mixedcos

Percentage Accurate: 66.7% → 99.3%

Time: 11.9s

Alternatives: 8

Speedup: 24.1×

• 31.4% 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: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))) (t_1 (* c (* x s))))
   (if (<= x 5e-64) (/ 1.0 (* t_1 t_1)) (/ (/ (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 t_1 = c * (x * s);
	double tmp;
	if (x <= 5e-64) {
		tmp = 1.0 / (t_1 * t_1);
	} 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) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = c * (x * s)
    if (x <= 5d-64) then
        tmp = 1.0d0 / (t_1 * t_1)
    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 t_1 = c * (x * s);
	double tmp;
	if (x <= 5e-64) {
		tmp = 1.0 / (t_1 * t_1);
	} 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)
	t_1 = c * (x * s)
	tmp = 0
	if x <= 5e-64:
		tmp = 1.0 / (t_1 * t_1)
	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))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 5e-64)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 5e-64)
		tmp = 1.0 / (t_1 * t_1);
	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]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e-64], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000033e-64

   1. Initial program 70.2%

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

      1. associate-/r* 70.2%

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

      2. remove-double-neg 70.2%

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

      3. distribute-lft-neg-out 70.2%

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

      4. distribute-lft-neg-out 70.2%

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

      5. distribute-rgt-neg-out 70.2%

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

      6. associate-/l/ 70.2%

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

      7. distribute-rgt-neg-out 70.2%

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

      8. distribute-lft-neg-out 70.2%

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

      9. associate-*l* 71.7%

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

      10. distribute-lft-neg-in 71.7%

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

      11. distribute-lft-neg-out 71.7%

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

      12. remove-double-neg 71.7%

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

      13. associate-*r* 70.7%

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

      14. *-commutative 70.7%

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

      15. associate-*r* 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in x around 0 58.6%

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

      1. unpow2 58.6%

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

      2. unpow2 58.6%

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

      3. associate-*r* 58.2%

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

      4. *-commutative 58.2%

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

      5. associate-*r* 56.6%

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

      6. unpow2 56.6%

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

   6. Simplified 56.6%

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

      1. add-sqr-sqrt 56.6%

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

      2. pow2 56.6%

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

      3. sqrt-div 56.5%

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

      4. metadata-eval 56.5%

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

      5. sqrt-prod 56.5%

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

      6. sqrt-prod 32.8%

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

      7. add-sqr-sqrt 62.6%

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

      8. unswap-sqr 76.7%

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

      9. sqrt-prod 54.7%

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

      10. add-sqr-sqrt 85.4%

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

      11. associate-*r* 85.2%

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

      12. *-commutative 85.2%

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

      13. associate-*r* 85.5%

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

      14. pow2 85.5%

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

      15. associate-/r* 85.6%

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

      16. associate-/r* 85.5%

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

   8. Applied egg-rr 85.5%

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

      1. associate-/l/ 85.6%

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

      2. associate-/l/ 85.5%

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

      3. frac-times 85.5%

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

      4. metadata-eval 85.5%

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

      5. *-commutative 85.5%

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

      6. *-commutative 85.5%

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

   10. Applied egg-rr 85.5%

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

   if 5.00000000000000033e-64 < x

   1. Initial program 60.9%

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

      1. associate-/r* 60.0%

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

      2. remove-double-neg 60.0%

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

      3. distribute-lft-neg-out 60.0%

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

      4. distribute-lft-neg-out 60.0%

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

      5. distribute-rgt-neg-out 60.0%

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

      6. associate-/l/ 60.9%

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

      7. distribute-rgt-neg-out 60.9%

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

      8. distribute-lft-neg-out 60.9%

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

      9. associate-*l* 60.8%

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

      10. distribute-lft-neg-in 60.8%

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

      11. distribute-lft-neg-out 60.8%

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

      12. remove-double-neg 60.8%

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

      13. associate-*r* 62.1%

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

      14. *-commutative 62.1%

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

      15. associate-*r* 60.9%

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

   3. Simplified 60.9%

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

      1. associate-/r* 61.3%

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

      2. swap-sqr 91.9%

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

      3. associate-/r* 82.9%

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

      4. associate-/r* 82.9%

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

      5. associate-/r* 83.0%

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

      6. *-un-lft-identity 83.0%

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

      7. add-sqr-sqrt 82.9%

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

      8. times-frac 82.9%

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

   5. Applied egg-rr 97.2%

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

      1. associate-*l/ 97.2%

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

      2. *-un-lft-identity 97.2%

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

      3. associate-*r* 93.7%

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

      4. *-commutative 93.7%

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

      5. associate-*r* 93.7%

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

      6. associate-*r* 92.4%

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

      7. *-commutative 92.4%

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

      8. associate-*r* 96.0%

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

   7. Applied egg-rr 96.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 2: 93.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+208}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{s \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (* c (* x s))))
   (if (<= x 1.2e-17)
     (/ 1.0 (* t_1 t_1))
     (if (<= x 8.5e+208)
       (/ t_0 (* x (* x (* c (* s (* s c))))))
       (/ (/ t_0 (* s (* x s))) (* 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 <= 1.2e-17) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x <= 8.5e+208) {
		tmp = t_0 / (x * (x * (c * (s * (s * c)))));
	} else {
		tmp = (t_0 / (s * (x * s))) / (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 <= 1.2d-17) then
        tmp = 1.0d0 / (t_1 * t_1)
    else if (x <= 8.5d+208) then
        tmp = t_0 / (x * (x * (c * (s * (s * c)))))
    else
        tmp = (t_0 / (s * (x * s))) / (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 <= 1.2e-17) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x <= 8.5e+208) {
		tmp = t_0 / (x * (x * (c * (s * (s * c)))));
	} else {
		tmp = (t_0 / (s * (x * s))) / (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 <= 1.2e-17:
		tmp = 1.0 / (t_1 * t_1)
	elif x <= 8.5e+208:
		tmp = t_0 / (x * (x * (c * (s * (s * c)))))
	else:
		tmp = (t_0 / (s * (x * s))) / (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 <= 1.2e-17)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	elseif (x <= 8.5e+208)
		tmp = Float64(t_0 / Float64(x * Float64(x * Float64(c * Float64(s * Float64(s * c))))));
	else
		tmp = Float64(Float64(t_0 / Float64(s * Float64(x * s))) / 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)
	t_0 = cos((x * 2.0));
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 1.2e-17)
		tmp = 1.0 / (t_1 * t_1);
	elseif (x <= 8.5e+208)
		tmp = t_0 / (x * (x * (c * (s * (s * c)))));
	else
		tmp = (t_0 / (s * (x * s))) / (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, 1.2e-17], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e+208], N[(t$95$0 / N[(x * N[(x * N[(c * N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.19999999999999993e-17

   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* 70.5%

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

      2. remove-double-neg 70.5%

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

      3. distribute-lft-neg-out 70.5%

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

      4. distribute-lft-neg-out 70.5%

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

      5. distribute-rgt-neg-out 70.5%

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

      6. associate-/l/ 70.4%

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

      7. distribute-rgt-neg-out 70.4%

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

      8. distribute-lft-neg-out 70.4%

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

      9. associate-*l* 71.9%

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

      10. distribute-lft-neg-in 71.9%

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

      11. distribute-lft-neg-out 71.9%

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

      12. remove-double-neg 71.9%

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

      13. associate-*r* 70.9%

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

      14. *-commutative 70.9%

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

      15. associate-*r* 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. associate-*r* 59.1%

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

      4. *-commutative 59.1%

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

      5. associate-*r* 57.6%

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

      6. unpow2 57.6%

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

   6. Simplified 57.6%

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

      1. add-sqr-sqrt 57.6%

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

      2. pow2 57.6%

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

      3. sqrt-div 57.6%

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

      4. metadata-eval 57.6%

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

      5. sqrt-prod 57.6%

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

      6. sqrt-prod 33.6%

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

      7. add-sqr-sqrt 63.8%

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

      8. unswap-sqr 77.0%

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

      9. sqrt-prod 53.6%

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

      10. add-sqr-sqrt 86.3%

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

      11. associate-*r* 86.1%

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

      12. *-commutative 86.1%

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

      13. associate-*r* 86.3%

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

      14. pow2 86.3%

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

      15. associate-/r* 86.4%

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

      16. associate-/r* 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. associate-/l/ 86.4%

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

      2. associate-/l/ 86.3%

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

      3. frac-times 86.4%

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

      4. metadata-eval 86.4%

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

      5. *-commutative 86.4%

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

      6. *-commutative 86.4%

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

   10. Applied egg-rr 86.4%

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

   if 1.19999999999999993e-17 < x < 8.4999999999999992e208

   1. Initial program 58.8%

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

      1. associate-/r* 56.7%

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

      2. remove-double-neg 56.7%

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

      3. distribute-lft-neg-out 56.7%

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

      4. distribute-lft-neg-out 56.7%

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

      5. distribute-rgt-neg-out 56.7%

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

      6. associate-/l/ 58.8%

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

      7. distribute-rgt-neg-out 58.8%

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

      8. distribute-lft-neg-out 58.8%

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

      9. associate-*l* 56.8%

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

      10. distribute-lft-neg-in 56.8%

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

      11. distribute-lft-neg-out 56.8%

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

      12. remove-double-neg 56.8%

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

      13. associate-*r* 58.7%

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

      14. *-commutative 58.7%

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

      15. associate-*r* 58.8%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in c around 0 65.1%

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

      1. unpow2 65.1%

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

      2. associate-*l* 90.2%

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

   6. Simplified 90.2%

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

   if 8.4999999999999992e208 < x

   1. Initial program 58.7%

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

      1. associate-/r* 60.4%

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

      2. remove-double-neg 60.4%

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

      3. distribute-lft-neg-out 60.4%

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

      4. distribute-lft-neg-out 60.4%

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

      5. distribute-rgt-neg-out 60.4%

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

      6. associate-/l/ 58.7%

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

      7. distribute-rgt-neg-out 58.7%

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

      8. distribute-lft-neg-out 58.7%

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

      9. associate-*l* 64.1%

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

      10. distribute-lft-neg-in 64.1%

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

      11. distribute-lft-neg-out 64.1%

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

      12. remove-double-neg 64.1%

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

      13. associate-*r* 64.2%

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

      14. *-commutative 64.2%

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

      15. associate-*r* 58.6%

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

   3. Simplified 58.6%

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

      1. swap-sqr 73.9%

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

      2. swap-sqr 58.6%

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

      3. associate-*r* 70.2%

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

      4. *-un-lft-identity 70.2%

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

      5. associate-*r* 63.0%

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

      6. associate-*r* 51.2%

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

      7. *-commutative 51.2%

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

      8. associate-*r* 51.6%

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

      9. associate-*r* 58.7%

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

      10. *-commutative 58.7%

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

      11. associate-*r* 64.1%

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

      12. times-frac 65.8%

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

      13. associate-*l* 84.0%

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

      14. *-commutative 84.0%

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

      15. *-commutative 84.0%

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

      16. associate-*l* 94.2%

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

   5. Applied egg-rr 94.2%

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

      1. associate-*l/ 94.2%

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

      2. *-lft-identity 94.2%

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

      3. *-commutative 94.2%

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

   7. Simplified 94.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+208}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 3: 94.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 10^{-42}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\frac{t_0}{\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{c}}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(x \cdot s\right)}\\ \end{array} \end{array} \]

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 1e-42)
     (/ 1.0 (* t_1 t_1))
     (if (<= x 2e+92)
       (/ t_0 (* (* x x) (* (* s c) (* s c))))
       (/ (/ t_0 c) (* (* s (* x c)) (* x s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 1e-42) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x <= 2e+92) {
		tmp = t_0 / ((x * x) * ((s * c) * (s * c)));
	} else {
		tmp = (t_0 / c) / ((s * (x * c)) * (x * s));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    t_1 = c * (x * s)
    if (x <= 1d-42) then
        tmp = 1.0d0 / (t_1 * t_1)
    else if (x <= 2d+92) then
        tmp = t_0 / ((x * x) * ((s * c) * (s * c)))
    else
        tmp = (t_0 / c) / ((s * (x * c)) * (x * s))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 1e-42) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x <= 2e+92) {
		tmp = t_0 / ((x * x) * ((s * c) * (s * c)));
	} else {
		tmp = (t_0 / c) / ((s * (x * c)) * (x * s));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	t_1 = c * (x * s)
	tmp = 0
	if x <= 1e-42:
		tmp = 1.0 / (t_1 * t_1)
	elif x <= 2e+92:
		tmp = t_0 / ((x * x) * ((s * c) * (s * c)))
	else:
		tmp = (t_0 / c) / ((s * (x * c)) * (x * s))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 1e-42)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	elseif (x <= 2e+92)
		tmp = Float64(t_0 / Float64(Float64(x * x) * Float64(Float64(s * c) * Float64(s * c))));
	else
		tmp = Float64(Float64(t_0 / c) / Float64(Float64(s * Float64(x * c)) * Float64(x * s)));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 1e-42)
		tmp = 1.0 / (t_1 * t_1);
	elseif (x <= 2e+92)
		tmp = t_0 / ((x * x) * ((s * c) * (s * c)));
	else
		tmp = (t_0 / c) / ((s * (x * c)) * (x * s));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := 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, 1e-42], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+92], N[(t$95$0 / N[(N[(x * x), $MachinePrecision] * N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / c), $MachinePrecision] / N[(N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.00000000000000004e-42

   1. Initial program 70.7%

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

      1. associate-/r* 70.7%

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

      2. remove-double-neg 70.7%

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

      3. distribute-lft-neg-out 70.7%

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

      4. distribute-lft-neg-out 70.7%

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

      5. distribute-rgt-neg-out 70.7%

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

      6. associate-/l/ 70.7%

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

      7. distribute-rgt-neg-out 70.7%

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

      8. distribute-lft-neg-out 70.7%

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

      9. associate-*l* 72.2%

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

      10. distribute-lft-neg-in 72.2%

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

      11. distribute-lft-neg-out 72.2%

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

      12. remove-double-neg 72.2%

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

      13. associate-*r* 71.2%

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

      14. *-commutative 71.2%

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

      15. associate-*r* 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

      2. unpow2 59.3%

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

      3. associate-*r* 58.9%

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

      4. *-commutative 58.9%

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

      5. associate-*r* 57.3%

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

      6. unpow2 57.3%

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

   6. Simplified 57.3%

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

      1. add-sqr-sqrt 57.3%

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

      2. pow2 57.3%

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

      3. sqrt-div 57.3%

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

      4. metadata-eval 57.3%

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

      5. sqrt-prod 57.3%

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

      6. sqrt-prod 32.8%

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

      7. add-sqr-sqrt 63.2%

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

      8. unswap-sqr 77.1%

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

      9. sqrt-prod 54.4%

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

      10. add-sqr-sqrt 85.7%

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

      11. associate-*r* 85.5%

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

      12. *-commutative 85.5%

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

      13. associate-*r* 85.7%

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

      14. pow2 85.7%

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

      15. associate-/r* 85.8%

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

      16. associate-/r* 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. associate-/l/ 85.8%

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

      2. associate-/l/ 85.7%

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

      3. frac-times 85.8%

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

      4. metadata-eval 85.8%

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

      5. *-commutative 85.8%

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

      6. *-commutative 85.8%

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

   10. Applied egg-rr 85.8%

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

   if 1.00000000000000004e-42 < x < 2.0000000000000001e92

   1. Initial program 59.7%

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

      1. associate-/r* 59.7%

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

      2. *-commutative 59.7%

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

      3. associate-*l* 59.7%

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

      4. unpow2 59.7%

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

      5. unpow2 59.7%

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

      6. associate-*r* 60.1%

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

      7. associate-/r* 72.1%

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

      8. associate-/l/ 72.2%

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

      9. associate-/l/ 72.1%

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

      10. *-commutative 72.1%

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

      11. associate-*l* 72.4%

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

      12. unpow2 72.4%

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

      13. associate-*l* 59.7%

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

      14. unpow2 59.7%

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

      15. unswap-sqr 99.5%

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

      16. *-commutative 99.5%

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

      17. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   if 2.0000000000000001e92 < x

   1. Initial program 59.1%

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

      1. associate-/r* 57.5%

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

      2. remove-double-neg 57.5%

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

      3. distribute-lft-neg-out 57.5%

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

      4. distribute-lft-neg-out 57.5%

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

      5. distribute-rgt-neg-out 57.5%

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

      6. associate-/l/ 59.1%

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

      7. distribute-rgt-neg-out 59.1%

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

      8. distribute-lft-neg-out 59.1%

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

      9. associate-*l* 61.2%

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

      10. distribute-lft-neg-in 61.2%

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

      11. distribute-lft-neg-out 61.2%

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

      12. remove-double-neg 61.2%

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

      13. associate-*r* 63.5%

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

      14. *-commutative 63.5%

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

      15. associate-*r* 59.1%

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

   3. Simplified 59.1%

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

      1. associate-/r* 59.8%

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

      2. swap-sqr 86.1%

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

      3. associate-/r* 70.2%

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

      4. associate-/r* 70.2%

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

      5. associate-/r* 70.2%

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

      6. *-un-lft-identity 70.2%

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

      7. add-sqr-sqrt 70.2%

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

      8. times-frac 70.2%

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

   5. Applied egg-rr 97.5%

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

      1. associate-/r* 97.5%

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

      2. frac-times 95.5%

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

      3. *-un-lft-identity 95.5%

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

      4. associate-*r* 89.2%

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

      5. *-commutative 89.2%

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

      6. associate-*r* 93.3%

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

   7. Applied egg-rr 93.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-42}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 4: 88.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= x 7e-18)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (* x 2.0)) (* x (* x (* c (* c (* s s)))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 7e-18) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 7d-18) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x * 2.0d0)) / (x * (x * (c * (c * (s * s)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 7e-18) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 7e-18:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 7e-18)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(c * Float64(c * Float64(s * s))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 7e-18)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x * 2.0)) / (x * (x * (c * (c * (s * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7e-18], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.9999999999999997e-18

   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* 70.5%

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

      2. remove-double-neg 70.5%

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

      3. distribute-lft-neg-out 70.5%

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

      4. distribute-lft-neg-out 70.5%

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

      5. distribute-rgt-neg-out 70.5%

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

      6. associate-/l/ 70.4%

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

      7. distribute-rgt-neg-out 70.4%

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

      8. distribute-lft-neg-out 70.4%

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

      9. associate-*l* 71.9%

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

      10. distribute-lft-neg-in 71.9%

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

      11. distribute-lft-neg-out 71.9%

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

      12. remove-double-neg 71.9%

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

      13. associate-*r* 70.9%

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

      14. *-commutative 70.9%

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

      15. associate-*r* 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. associate-*r* 59.1%

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

      4. *-commutative 59.1%

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

      5. associate-*r* 57.6%

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

      6. unpow2 57.6%

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

   6. Simplified 57.6%

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

      1. add-sqr-sqrt 57.6%

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

      2. pow2 57.6%

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

      3. sqrt-div 57.6%

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

      4. metadata-eval 57.6%

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

      5. sqrt-prod 57.6%

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

      6. sqrt-prod 33.6%

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

      7. add-sqr-sqrt 63.8%

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

      8. unswap-sqr 77.0%

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

      9. sqrt-prod 53.6%

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

      10. add-sqr-sqrt 86.3%

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

      11. associate-*r* 86.1%

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

      12. *-commutative 86.1%

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

      13. associate-*r* 86.3%

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

      14. pow2 86.3%

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

      15. associate-/r* 86.4%

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

      16. associate-/r* 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. associate-/l/ 86.4%

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

      2. associate-/l/ 86.3%

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

      3. frac-times 86.4%

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

      4. metadata-eval 86.4%

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

      5. *-commutative 86.4%

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

      6. *-commutative 86.4%

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

   10. Applied egg-rr 86.4%

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

   if 6.9999999999999997e-18 < x

   1. Initial program 58.8%

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

      1. associate-/r* 57.7%

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

      2. remove-double-neg 57.7%

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

      3. distribute-lft-neg-out 57.7%

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

      4. distribute-lft-neg-out 57.7%

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

      5. distribute-rgt-neg-out 57.7%

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

      6. associate-/l/ 58.8%

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

      7. distribute-rgt-neg-out 58.8%

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

      8. distribute-lft-neg-out 58.8%

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

      9. associate-*l* 58.7%

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

      10. distribute-lft-neg-in 58.7%

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

      11. distribute-lft-neg-out 58.7%

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

      12. remove-double-neg 58.7%

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

      13. associate-*r* 60.1%

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

      14. *-commutative 60.1%

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

      15. associate-*r* 58.7%

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

   3. Simplified 66.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 92.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* c (* x s))))
   (if (<= x 1.2e-17)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (* x 2.0)) (* x (* x (* c (* s (* s c)))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1.2e-17) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x * 2.0)) / (x * (x * (c * (s * (s * c)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 1.2d-17) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x * 2.0d0)) / (x * (x * (c * (s * (s * c)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1.2e-17) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x * 2.0)) / (x * (x * (c * (s * (s * c)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 1.2e-17:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x * 2.0)) / (x * (x * (c * (s * (s * c)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 1.2e-17)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(x * Float64(c * Float64(s * Float64(s * c))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 1.2e-17)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x * 2.0)) / (x * (x * (c * (s * (s * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.2e-17], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c * N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999993e-17

   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* 70.5%

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

      2. remove-double-neg 70.5%

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

      3. distribute-lft-neg-out 70.5%

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

      4. distribute-lft-neg-out 70.5%

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

      5. distribute-rgt-neg-out 70.5%

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

      6. associate-/l/ 70.4%

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

      7. distribute-rgt-neg-out 70.4%

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

      8. distribute-lft-neg-out 70.4%

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

      9. associate-*l* 71.9%

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

      10. distribute-lft-neg-in 71.9%

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

      11. distribute-lft-neg-out 71.9%

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

      12. remove-double-neg 71.9%

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

      13. associate-*r* 70.9%

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

      14. *-commutative 70.9%

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

      15. associate-*r* 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in x around 0 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. associate-*r* 59.1%

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

      4. *-commutative 59.1%

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

      5. associate-*r* 57.6%

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

      6. unpow2 57.6%

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

   6. Simplified 57.6%

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

      1. add-sqr-sqrt 57.6%

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

      2. pow2 57.6%

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

      3. sqrt-div 57.6%

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

      4. metadata-eval 57.6%

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

      5. sqrt-prod 57.6%

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

      6. sqrt-prod 33.6%

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

      7. add-sqr-sqrt 63.8%

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

      8. unswap-sqr 77.0%

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

      9. sqrt-prod 53.6%

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

      10. add-sqr-sqrt 86.3%

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

      11. associate-*r* 86.1%

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

      12. *-commutative 86.1%

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

      13. associate-*r* 86.3%

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

      14. pow2 86.3%

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

      15. associate-/r* 86.4%

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

      16. associate-/r* 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. associate-/l/ 86.4%

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

      2. associate-/l/ 86.3%

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

      3. frac-times 86.4%

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

      4. metadata-eval 86.4%

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

      5. *-commutative 86.4%

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

      6. *-commutative 86.4%

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

   10. Applied egg-rr 86.4%

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

   if 1.19999999999999993e-17 < x

   1. Initial program 58.8%

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

      1. associate-/r* 57.7%

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

      2. remove-double-neg 57.7%

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

      3. distribute-lft-neg-out 57.7%

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

      4. distribute-lft-neg-out 57.7%

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

      5. distribute-rgt-neg-out 57.7%

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

      6. associate-/l/ 58.8%

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

      7. distribute-rgt-neg-out 58.8%

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

      8. distribute-lft-neg-out 58.8%

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

      9. associate-*l* 58.7%

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

      10. distribute-lft-neg-in 58.7%

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

      11. distribute-lft-neg-out 58.7%

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

      12. remove-double-neg 58.7%

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

      13. associate-*r* 60.1%

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

      14. *-commutative 60.1%

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

      15. associate-*r* 58.7%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in c around 0 66.4%

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

      1. unpow2 66.4%

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

      2. associate-*l* 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 69.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

   1. associate-/r* 67.1%

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

   2. remove-double-neg 67.1%

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

   3. distribute-lft-neg-out 67.1%

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

   4. distribute-lft-neg-out 67.1%

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

   5. distribute-rgt-neg-out 67.1%

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

   6. associate-/l/ 67.4%

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

   7. distribute-rgt-neg-out 67.4%

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

   8. distribute-lft-neg-out 67.4%

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

   9. associate-*l* 68.4%

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

   10. distribute-lft-neg-in 68.4%

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

   11. distribute-lft-neg-out 68.4%

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

   12. remove-double-neg 68.4%

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

   13. associate-*r* 68.1%

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

   14. *-commutative 68.1%

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

   15. associate-*r* 67.1%

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

3. Simplified 67.1%

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

4. Taylor expanded in x around 0 54.3%

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

   1. unpow2 54.3%

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

   2. associate-/r* 53.9%

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

   3. unpow2 53.9%

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

   4. unpow2 53.9%

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

   5. swap-sqr 68.0%

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

   6. unpow2 68.0%

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

   7. associate-/r* 68.4%

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

   8. unpow2 68.4%

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

   9. swap-sqr 79.0%

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

   10. unpow2 79.0%

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

   11. associate-*r* 78.8%

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

   12. *-commutative 78.8%

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

   13. associate-*l* 79.0%

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

6. Simplified 79.0%

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

   1. unpow2 79.0%

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

   2. associate-*r* 77.8%

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

   3. associate-*r* 78.8%

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

   4. unswap-sqr 67.4%

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

   5. associate-*l* 69.8%

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

   6. *-commutative 69.8%

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

   7. *-commutative 69.8%

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

8. Applied egg-rr 69.8%

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

9. Final simplification 69.8%

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

Alternative 7: 78.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

   1. associate-/r* 67.1%

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

   2. remove-double-neg 67.1%

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

   3. distribute-lft-neg-out 67.1%

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

   4. distribute-lft-neg-out 67.1%

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

   5. distribute-rgt-neg-out 67.1%

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

   6. associate-/l/ 67.4%

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

   7. distribute-rgt-neg-out 67.4%

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

   8. distribute-lft-neg-out 67.4%

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

   9. associate-*l* 68.4%

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

   10. distribute-lft-neg-in 68.4%

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

   11. distribute-lft-neg-out 68.4%

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

   12. remove-double-neg 68.4%

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

   13. associate-*r* 68.1%

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

   14. *-commutative 68.1%

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

   15. associate-*r* 67.1%

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

3. Simplified 67.1%

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

4. Taylor expanded in x around 0 54.3%

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

   1. unpow2 54.3%

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

   2. associate-/r* 53.9%

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

   3. unpow2 53.9%

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

   4. unpow2 53.9%

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

   5. swap-sqr 68.0%

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

   6. unpow2 68.0%

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

   7. associate-/r* 68.4%

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

   8. unpow2 68.4%

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

   9. swap-sqr 79.0%

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

   10. unpow2 79.0%

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

   11. associate-*r* 78.8%

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

   12. *-commutative 78.8%

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

   13. associate-*l* 79.0%

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

6. Simplified 79.0%

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

   1. associate-*r* 78.8%

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

   2. pow-prod-down 67.4%

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

   3. pow2 67.4%

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

   4. pow2 67.4%

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

   5. *-commutative 67.4%

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

   6. *-commutative 67.4%

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

   7. associate-*r* 65.4%

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

   8. associate-*r* 70.2%

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

   9. add-sqr-sqrt 70.2%

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

   10. associate-*r* 65.4%

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

   11. associate-*r* 65.4%

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

   12. *-commutative 65.4%

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

   13. sqrt-prod 65.4%

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

   14. sqrt-prod 32.3%

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

   15. add-sqr-sqrt 55.3%

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

   16. sqrt-prod 31.6%

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

   17. add-sqr-sqrt 57.7%

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

   18. *-commutative 57.7%

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

   19. associate-*r* 53.0%

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

   20. associate-*r* 54.0%

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

8. Applied egg-rr 78.8%

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

9. Final simplification 78.8%

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

Alternative 8: 79.4% accurate, 24.1× speedup

Math

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

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 (* c (* x s)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

   1. associate-/r* 67.1%

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

   2. remove-double-neg 67.1%

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

   3. distribute-lft-neg-out 67.1%

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

   4. distribute-lft-neg-out 67.1%

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

   5. distribute-rgt-neg-out 67.1%

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

   6. associate-/l/ 67.4%

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

   7. distribute-rgt-neg-out 67.4%

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

   8. distribute-lft-neg-out 67.4%

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

   9. associate-*l* 68.4%

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

   10. distribute-lft-neg-in 68.4%

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

   11. distribute-lft-neg-out 68.4%

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

   12. remove-double-neg 68.4%

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

   13. associate-*r* 68.1%

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

   14. *-commutative 68.1%

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

   15. associate-*r* 67.1%

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

3. Simplified 67.1%

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

4. Taylor expanded in x around 0 54.3%

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

   1. unpow2 54.3%

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

   2. unpow2 54.3%

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

   3. associate-*r* 54.0%

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

   4. *-commutative 54.0%

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

   5. associate-*r* 52.9%

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

   6. unpow2 52.9%

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

6. Simplified 52.9%

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

   1. add-sqr-sqrt 52.9%

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

   2. pow2 52.9%

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

   3. sqrt-div 52.9%

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

   4. metadata-eval 52.9%

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

   5. sqrt-prod 52.9%

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

   6. sqrt-prod 30.9%

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

   7. add-sqr-sqrt 60.3%

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

   8. unswap-sqr 71.9%

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

   9. sqrt-prod 45.6%

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

   10. add-sqr-sqrt 78.9%

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

   11. associate-*r* 78.7%

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

   12. *-commutative 78.7%

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

   13. associate-*r* 78.9%

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

   14. pow2 78.9%

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

   15. associate-/r* 78.9%

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

   16. associate-/r* 78.9%

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

8. Applied egg-rr 78.9%

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

   1. associate-/l/ 78.9%

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

   2. associate-/l/ 78.9%

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

   3. frac-times 79.0%

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

   4. metadata-eval 79.0%

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

   5. *-commutative 79.0%

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

   6. *-commutative 79.0%

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

10. Applied egg-rr 79.0%

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

11. Final simplification 79.0%

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

Reproduce

herbie shell --seed 2023273 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))