mixedcos

Percentage Accurate: 66.9% → 99.6%

Time: 12.4s

Alternatives: 13

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% 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: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)\\ \end{array} \end{array} \]

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 3.6e-31)
   (/ (/ (/ 1.0 c) (* x s)) (* c (* x s)))
   (* (pow (* s (* x c)) -2.0) (cos (* x 2.0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.6e-31) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = pow((s * (x * c)), -2.0) * cos((x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 3.6d-31) then
        tmp = ((1.0d0 / c) / (x * s)) / (c * (x * s))
    else
        tmp = ((s * (x * c)) ** (-2.0d0)) * cos((x * 2.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.6e-31) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = Math.pow((s * (x * c)), -2.0) * Math.cos((x * 2.0));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 3.6e-31:
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s))
	else:
		tmp = math.pow((s * (x * c)), -2.0) * math.cos((x * 2.0))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 3.6e-31)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	else
		tmp = Float64((Float64(s * Float64(x * c)) ^ -2.0) * cos(Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 3.6e-31)
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	else
		tmp = ((s * (x * c)) ^ -2.0) * cos((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 3.6e-31], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.60000000000000004e-31

   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. associate-/r* 69.9%

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

      2. unpow2 69.9%

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

      3. *-commutative 69.9%

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

      4. unpow2 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around 0 62.9%

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

      1. unpow2 62.9%

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

   6. Simplified 62.9%

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

      1. add-sqr-sqrt 62.9%

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

   8. Applied egg-rr 82.3%

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

      1. un-div-inv 82.4%

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

      2. associate-*l* 80.7%

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

      3. associate-*l* 82.1%

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

      4. associate-/r* 82.1%

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

   10. Applied egg-rr 82.1%

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

   if 3.60000000000000004e-31 < x

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*r* 60.5%

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

      3. associate-*r* 60.4%

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

      4. unpow2 60.4%

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

      5. unswap-sqr 73.7%

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

      6. unpow2 73.7%

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

      7. swap-sqr 90.2%

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

      8. *-commutative 90.2%

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

      9. *-commutative 90.2%

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

      10. *-commutative 90.2%

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

      11. *-commutative 90.2%

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

   3. Simplified 90.2%

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

      1. div-inv 90.3%

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

      2. *-commutative 90.3%

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

      3. pow2 90.3%

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

      4. pow-flip 91.0%

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

      5. metadata-eval 91.0%

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

   5. Applied egg-rr 91.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)\\ \end{array} \]

Alternative 2: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= x 2e-33)
     (/ (/ (/ 1.0 c) (* x s)) (* c (* x s)))
     (if (<= x 9.5e+109)
       (/ t_0 (* s (* (* x x) (* c (* c s)))))
       (/ t_0 (* s (* s (* (* x c) (* x c)))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (x <= 2e-33) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else if (x <= 9.5e+109) {
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	} else {
		tmp = t_0 / (s * (s * ((x * c) * (x * c))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (x <= 2d-33) then
        tmp = ((1.0d0 / c) / (x * s)) / (c * (x * s))
    else if (x <= 9.5d+109) then
        tmp = t_0 / (s * ((x * x) * (c * (c * s))))
    else
        tmp = t_0 / (s * (s * ((x * c) * (x * c))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 2e-33) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else if (x <= 9.5e+109) {
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	} else {
		tmp = t_0 / (s * (s * ((x * c) * (x * c))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 2e-33:
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s))
	elif x <= 9.5e+109:
		tmp = t_0 / (s * ((x * x) * (c * (c * s))))
	else:
		tmp = t_0 / (s * (s * ((x * c) * (x * c))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 2e-33)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	elseif (x <= 9.5e+109)
		tmp = Float64(t_0 / Float64(s * Float64(Float64(x * x) * Float64(c * Float64(c * s)))));
	else
		tmp = Float64(t_0 / Float64(s * Float64(s * Float64(Float64(x * c) * Float64(x * c)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 2e-33)
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	elseif (x <= 9.5e+109)
		tmp = t_0 / (s * ((x * x) * (c * (c * s))));
	else
		tmp = t_0 / (s * (s * ((x * c) * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2e-33], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+109], N[(t$95$0 / N[(s * N[(N[(x * x), $MachinePrecision] * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(s * N[(s * N[(N[(x * c), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.0000000000000001e-33

   1. Initial program 69.6%

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

      1. associate-/r* 69.8%

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

      2. unpow2 69.8%

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

      3. *-commutative 69.8%

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

      4. unpow2 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in x around 0 62.7%

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

      1. unpow2 62.7%

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

   6. Simplified 62.7%

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

      1. add-sqr-sqrt 62.7%

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

   8. Applied egg-rr 82.2%

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

      1. un-div-inv 82.3%

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

      2. associate-*l* 80.6%

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

      3. associate-*l* 82.0%

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

      4. associate-/r* 82.0%

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

   10. Applied egg-rr 82.0%

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

   if 2.0000000000000001e-33 < x < 9.49999999999999972e109

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*l* 66.4%

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

      3. associate-*r* 69.5%

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

      4. *-commutative 69.5%

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

      5. unpow2 69.5%

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

      6. associate-*r* 73.6%

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

      7. associate-*r* 76.6%

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

      8. *-commutative 76.6%

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

      9. unpow2 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in c around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. unpow2 76.6%

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

      3. associate-*l* 81.9%

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

   6. Simplified 81.9%

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

   if 9.49999999999999972e109 < x

   1. Initial program 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-*l* 57.2%

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

      3. associate-*r* 57.2%

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

      4. *-commutative 57.2%

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

      5. unpow2 57.2%

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

      6. associate-*r* 68.4%

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

      7. associate-*r* 68.4%

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

      8. *-commutative 68.4%

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

      9. unpow2 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in x around 0 63.9%

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

      1. *-commutative 63.9%

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

      2. unpow2 63.9%

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

      3. unpow2 63.9%

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

      4. swap-sqr 87.5%

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

      5. unpow2 87.5%

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

      6. *-commutative 87.5%

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

   6. Simplified 87.5%

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

      1. unpow2 87.5%

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

      2. *-commutative 87.5%

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

      3. *-commutative 87.5%

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

   8. Applied egg-rr 87.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 3: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{t_1}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{t_0}{t_1 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (* c (* x s))))
   (if (<= x 3.1e-34)
     (/ (/ (/ 1.0 c) (* x s)) t_1)
     (if (<= x 1.5e+110)
       (/ t_0 (* t_1 (* s (* x c))))
       (/ t_0 (* s (* s (* (* x c) (* x c)))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 3.1e-34) {
		tmp = ((1.0 / c) / (x * s)) / t_1;
	} else if (x <= 1.5e+110) {
		tmp = t_0 / (t_1 * (s * (x * c)));
	} else {
		tmp = t_0 / (s * (s * ((x * c) * (x * c))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    t_1 = c * (x * s)
    if (x <= 3.1d-34) then
        tmp = ((1.0d0 / c) / (x * s)) / t_1
    else if (x <= 1.5d+110) then
        tmp = t_0 / (t_1 * (s * (x * c)))
    else
        tmp = t_0 / (s * (s * ((x * c) * (x * c))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 3.1e-34) {
		tmp = ((1.0 / c) / (x * s)) / t_1;
	} else if (x <= 1.5e+110) {
		tmp = t_0 / (t_1 * (s * (x * c)));
	} else {
		tmp = t_0 / (s * (s * ((x * c) * (x * c))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	t_1 = c * (x * s)
	tmp = 0
	if x <= 3.1e-34:
		tmp = ((1.0 / c) / (x * s)) / t_1
	elif x <= 1.5e+110:
		tmp = t_0 / (t_1 * (s * (x * c)))
	else:
		tmp = t_0 / (s * (s * ((x * c) * (x * c))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 3.1e-34)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / t_1);
	elseif (x <= 1.5e+110)
		tmp = Float64(t_0 / Float64(t_1 * Float64(s * Float64(x * c))));
	else
		tmp = Float64(t_0 / Float64(s * Float64(s * Float64(Float64(x * c) * Float64(x * c)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 3.1e-34)
		tmp = ((1.0 / c) / (x * s)) / t_1;
	elseif (x <= 1.5e+110)
		tmp = t_0 / (t_1 * (s * (x * c)));
	else
		tmp = t_0 / (s * (s * ((x * c) * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.1e-34], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 1.5e+110], N[(t$95$0 / N[(t$95$1 * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(s * N[(s * N[(N[(x * c), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.0999999999999998e-34

   1. Initial program 69.6%

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

      1. associate-/r* 69.8%

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

      2. unpow2 69.8%

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

      3. *-commutative 69.8%

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

      4. unpow2 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in x around 0 62.7%

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

      1. unpow2 62.7%

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

   6. Simplified 62.7%

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

      1. add-sqr-sqrt 62.7%

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

   8. Applied egg-rr 82.2%

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

      1. un-div-inv 82.3%

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

      2. associate-*l* 80.6%

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

      3. associate-*l* 82.0%

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

      4. associate-/r* 82.0%

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

   10. Applied egg-rr 82.0%

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

   if 3.0999999999999998e-34 < x < 1.50000000000000004e110

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*r* 66.4%

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

      3. associate-*r* 69.4%

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

      4. unpow2 69.4%

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

      5. unswap-sqr 70.9%

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

      6. unpow2 70.9%

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

      7. swap-sqr 88.0%

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

      8. *-commutative 88.0%

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

      9. *-commutative 88.0%

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

      10. *-commutative 88.0%

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

      11. *-commutative 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in s around 0 88.1%

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

   if 1.50000000000000004e110 < x

   1. Initial program 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-*l* 57.2%

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

      3. associate-*r* 57.2%

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

      4. *-commutative 57.2%

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

      5. unpow2 57.2%

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

      6. associate-*r* 68.4%

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

      7. associate-*r* 68.4%

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

      8. *-commutative 68.4%

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

      9. unpow2 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in x around 0 63.9%

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

      1. *-commutative 63.9%

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

      2. unpow2 63.9%

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

      3. unpow2 63.9%

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

      4. swap-sqr 87.5%

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

      5. unpow2 87.5%

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

      6. *-commutative 87.5%

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

   6. Simplified 87.5%

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

      1. unpow2 87.5%

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

      2. *-commutative 87.5%

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

      3. *-commutative 87.5%

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

   8. Applied egg-rr 87.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 4: 98.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

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 3.4e-21)
   (/ (/ (/ 1.0 c) (* x s)) (* c (* x s)))
   (* (/ 1.0 s) (/ (cos (* x 2.0)) (* (* 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 tmp;
	if (x <= 3.4e-21) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = (1.0 / s) * (cos((x * 2.0)) / ((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) :: tmp
    if (x <= 3.4d-21) then
        tmp = ((1.0d0 / c) / (x * s)) / (c * (x * s))
    else
        tmp = (1.0d0 / s) * (cos((x * 2.0d0)) / ((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 tmp;
	if (x <= 3.4e-21) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = (1.0 / s) * (Math.cos((x * 2.0)) / ((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):
	tmp = 0
	if x <= 3.4e-21:
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s))
	else:
		tmp = (1.0 / s) * (math.cos((x * 2.0)) / ((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)
	tmp = 0.0
	if (x <= 3.4e-21)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(Float64(1.0 / s) * Float64(cos(Float64(x * 2.0)) / 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)
	tmp = 0.0;
	if (x <= 3.4e-21)
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	else
		tmp = (1.0 / s) * (cos((x * 2.0)) / ((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_] := If[LessEqual[x, 3.4e-21], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / s), $MachinePrecision] * N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 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 < 3.4e-21

   1. Initial program 69.9%

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

      1. associate-/r* 70.1%

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

      2. unpow2 70.1%

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

      3. *-commutative 70.1%

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

      4. unpow2 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 63.1%

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

      1. unpow2 63.1%

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

   6. Simplified 63.1%

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

      1. add-sqr-sqrt 63.1%

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

   8. Applied egg-rr 82.4%

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

      1. un-div-inv 82.4%

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

      2. associate-*l* 80.8%

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

      3. associate-*l* 82.2%

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

      4. associate-/r* 82.2%

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

   10. Applied egg-rr 82.2%

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

   if 3.4e-21 < x

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-*r* 59.9%

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

      3. associate-*r* 59.8%

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

      4. unpow2 59.8%

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

      5. unswap-sqr 73.4%

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

      6. unpow2 73.4%

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

      7. swap-sqr 90.1%

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

      8. *-commutative 90.1%

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

      9. *-commutative 90.1%

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

      10. *-commutative 90.1%

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

      11. *-commutative 90.1%

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

   3. Simplified 90.1%

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

      1. *-un-lft-identity 90.1%

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

      2. associate-*l* 87.4%

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

      3. times-frac 87.5%

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

      4. *-commutative 87.5%

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

   5. Applied egg-rr 87.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{t_1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

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 (* c (* x s))) (t_1 (cos (* x 2.0))))
   (if (<= x 1.6e+29)
     (* (/ 1.0 t_0) (/ t_1 t_0))
     (* (/ 1.0 s) (/ t_1 (* (* 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 = c * (x * s);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (x <= 1.6e+29) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = (1.0 / s) * (t_1 / ((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) :: t_1
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = cos((x * 2.0d0))
    if (x <= 1.6d+29) then
        tmp = (1.0d0 / t_0) * (t_1 / t_0)
    else
        tmp = (1.0d0 / s) * (t_1 / ((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 = c * (x * s);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 1.6e+29) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = (1.0 / s) * (t_1 / ((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 = c * (x * s)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if x <= 1.6e+29:
		tmp = (1.0 / t_0) * (t_1 / t_0)
	else:
		tmp = (1.0 / s) * (t_1 / ((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 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 1.6e+29)
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_1 / t_0));
	else
		tmp = Float64(Float64(1.0 / s) * Float64(t_1 / 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 = c * (x * s);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 1.6e+29)
		tmp = (1.0 / t_0) * (t_1 / t_0);
	else
		tmp = (1.0 / s) * (t_1 / ((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[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.6e+29], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / s), $MachinePrecision] * N[(t$95$1 / 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 < 1.59999999999999993e29

   1. Initial program 69.4%

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

      1. unpow2 69.4%

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

      2. *-commutative 69.4%

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

      3. unpow2 69.4%

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

   3. Simplified 69.4%

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

      1. *-un-lft-identity 69.4%

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

      2. add-sqr-sqrt 69.3%

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

      3. times-frac 69.3%

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

      4. sqrt-prod 69.3%

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

      5. sqrt-prod 37.5%

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

      6. add-sqr-sqrt 47.9%

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

      7. associate-*r* 46.2%

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

      8. sqrt-prod 46.2%

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

      9. sqrt-prod 15.5%

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

      10. add-sqr-sqrt 47.9%

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

      11. sqrt-prod 29.7%

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

      12. add-sqr-sqrt 50.1%

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

      13. *-commutative 50.1%

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

   5. Applied egg-rr 97.5%

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

   if 1.59999999999999993e29 < x

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-*r* 59.6%

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

      3. associate-*r* 57.9%

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

      4. unpow2 57.9%

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

      5. unswap-sqr 73.4%

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

      6. unpow2 73.4%

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

      7. swap-sqr 88.3%

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

      8. *-commutative 88.3%

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

      9. *-commutative 88.3%

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

      10. *-commutative 88.3%

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

      11. *-commutative 88.3%

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

   3. Simplified 88.3%

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

      1. *-un-lft-identity 88.3%

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

      2. associate-*l* 88.3%

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

      3. times-frac 88.4%

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

      4. *-commutative 88.4%

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

   5. Applied egg-rr 88.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 6: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

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 4.2e-58)
     (/ (/ (/ 1.0 c) (* x s)) (* c (* x s)))
     (* (/ (cos (* x 2.0)) t_0) (/ 1.0 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 <= 4.2e-58) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = (cos((x * 2.0)) / t_0) * (1.0 / 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 <= 4.2d-58) then
        tmp = ((1.0d0 / c) / (x * s)) / (c * (x * s))
    else
        tmp = (cos((x * 2.0d0)) / t_0) * (1.0d0 / 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 <= 4.2e-58) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = (Math.cos((x * 2.0)) / t_0) * (1.0 / 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 <= 4.2e-58:
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s))
	else:
		tmp = (math.cos((x * 2.0)) / t_0) * (1.0 / 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 <= 4.2e-58)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / t_0) * Float64(1.0 / 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 <= 4.2e-58)
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	else
		tmp = (cos((x * 2.0)) / t_0) * (1.0 / 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, 4.2e-58], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999975e-58

   1. Initial program 69.2%

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

      1. associate-/r* 69.1%

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

      2. unpow2 69.1%

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

      3. *-commutative 69.1%

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

      4. unpow2 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in x around 0 61.8%

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

      1. unpow2 61.8%

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

   6. Simplified 61.8%

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

      1. add-sqr-sqrt 61.8%

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

   8. Applied egg-rr 81.5%

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

      1. un-div-inv 81.6%

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

      2. associate-*l* 79.8%

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

      3. associate-*l* 81.3%

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

      4. associate-/r* 81.3%

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

   10. Applied egg-rr 81.3%

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

   if 4.19999999999999975e-58 < x

   1. Initial program 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-*r* 62.8%

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

      3. associate-*r* 62.7%

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

      4. unpow2 62.7%

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

      5. unswap-sqr 74.7%

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

      6. unpow2 74.7%

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

      7. swap-sqr 90.7%

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

      8. *-commutative 90.7%

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

      9. *-commutative 90.7%

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

      10. *-commutative 90.7%

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

      11. *-commutative 90.7%

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

   3. Simplified 90.7%

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

      1. associate-/r* 91.9%

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

      2. div-inv 91.9%

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

      3. *-commutative 91.9%

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

   5. Applied egg-rr 91.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 7: 93.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0145:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.0145)
   (/ (/ (/ 1.0 c) (* x s)) (* c (* x s)))
   (/ (cos (* x 2.0)) (* s (* s (* x (* c (* x c))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0145) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.0145d0) then
        tmp = ((1.0d0 / c) / (x * s)) / (c * (x * s))
    else
        tmp = cos((x * 2.0d0)) / (s * (s * (x * (c * (x * c)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.0145) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = Math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.0145:
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s))
	else:
		tmp = math.cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.0145)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(s * Float64(x * Float64(c * Float64(x * c))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.0145)
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	else
		tmp = cos((x * 2.0)) / (s * (s * (x * (c * (x * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.0145], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(s * N[(x * N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0145000000000000007

   1. Initial program 70.4%

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

      1. associate-/r* 70.6%

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

      2. unpow2 70.6%

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

      3. *-commutative 70.6%

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

      4. unpow2 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in x around 0 63.7%

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

      1. unpow2 63.7%

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

   6. Simplified 63.7%

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

      1. add-sqr-sqrt 63.7%

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

   8. Applied egg-rr 82.7%

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

      1. un-div-inv 82.7%

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

      2. associate-*l* 81.1%

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

      3. associate-*l* 82.5%

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

      4. associate-/r* 82.5%

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

   10. Applied egg-rr 82.5%

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

   if 0.0145000000000000007 < x

   1. Initial program 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-*l* 58.2%

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

      3. associate-*r* 59.5%

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

      4. *-commutative 59.5%

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

      5. unpow2 59.5%

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

      6. associate-*r* 68.4%

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

      7. associate-*r* 69.8%

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

      8. *-commutative 69.8%

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

      9. unpow2 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in x around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. associate-*r* 67.4%

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

      4. unpow2 67.4%

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

      5. associate-*r* 78.6%

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

      6. *-commutative 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0145:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.011:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 0.011)
   (/ (/ (/ 1.0 c) (* x s)) (* c (* x s)))
   (/ (cos (* x 2.0)) (* s (* s (* (* x c) (* x c)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.011) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = cos((x * 2.0)) / (s * (s * ((x * c) * (x * c))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 0.011d0) then
        tmp = ((1.0d0 / c) / (x * s)) / (c * (x * s))
    else
        tmp = cos((x * 2.0d0)) / (s * (s * ((x * c) * (x * c))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 0.011) {
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	} else {
		tmp = Math.cos((x * 2.0)) / (s * (s * ((x * c) * (x * c))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 0.011:
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s))
	else:
		tmp = math.cos((x * 2.0)) / (s * (s * ((x * c) * (x * c))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 0.011)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s * Float64(s * Float64(Float64(x * c) * Float64(x * c)))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 0.011)
		tmp = ((1.0 / c) / (x * s)) / (c * (x * s));
	else
		tmp = cos((x * 2.0)) / (s * (s * ((x * c) * (x * c))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 0.011], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(s * N[(N[(x * c), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.010999999999999999

   1. Initial program 70.4%

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

      1. associate-/r* 70.6%

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

      2. unpow2 70.6%

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

      3. *-commutative 70.6%

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

      4. unpow2 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in x around 0 63.7%

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

      1. unpow2 63.7%

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

   6. Simplified 63.7%

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

      1. add-sqr-sqrt 63.7%

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

   8. Applied egg-rr 82.7%

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

      1. un-div-inv 82.7%

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

      2. associate-*l* 81.1%

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

      3. associate-*l* 82.5%

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

      4. associate-/r* 82.5%

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

   10. Applied egg-rr 82.5%

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

   if 0.010999999999999999 < x

   1. Initial program 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-*l* 58.2%

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

      3. associate-*r* 59.5%

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

      4. *-commutative 59.5%

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

      5. unpow2 59.5%

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

      6. associate-*r* 68.4%

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

      7. associate-*r* 69.8%

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

      8. *-commutative 69.8%

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

      9. unpow2 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in x around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. unpow2 65.5%

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

      4. swap-sqr 81.0%

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

      5. unpow2 81.0%

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

      6. *-commutative 81.0%

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

   6. Simplified 81.0%

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

      1. unpow2 81.0%

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

      2. *-commutative 81.0%

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

      3. *-commutative 81.0%

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

   8. Applied egg-rr 81.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.011:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 9: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

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 3.5e-31)
     (/ (/ (/ 1.0 c) (* x s)) (* 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 <= 3.5e-31) {
		tmp = ((1.0 / c) / (x * s)) / (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 <= 3.5d-31) then
        tmp = ((1.0d0 / c) / (x * s)) / (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 <= 3.5e-31) {
		tmp = ((1.0 / c) / (x * s)) / (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 <= 3.5e-31:
		tmp = ((1.0 / c) / (x * s)) / (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 <= 3.5e-31)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s)) / 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 <= 3.5e-31)
		tmp = ((1.0 / c) / (x * s)) / (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, 3.5e-31], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $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 < 3.49999999999999985e-31

   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. associate-/r* 69.9%

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

      2. unpow2 69.9%

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

      3. *-commutative 69.9%

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

      4. unpow2 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around 0 62.9%

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

      1. unpow2 62.9%

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

   6. Simplified 62.9%

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

      1. add-sqr-sqrt 62.9%

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

   8. Applied egg-rr 82.3%

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

      1. un-div-inv 82.4%

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

      2. associate-*l* 80.7%

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

      3. associate-*l* 82.1%

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

      4. associate-/r* 82.1%

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

   10. Applied egg-rr 82.1%

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

   if 3.49999999999999985e-31 < x

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*r* 60.5%

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

      3. associate-*r* 60.4%

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

      4. unpow2 60.4%

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

      5. unswap-sqr 73.7%

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

      6. unpow2 73.7%

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

      7. swap-sqr 90.2%

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

      8. *-commutative 90.2%

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

      9. *-commutative 90.2%

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

      10. *-commutative 90.2%

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

      11. *-commutative 90.2%

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

   3. Simplified 90.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 10: 55.7% accurate, 24.1× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. *-commutative 67.6%

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

   2. associate-*r* 63.9%

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

   3. associate-*r* 63.7%

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

   4. unpow2 63.7%

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

   5. unswap-sqr 77.8%

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

   6. unpow2 77.8%

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

   7. swap-sqr 95.2%

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

   8. *-commutative 95.2%

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

   9. *-commutative 95.2%

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

   10. *-commutative 95.2%

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

   11. *-commutative 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in x around 0 55.9%

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

   1. unpow2 55.9%

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

   2. unpow2 55.9%

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

   3. *-commutative 55.9%

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

   4. unpow2 55.9%

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

6. Simplified 55.9%

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

7. Final simplification 55.9%

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

Alternative 11: 76.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. *-commutative 67.6%

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

   2. associate-*r* 63.9%

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

   3. associate-*r* 63.7%

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

   4. unpow2 63.7%

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

   5. unswap-sqr 77.8%

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

   6. unpow2 77.8%

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

   7. swap-sqr 95.2%

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

   8. *-commutative 95.2%

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

   9. *-commutative 95.2%

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

   10. *-commutative 95.2%

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

   11. *-commutative 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in x around 0 55.9%

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

   1. unpow2 55.9%

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

   2. associate-*r* 55.9%

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

   3. *-commutative 55.9%

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

   4. associate-*r* 55.6%

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

   5. *-commutative 55.6%

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

   6. unpow2 55.6%

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

   7. unpow2 55.6%

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

   8. swap-sqr 65.1%

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

   9. swap-sqr 75.4%

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

   10. unpow2 75.4%

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

   11. associate-*r* 74.9%

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

   12. *-commutative 74.9%

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

   13. *-commutative 74.9%

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

   14. *-commutative 74.9%

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

6. Simplified 74.9%

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

   1. unpow2 74.9%

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

   2. associate-*r* 74.2%

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

   3. associate-*r* 75.5%

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

8. Applied egg-rr 75.5%

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

9. Taylor expanded in c around 0 74.7%

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

10. Final simplification 74.7%

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

Alternative 12: 79.0% accurate, 24.1× speedup

Math

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

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 67.6%

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

   1. *-commutative 67.6%

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

   2. associate-*r* 63.9%

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

   3. associate-*r* 63.7%

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

   4. unpow2 63.7%

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

   5. unswap-sqr 77.8%

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

   6. unpow2 77.8%

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

   7. swap-sqr 95.2%

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

   8. *-commutative 95.2%

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

   9. *-commutative 95.2%

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

   10. *-commutative 95.2%

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

   11. *-commutative 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in x around 0 55.9%

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

   1. unpow2 55.9%

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

   2. associate-*r* 55.9%

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

   3. *-commutative 55.9%

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

   4. associate-*r* 55.6%

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

   5. *-commutative 55.6%

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

   6. unpow2 55.6%

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

   7. unpow2 55.6%

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

   8. swap-sqr 65.1%

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

   9. swap-sqr 75.4%

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

   10. unpow2 75.4%

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

   11. associate-*r* 74.9%

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

   12. *-commutative 74.9%

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

   13. *-commutative 74.9%

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

   14. *-commutative 74.9%

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

6. Simplified 74.9%

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

   1. unpow2 74.9%

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

   2. associate-*r* 74.2%

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

   3. associate-*r* 75.5%

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

8. Applied egg-rr 75.5%

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

9. Final simplification 75.5%

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

Alternative 13: 80.3% 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 67.6%

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

   1. associate-/r* 67.6%

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

   2. unpow2 67.6%

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

   3. *-commutative 67.6%

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

   4. unpow2 67.6%

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

3. Simplified 67.6%

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

4. Taylor expanded in x around 0 58.5%

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

   1. unpow2 58.5%

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

6. Simplified 58.5%

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

   1. add-sqr-sqrt 58.5%

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

8. Applied egg-rr 75.6%

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

   1. un-div-inv 75.6%

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

   2. associate-*l* 74.3%

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

   3. associate-*l* 75.4%

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

   4. associate-/r* 75.4%

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

10. Applied egg-rr 75.4%

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

11. Final simplification 75.4%

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

Reproduce

herbie shell --seed 2023224 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))