

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

   3. Simplified 99.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 8: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\frac{t_1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

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))) (t_1 (cos (* x 2.0))))
   (if (<= x 7.2e+117)
     (/ (/ (/ t_1 c) (* x s)) (* c (* x s)))
     (/ t_1 (* 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 t_1 = cos((x * 2.0));
	double tmp;
	if (x <= 7.2e+117) {
		tmp = ((t_1 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = t_1 / (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) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = cos((x * 2.0d0))
    if (x <= 7.2d+117) then
        tmp = ((t_1 / c) / (x * s)) / (c * (x * s))
    else
        tmp = t_1 / (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 t_1 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 7.2e+117) {
		tmp = ((t_1 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = t_1 / (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)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if x <= 7.2e+117:
		tmp = ((t_1 / c) / (x * s)) / (c * (x * s))
	else:
		tmp = t_1 / (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))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 7.2e+117)
		tmp = Float64(Float64(Float64(t_1 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(t_1 / 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);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 7.2e+117)
		tmp = ((t_1 / c) / (x * s)) / (c * (x * s));
	else
		tmp = t_1 / (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]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 7.2e+117], N[(N[(N[(t$95$1 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000025e117

   1. Initial program 64.0%

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

      1. unpow2 64.0%

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

      2. *-commutative 64.0%

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

      3. unpow2 64.0%

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

   3. Simplified 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. add-sqr-sqrt 63.9%

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

      3. times-frac 64.0%

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

      4. sqrt-prod 64.0%

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

      5. sqrt-prod 27.3%

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

      6. add-sqr-sqrt 39.9%

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

      7. associate-*r* 37.5%

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

      8. sqrt-prod 37.5%

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

      9. sqrt-prod 15.9%

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

      10. add-sqr-sqrt 40.6%

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

      11. sqrt-prod 20.7%

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

      12. add-sqr-sqrt 44.6%

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

      13. *-commutative 44.6%

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

   5. Applied egg-rr 96.3%

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

      1. associate-*l/ 96.3%

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

      2. *-un-lft-identity 96.3%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in x around inf 96.3%

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

      1. associate-/r* 96.3%

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

      2. *-commutative 96.3%

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

   10. Simplified 96.3%

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

   if 7.20000000000000025e117 < x

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*r* 57.2%

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

      3. associate-*r* 57.0%

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

      4. unpow2 57.0%

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

      5. unswap-sqr 87.8%

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

      6. unpow2 87.8%

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

      7. swap-sqr 99.1%

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

      8. *-commutative 99.1%

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

      9. *-commutative 99.1%

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

      10. *-commutative 99.1%

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

      11. *-commutative 99.1%

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

   3. Simplified 99.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 9: 55.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.8%

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

   1. *-commutative 64.8%

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

   2. associate-*r* 59.5%

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

   3. associate-*r* 58.8%

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

   4. unpow2 58.8%

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

   5. unswap-sqr 72.4%

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

   6. unpow2 72.4%

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

   7. swap-sqr 97.6%

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

   8. *-commutative 97.6%

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

   9. *-commutative 97.6%

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

   10. *-commutative 97.6%

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

   11. *-commutative 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in x around 0 51.8%

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

   1. unpow2 51.8%

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

   2. unpow2 51.8%

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

   3. *-commutative 51.8%

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

   4. unpow2 51.8%

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

6. Simplified 51.8%

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

7. Final simplification 51.8%

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

Alternative 10: 79.4% 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 64.8%

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

   1. unpow2 64.8%

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

   2. *-commutative 64.8%

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

   3. unpow2 64.8%

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

3. Simplified 64.8%

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

   1. *-un-lft-identity 64.8%

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

   2. add-sqr-sqrt 64.7%

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

   3. times-frac 64.7%

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

   4. sqrt-prod 64.8%

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

   5. sqrt-prod 27.6%

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

   6. add-sqr-sqrt 42.4%

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

   7. associate-*r* 39.8%

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

   8. sqrt-prod 39.8%

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

   9. sqrt-prod 21.7%

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

   10. add-sqr-sqrt 43.0%

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

   11. sqrt-prod 21.3%

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

   12. add-sqr-sqrt 46.3%

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

   13. *-commutative 46.3%

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

5. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. *-un-lft-identity 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in x around 0 51.5%

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

   1. associate-*r* 51.7%

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

   2. *-commutative 51.7%

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

   3. associate-*r* 51.8%

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

   4. associate-/r* 51.7%

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

   5. unpow2 51.7%

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

   6. associate-/r* 51.7%

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

   7. *-lft-identity 51.7%

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

   8. associate-*l/ 51.7%

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

   9. *-commutative 51.7%

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

   10. unpow2 51.7%

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

   11. unpow2 51.7%

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

   12. swap-sqr 64.4%

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

   13. times-frac 76.8%

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

   14. unpow2 76.8%

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

   15. *-commutative 76.8%

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

   16. associate-/r* 77.6%

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

10. Simplified 77.6%

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

    1. unpow2 77.6%

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

    2. clear-num 77.6%

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

    3. clear-num 77.5%

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

    4. frac-times 77.5%

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

    5. metadata-eval 77.5%

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

    6. div-inv 77.5%

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

    7. clear-num 77.6%

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

    8. div-inv 77.6%

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

    9. clear-num 77.6%

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

    10. /-rgt-identity 77.6%

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

    11. div-inv 77.6%

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

    12. clear-num 77.6%

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

    13. div-inv 77.6%

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

    14. clear-num 77.6%

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

    15. /-rgt-identity 77.6%

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

12. Applied egg-rr 77.6%

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

13. Final simplification 77.6%

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

Alternative 11: 80.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * 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 = c * (x * 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 = c * (x * 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 = c * (x * 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(c * Float64(x * s))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * 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[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.8%

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

   1. unpow2 64.8%

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

   2. *-commutative 64.8%

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

   3. unpow2 64.8%

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

3. Simplified 64.8%

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

   1. *-un-lft-identity 64.8%

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

   2. add-sqr-sqrt 64.7%

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

   3. times-frac 64.7%

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

   4. sqrt-prod 64.8%

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

   5. sqrt-prod 27.6%

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

   6. add-sqr-sqrt 42.4%

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

   7. associate-*r* 39.8%

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

   8. sqrt-prod 39.8%

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

   9. sqrt-prod 21.7%

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

   10. add-sqr-sqrt 43.0%

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

   11. sqrt-prod 21.3%

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

   12. add-sqr-sqrt 46.3%

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

   13. *-commutative 46.3%

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

5. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. *-un-lft-identity 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in x around 0 76.8%

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

9. Final simplification 76.8%

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

Alternative 12: 80.6% accurate, 24.1× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.8%

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

   1. unpow2 64.8%

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

   2. *-commutative 64.8%

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

   3. unpow2 64.8%

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

3. Simplified 64.8%

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

   1. *-un-lft-identity 64.8%

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

   2. add-sqr-sqrt 64.7%

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

   3. times-frac 64.7%

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

   4. sqrt-prod 64.8%

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

   5. sqrt-prod 27.6%

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

   6. add-sqr-sqrt 42.4%

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

   7. associate-*r* 39.8%

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

   8. sqrt-prod 39.8%

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

   9. sqrt-prod 21.7%

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

   10. add-sqr-sqrt 43.0%

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

   11. sqrt-prod 21.3%

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

   12. add-sqr-sqrt 46.3%

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

   13. *-commutative 46.3%

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

5. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. *-un-lft-identity 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in x around 0 76.8%

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

   1. associate-/r* 76.8%

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

10. Simplified 76.8%

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

11. Final simplification 76.8%

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

Alternative 13: 80.6% accurate, 24.1× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.8%

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

   1. unpow2 64.8%

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

   2. *-commutative 64.8%

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

   3. unpow2 64.8%

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

3. Simplified 64.8%

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

   1. *-un-lft-identity 64.8%

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

   2. add-sqr-sqrt 64.7%

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

   3. times-frac 64.7%

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

   4. sqrt-prod 64.8%

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

   5. sqrt-prod 27.6%

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

   6. add-sqr-sqrt 42.4%

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

   7. associate-*r* 39.8%

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

   8. sqrt-prod 39.8%

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

   9. sqrt-prod 21.7%

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

   10. add-sqr-sqrt 43.0%

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

   11. sqrt-prod 21.3%

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

   12. add-sqr-sqrt 46.3%

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

   13. *-commutative 46.3%

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

5. Applied egg-rr 96.4%

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

   1. associate-*l/ 96.4%

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

   2. *-un-lft-identity 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in x around inf 96.4%

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

   1. associate-/r* 96.4%

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

   2. *-commutative 96.4%

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

10. Simplified 96.4%

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

11. Taylor expanded in x around 0 75.3%

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

    1. associate-/r* 75.3%

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

    2. associate-/l/ 76.8%

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

    3. associate-/l/ 76.8%

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

    4. *-commutative 76.8%

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

13. Simplified 76.8%

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

14. Final simplification 76.8%

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

Reproduce

herbie shell --seed 2023217 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))