mixedcos

Percentage Accurate: 66.5% → 98.1%

Time: 20.2s

Alternatives: 15

Speedup: 24.1×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq 5.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 10^{+125}:\\ \;\;\;\;\frac{\frac{t_0}{c \cdot s}}{x \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)}\\ \end{array} \end{array} \]

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 (* 2.0 x))))
   (if (<= x 5.2e-18)
     (/ -1.0 (* (* c (* x s)) (* c (* x (- s)))))
     (if (<= x 1e+125)
       (/ (/ t_0 (* c s)) (* x (* x (* c s))))
       (/ t_0 (* (* (* x c) (* x c)) (* s s)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (x <= 5.2e-18) {
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)));
	} else if (x <= 1e+125) {
		tmp = (t_0 / (c * s)) / (x * (x * (c * s)));
	} else {
		tmp = t_0 / (((x * c) * (x * c)) * (s * s));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    if (x <= 5.2d-18) then
        tmp = (-1.0d0) / ((c * (x * s)) * (c * (x * -s)))
    else if (x <= 1d+125) then
        tmp = (t_0 / (c * s)) / (x * (x * (c * s)))
    else
        tmp = t_0 / (((x * c) * (x * c)) * (s * s))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((2.0 * x));
	double tmp;
	if (x <= 5.2e-18) {
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)));
	} else if (x <= 1e+125) {
		tmp = (t_0 / (c * s)) / (x * (x * (c * s)));
	} else {
		tmp = t_0 / (((x * c) * (x * c)) * (s * s));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if x <= 5.2e-18:
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)))
	elif x <= 1e+125:
		tmp = (t_0 / (c * s)) / (x * (x * (c * s)))
	else:
		tmp = t_0 / (((x * c) * (x * c)) * (s * s))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (x <= 5.2e-18)
		tmp = Float64(-1.0 / Float64(Float64(c * Float64(x * s)) * Float64(c * Float64(x * Float64(-s)))));
	elseif (x <= 1e+125)
		tmp = Float64(Float64(t_0 / Float64(c * s)) / Float64(x * Float64(x * Float64(c * s))));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(x * c) * Float64(x * c)) * Float64(s * s)));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((2.0 * x));
	tmp = 0.0;
	if (x <= 5.2e-18)
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)));
	elseif (x <= 1e+125)
		tmp = (t_0 / (c * s)) / (x * (x * (c * s)));
	else
		tmp = t_0 / (((x * c) * (x * c)) * (s * s));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5.2e-18], N[(-1.0 / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(c * N[(x * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+125], N[(N[(t$95$0 / N[(c * s), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(x * c), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.2000000000000001e-18

   1. Initial program 64.5%

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

      1. associate-/r* 64.0%

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

      2. remove-double-neg 64.0%

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

      3. distribute-lft-neg-out 64.0%

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

      4. distribute-lft-neg-out 64.0%

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

      5. distribute-rgt-neg-out 64.0%

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

      6. associate-/l/ 64.5%

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

      7. distribute-rgt-neg-out 64.5%

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

      8. distribute-lft-neg-out 64.5%

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

      9. associate-*l* 67.3%

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

      10. distribute-lft-neg-in 67.3%

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

      11. distribute-lft-neg-out 67.3%

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

      12. remove-double-neg 67.3%

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

      13. associate-*r* 67.1%

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

      14. *-commutative 67.1%

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

      15. associate-*r* 65.2%

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

   3. Simplified 65.2%

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

      1. associate-/r* 65.2%

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

      2. swap-sqr 85.5%

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

      3. associate-/r* 74.1%

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

      4. associate-/r* 73.7%

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

      5. associate-/r* 74.6%

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

      6. *-un-lft-identity 74.6%

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

      7. add-sqr-sqrt 74.5%

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

      8. times-frac 74.5%

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

   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. Taylor expanded in x around 0 85.0%

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

      1. associate-/r* 85.0%

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

   8. Simplified 85.0%

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

      1. associate-/r* 85.0%

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

      2. frac-2neg 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

      5. frac-times 85.1%

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

      6. metadata-eval 85.1%

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

      7. *-commutative 85.1%

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

   10. Applied egg-rr 85.1%

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

   if 5.2000000000000001e-18 < x < 9.9999999999999992e124

   1. Initial program 68.2%

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

      1. associate-/r* 68.2%

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

      2. remove-double-neg 68.2%

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

      3. distribute-lft-neg-out 68.2%

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

      4. distribute-lft-neg-out 68.2%

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

      5. distribute-rgt-neg-out 68.2%

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

      6. associate-/l/ 68.2%

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

      7. distribute-rgt-neg-out 68.2%

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

      8. distribute-lft-neg-out 68.2%

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

      9. associate-*l* 68.3%

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

      10. distribute-lft-neg-in 68.3%

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

      11. distribute-lft-neg-out 68.3%

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

      12. remove-double-neg 68.3%

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

      13. associate-*r* 60.3%

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

      14. *-commutative 60.3%

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

      15. associate-*r* 60.3%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in x around 0 68.2%

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

      1. unpow2 68.2%

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

      2. unpow2 68.2%

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

      3. unpow2 68.2%

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

      4. swap-sqr 68.2%

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

      5. swap-sqr 95.5%

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

      6. unpow2 95.5%

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

      7. associate-*r* 99.7%

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

      8. *-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unswap-sqr 91.5%

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

      5. associate-*r* 99.6%

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

      6. times-frac 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. associate-*l* 99.6%

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

   10. Simplified 99.6%

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

   if 9.9999999999999992e124 < x

   1. Initial program 65.4%

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

      1. associate-/r* 63.6%

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

      2. remove-double-neg 63.6%

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

      3. distribute-lft-neg-out 63.6%

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

      4. distribute-lft-neg-out 63.6%

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

      5. distribute-rgt-neg-out 63.6%

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

      6. associate-/l/ 65.4%

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

      7. distribute-rgt-neg-out 65.4%

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

      8. distribute-lft-neg-out 65.4%

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

      9. associate-*l* 67.3%

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

      10. distribute-lft-neg-in 67.3%

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

      11. distribute-lft-neg-out 67.3%

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

      12. remove-double-neg 67.3%

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

      13. associate-*r* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*r* 63.5%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around 0 52.9%

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

      1. unpow2 52.9%

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

      2. unpow2 52.9%

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

      3. unpow2 52.9%

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

      4. swap-sqr 87.5%

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

      5. swap-sqr 99.5%

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

      6. unpow2 99.5%

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

      7. associate-*r* 97.2%

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

      8. *-commutative 97.2%

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

   6. Simplified 97.2%

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

      1. unpow2 97.2%

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

      2. associate-*r* 92.8%

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

      3. associate-*r* 95.2%

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

      4. swap-sqr 72.8%

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

   8. Applied egg-rr 72.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 10^{+125}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot s}}{x \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)}\\ \end{array} \]

Alternative 2: 97.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return math.cos((2.0 * x)) / math.pow((x * (c * s)), 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 70.4%

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

4. Taylor expanded in x around 0 58.6%

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

   1. unpow2 58.6%

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

   2. unpow2 58.6%

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

   3. unpow2 58.6%

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

   4. swap-sqr 76.5%

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

   5. swap-sqr 97.5%

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

   6. unpow2 97.5%

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

   7. associate-*r* 97.7%

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

   8. *-commutative 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 3: 96.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return math.cos((2.0 * x)) / math.pow((c * (x * s)), 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 70.4%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

   3. associate-*r* 66.2%

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

   4. associate-*l* 65.0%

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

   5. *-commutative 65.0%

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

   6. add-sqr-sqrt 64.9%

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

   7. pow2 64.9%

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

   8. sqrt-prod 64.9%

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

   9. sqrt-prod 35.3%

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

   10. add-sqr-sqrt 70.6%

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

   11. associate-*r* 63.4%

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

   12. sqrt-prod 65.8%

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

   13. sqrt-prod 35.5%

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

   14. add-sqr-sqrt 78.0%

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

   15. sqrt-prod 47.5%

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

   16. add-sqr-sqrt 97.5%

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

   17. *-commutative 97.5%

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

5. Applied egg-rr 97.5%

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

6. Final simplification 97.5%

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

Alternative 4: 89.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 0.01)
     (/ (+ 1.0 (* -2.0 (* x x))) (* t_0 t_0))
     (/ (cos (* 2.0 x)) (* 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 <= 0.01) {
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0);
	} else {
		tmp = cos((2.0 * x)) / (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 <= 0.01d0) then
        tmp = (1.0d0 + ((-2.0d0) * (x * x))) / (t_0 * t_0)
    else
        tmp = cos((2.0d0 * x)) / (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 <= 0.01) {
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0);
	} else {
		tmp = Math.cos((2.0 * x)) / (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 <= 0.01:
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0)
	else:
		tmp = math.cos((2.0 * x)) / (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 <= 0.01)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x * x))) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / 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 <= 0.01)
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0);
	else
		tmp = cos((2.0 * x)) / (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, 0.01], N[(N[(1.0 + N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $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 < 0.0100000000000000002

   1. Initial program 64.4%

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

      1. associate-/r* 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*l* 58.1%

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

      4. unpow2 58.1%

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

      5. unpow2 58.1%

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

      6. associate-*r* 63.7%

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

      7. associate-/r* 65.6%

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

      8. associate-/l/ 65.6%

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

      9. associate-/l/ 65.6%

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

      10. *-commutative 65.6%

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

      11. associate-*l* 64.8%

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

      12. unpow2 64.8%

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

      13. associate-*l* 59.4%

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

      14. unpow2 59.4%

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

      15. unswap-sqr 74.8%

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

      16. *-commutative 74.8%

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

      17. *-commutative 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in x around 0 60.0%

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

      1. unpow2 60.0%

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

   6. Simplified 60.0%

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

      1. unswap-sqr 74.6%

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

      2. unpow2 74.6%

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

      3. *-commutative 74.6%

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

      4. associate-*r* 75.2%

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

      5. *-commutative 75.2%

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

   8. Applied egg-rr 75.2%

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

      1. unpow2 75.2%

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

   10. Applied egg-rr 75.2%

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

   if 0.0100000000000000002 < x

   1. Initial program 66.8%

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

      1. associate-/r* 65.7%

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

      2. remove-double-neg 65.7%

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

      3. distribute-lft-neg-out 65.7%

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

      4. distribute-lft-neg-out 65.7%

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

      5. distribute-rgt-neg-out 65.7%

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

      6. associate-/l/ 66.8%

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

      7. distribute-rgt-neg-out 66.8%

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

      8. distribute-lft-neg-out 66.8%

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

      9. associate-*l* 68.2%

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

      10. distribute-lft-neg-in 68.2%

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

      11. distribute-lft-neg-out 68.2%

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

      12. remove-double-neg 68.2%

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

      13. associate-*r* 64.1%

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

      14. *-commutative 64.1%

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

      15. associate-*r* 62.7%

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

   3. Simplified 66.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 5: 92.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 4.2e-17)
   (/ -1.0 (* (* c (* x s)) (* c (* x (- s)))))
   (/ (cos (* 2.0 x)) (* x (* x (* c (* s (* c s))))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.2e-17) {
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)));
	} else {
		tmp = cos((2.0 * x)) / (x * (x * (c * (s * (c * s)))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 4.2d-17) then
        tmp = (-1.0d0) / ((c * (x * s)) * (c * (x * -s)))
    else
        tmp = cos((2.0d0 * x)) / (x * (x * (c * (s * (c * s)))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4.2e-17) {
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)));
	} else {
		tmp = Math.cos((2.0 * x)) / (x * (x * (c * (s * (c * s)))));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 4.2e-17:
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)))
	else:
		tmp = math.cos((2.0 * x)) / (x * (x * (c * (s * (c * s)))))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 4.2e-17)
		tmp = Float64(-1.0 / Float64(Float64(c * Float64(x * s)) * Float64(c * Float64(x * Float64(-s)))));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(x * Float64(x * Float64(c * Float64(s * Float64(c * s))))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4.2e-17)
		tmp = -1.0 / ((c * (x * s)) * (c * (x * -s)));
	else
		tmp = cos((2.0 * x)) / (x * (x * (c * (s * (c * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 4.2e-17], N[(-1.0 / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(c * N[(x * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(c * N[(s * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.19999999999999984e-17

   1. Initial program 64.5%

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

      1. associate-/r* 64.0%

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

      2. remove-double-neg 64.0%

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

      3. distribute-lft-neg-out 64.0%

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

      4. distribute-lft-neg-out 64.0%

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

      5. distribute-rgt-neg-out 64.0%

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

      6. associate-/l/ 64.5%

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

      7. distribute-rgt-neg-out 64.5%

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

      8. distribute-lft-neg-out 64.5%

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

      9. associate-*l* 67.3%

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

      10. distribute-lft-neg-in 67.3%

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

      11. distribute-lft-neg-out 67.3%

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

      12. remove-double-neg 67.3%

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

      13. associate-*r* 67.1%

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

      14. *-commutative 67.1%

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

      15. associate-*r* 65.2%

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

   3. Simplified 65.2%

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

      1. associate-/r* 65.2%

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

      2. swap-sqr 85.5%

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

      3. associate-/r* 74.1%

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

      4. associate-/r* 73.7%

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

      5. associate-/r* 74.6%

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

      6. *-un-lft-identity 74.6%

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

      7. add-sqr-sqrt 74.5%

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

      8. times-frac 74.5%

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

   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. Taylor expanded in x around 0 85.0%

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

      1. associate-/r* 85.0%

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

   8. Simplified 85.0%

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

      1. associate-/r* 85.0%

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

      2. frac-2neg 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

      5. frac-times 85.1%

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

      6. metadata-eval 85.1%

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

      7. *-commutative 85.1%

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

   10. Applied egg-rr 85.1%

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

   if 4.19999999999999984e-17 < x

   1. Initial program 66.3%

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

      1. associate-/r* 65.2%

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

      2. remove-double-neg 65.2%

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

      3. distribute-lft-neg-out 65.2%

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

      4. distribute-lft-neg-out 65.2%

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

      5. distribute-rgt-neg-out 65.2%

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

      6. associate-/l/ 66.3%

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

      7. distribute-rgt-neg-out 66.3%

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

      8. distribute-lft-neg-out 66.3%

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

      9. associate-*l* 67.7%

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

      10. distribute-lft-neg-in 67.7%

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

      11. distribute-lft-neg-out 67.7%

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

      12. remove-double-neg 67.7%

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

      13. associate-*r* 63.7%

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

      14. *-commutative 63.7%

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

      15. associate-*r* 62.3%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in c around 0 65.9%

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

      1. unpow2 55.0%

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

      2. associate-*r* 62.0%

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

      3. *-commutative 62.0%

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

   6. Simplified 81.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 6: 95.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(t_0 \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= x 1.6e-60)
     (/ (/ 1.0 c) (* (* x s) t_0))
     (/ (cos (* 2.0 x)) (* s (* t_0 (* x c)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1.6e-60) {
		tmp = (1.0 / c) / ((x * s) * t_0);
	} else {
		tmp = cos((2.0 * x)) / (s * (t_0 * (x * c)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 1.6d-60) then
        tmp = (1.0d0 / c) / ((x * s) * t_0)
    else
        tmp = cos((2.0d0 * x)) / (s * (t_0 * (x * c)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1.6e-60) {
		tmp = (1.0 / c) / ((x * s) * t_0);
	} else {
		tmp = Math.cos((2.0 * x)) / (s * (t_0 * (x * c)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 1.6e-60:
		tmp = (1.0 / c) / ((x * s) * t_0)
	else:
		tmp = math.cos((2.0 * x)) / (s * (t_0 * (x * c)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 1.6e-60)
		tmp = Float64(Float64(1.0 / c) / Float64(Float64(x * s) * t_0));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(s * Float64(t_0 * Float64(x * c))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 1.6e-60)
		tmp = (1.0 / c) / ((x * s) * t_0);
	else
		tmp = cos((2.0 * x)) / (s * (t_0 * (x * c)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.6e-60], N[(N[(1.0 / c), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(s * N[(t$95$0 * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001e-60

   1. Initial program 64.4%

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

      1. associate-/r* 63.9%

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

      2. remove-double-neg 63.9%

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

      3. distribute-lft-neg-out 63.9%

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

      4. distribute-lft-neg-out 63.9%

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

      5. distribute-rgt-neg-out 63.9%

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

      6. associate-/l/ 64.4%

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

      7. distribute-rgt-neg-out 64.4%

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

      8. distribute-lft-neg-out 64.4%

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

      9. associate-*l* 66.8%

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

      10. distribute-lft-neg-in 66.8%

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

      11. distribute-lft-neg-out 66.8%

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

      12. remove-double-neg 66.8%

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

      13. associate-*r* 66.6%

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

      14. *-commutative 66.6%

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

      15. associate-*r* 64.6%

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

   3. Simplified 64.6%

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

      1. associate-/r* 64.6%

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

      2. swap-sqr 84.8%

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

      3. associate-/r* 72.9%

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

      4. associate-/r* 72.5%

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

      5. associate-/r* 73.4%

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

      6. *-un-lft-identity 73.4%

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

      7. add-sqr-sqrt 73.3%

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

      8. times-frac 73.3%

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

   5. Applied egg-rr 97.1%

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

   6. Taylor expanded in x around 0 84.3%

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

      1. associate-/r* 84.4%

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

   8. Simplified 84.4%

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

      1. frac-times 82.1%

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

      2. *-un-lft-identity 82.1%

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

      3. *-commutative 82.1%

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

      4. *-commutative 82.1%

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

   10. Applied egg-rr 82.1%

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

   if 1.6000000000000001e-60 < x

   1. Initial program 66.4%

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

      1. associate-/r* 65.5%

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

      2. remove-double-neg 65.5%

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

      3. distribute-lft-neg-out 65.5%

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

      4. distribute-lft-neg-out 65.5%

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

      5. distribute-rgt-neg-out 65.5%

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

      6. associate-/l/ 66.4%

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

      7. distribute-rgt-neg-out 66.4%

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

      8. distribute-lft-neg-out 66.4%

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

      9. associate-*l* 68.9%

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

      10. distribute-lft-neg-in 68.9%

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

      11. distribute-lft-neg-out 68.9%

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

      12. remove-double-neg 68.9%

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

      13. associate-*r* 65.5%

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

      14. *-commutative 65.5%

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

      15. associate-*r* 64.3%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in x around 0 59.4%

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

      1. unpow2 59.4%

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

      2. unpow2 59.4%

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

      3. unpow2 59.4%

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

      4. swap-sqr 78.9%

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

      5. swap-sqr 98.2%

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

      6. unpow2 98.2%

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

      7. associate-*r* 98.3%

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

      8. *-commutative 98.3%

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

   6. Simplified 98.3%

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

      1. unpow2 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*r* 97.0%

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

      4. associate-*r* 95.9%

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

      5. associate-*r* 93.4%

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

      6. *-commutative 93.4%

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

   8. Applied egg-rr 93.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 7: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{\frac{\cos \left(2 \cdot x\right)}{t_0}}{t_0} \end{array} \end{array} \]

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 (* c (* x s)))) (/ (/ (cos (* 2.0 x)) 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 (cos((2.0 * x)) / 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 = (cos((2.0d0 * x)) / 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 (Math.cos((2.0 * x)) / 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 (math.cos((2.0 * x)) / t_0) / t_0

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(Float64(cos(Float64(2.0 * x)) / 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 = (cos((2.0 * x)) / t_0) / t_0;
end

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 64.5%

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

   1. associate-/r* 64.5%

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

   2. swap-sqr 85.1%

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

   3. associate-/r* 73.6%

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

   4. associate-/r* 73.3%

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

   5. associate-/r* 73.9%

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

   6. *-un-lft-identity 73.9%

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

   7. add-sqr-sqrt 73.9%

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

   8. times-frac 73.9%

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

5. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

7. Applied egg-rr 97.4%

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

8. Final simplification 97.4%

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

Alternative 8: 78.9% accurate, 13.5× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 10^{-16}:\\ \;\;\;\;\frac{-1}{t_0 \cdot \left(c \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot t_0}}{x \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= x 1e-16)
     (/ -1.0 (* t_0 (* c (* x (- s)))))
     (if (<= x 3.6e+122)
       (/ (+ 1.0 (* -2.0 (* x x))) (* x (* x (* c (* s (* c s))))))
       (/ (/ 1.0 (* s t_0)) (* x c))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1e-16) {
		tmp = -1.0 / (t_0 * (c * (x * -s)));
	} else if (x <= 3.6e+122) {
		tmp = (1.0 + (-2.0 * (x * x))) / (x * (x * (c * (s * (c * s)))));
	} else {
		tmp = (1.0 / (s * t_0)) / (x * c);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 1d-16) then
        tmp = (-1.0d0) / (t_0 * (c * (x * -s)))
    else if (x <= 3.6d+122) then
        tmp = (1.0d0 + ((-2.0d0) * (x * x))) / (x * (x * (c * (s * (c * s)))))
    else
        tmp = (1.0d0 / (s * t_0)) / (x * c)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 1e-16) {
		tmp = -1.0 / (t_0 * (c * (x * -s)));
	} else if (x <= 3.6e+122) {
		tmp = (1.0 + (-2.0 * (x * x))) / (x * (x * (c * (s * (c * s)))));
	} else {
		tmp = (1.0 / (s * t_0)) / (x * c);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 1e-16:
		tmp = -1.0 / (t_0 * (c * (x * -s)))
	elif x <= 3.6e+122:
		tmp = (1.0 + (-2.0 * (x * x))) / (x * (x * (c * (s * (c * s)))))
	else:
		tmp = (1.0 / (s * t_0)) / (x * c)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 1e-16)
		tmp = Float64(-1.0 / Float64(t_0 * Float64(c * Float64(x * Float64(-s)))));
	elseif (x <= 3.6e+122)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x * x))) / Float64(x * Float64(x * Float64(c * Float64(s * Float64(c * s))))));
	else
		tmp = Float64(Float64(1.0 / Float64(s * t_0)) / Float64(x * c));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 1e-16)
		tmp = -1.0 / (t_0 * (c * (x * -s)));
	elseif (x <= 3.6e+122)
		tmp = (1.0 + (-2.0 * (x * x))) / (x * (x * (c * (s * (c * s)))));
	else
		tmp = (1.0 / (s * t_0)) / (x * c);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e-16], N[(-1.0 / N[(t$95$0 * N[(c * N[(x * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e+122], N[(N[(1.0 + N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(c * N[(s * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(s * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.9999999999999998e-17

   1. Initial program 64.5%

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

      1. associate-/r* 64.0%

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

      2. remove-double-neg 64.0%

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

      3. distribute-lft-neg-out 64.0%

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

      4. distribute-lft-neg-out 64.0%

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

      5. distribute-rgt-neg-out 64.0%

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

      6. associate-/l/ 64.5%

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

      7. distribute-rgt-neg-out 64.5%

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

      8. distribute-lft-neg-out 64.5%

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

      9. associate-*l* 67.3%

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

      10. distribute-lft-neg-in 67.3%

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

      11. distribute-lft-neg-out 67.3%

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

      12. remove-double-neg 67.3%

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

      13. associate-*r* 67.1%

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

      14. *-commutative 67.1%

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

      15. associate-*r* 65.2%

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

   3. Simplified 65.2%

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

      1. associate-/r* 65.2%

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

      2. swap-sqr 85.5%

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

      3. associate-/r* 74.1%

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

      4. associate-/r* 73.7%

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

      5. associate-/r* 74.6%

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

      6. *-un-lft-identity 74.6%

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

      7. add-sqr-sqrt 74.5%

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

      8. times-frac 74.5%

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

   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. Taylor expanded in x around 0 85.0%

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

      1. associate-/r* 85.0%

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

   8. Simplified 85.0%

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

      1. associate-/r* 85.0%

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

      2. frac-2neg 85.0%

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

      3. metadata-eval 85.0%

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

      4. *-commutative 85.0%

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

      5. frac-times 85.1%

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

      6. metadata-eval 85.1%

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

      7. *-commutative 85.1%

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

   10. Applied egg-rr 85.1%

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

   if 9.9999999999999998e-17 < x < 3.6000000000000003e122

   1. Initial program 71.4%

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

      1. associate-/r* 71.5%

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

      2. remove-double-neg 71.5%

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

      3. distribute-lft-neg-out 71.5%

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

      4. distribute-lft-neg-out 71.5%

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

      5. distribute-rgt-neg-out 71.5%

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

      6. associate-/l/ 71.4%

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

      7. distribute-rgt-neg-out 71.4%

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

      8. distribute-lft-neg-out 71.4%

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

      9. associate-*l* 71.6%

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

      10. distribute-lft-neg-in 71.6%

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

      11. distribute-lft-neg-out 71.6%

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

      12. remove-double-neg 71.6%

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

      13. associate-*r* 63.2%

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

      14. *-commutative 63.2%

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

      15. associate-*r* 63.2%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in x around 0 58.1%

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

      1. unpow2 63.0%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in c around 0 58.1%

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

      1. unpow2 45.6%

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

      2. associate-*r* 49.7%

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

      3. *-commutative 49.7%

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

   9. Simplified 63.0%

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

   if 3.6000000000000003e122 < x

   1. Initial program 63.8%

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

      1. associate-/r* 62.0%

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

      2. remove-double-neg 62.0%

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

      3. distribute-lft-neg-out 62.0%

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

      4. distribute-lft-neg-out 62.0%

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

      5. distribute-rgt-neg-out 62.0%

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

      6. associate-/l/ 63.8%

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

      7. distribute-rgt-neg-out 63.8%

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

      8. distribute-lft-neg-out 63.8%

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

      9. associate-*l* 65.7%

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

      10. distribute-lft-neg-in 65.7%

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

      11. distribute-lft-neg-out 65.7%

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

      12. remove-double-neg 65.7%

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

      13. associate-*r* 63.9%

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

      14. *-commutative 63.9%

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

      15. associate-*r* 61.9%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in x around 0 51.6%

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

      1. unpow2 51.6%

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

      2. unpow2 51.6%

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

      3. unpow2 51.6%

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

      4. swap-sqr 85.3%

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

      5. swap-sqr 99.5%

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

      6. unpow2 99.5%

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

      7. associate-*r* 97.3%

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

      8. *-commutative 97.3%

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

   6. Simplified 97.3%

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

      1. *-commutative 97.3%

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

      2. *-un-lft-identity 97.3%

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

      3. *-commutative 97.3%

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

      4. associate-*r* 99.5%

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

      5. pow2 99.5%

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

      6. frac-times 99.5%

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

      7. associate-*l/ 99.6%

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

      8. associate-*r* 97.4%

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

      9. *-commutative 97.4%

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

      10. associate-*r* 95.5%

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

      11. times-frac 93.2%

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

      12. *-commutative 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. associate-*l/ 93.2%

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

      2. *-lft-identity 93.2%

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

      3. associate-/l/ 93.1%

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

      4. *-commutative 93.1%

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

      5. *-commutative 93.1%

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

   10. Simplified 93.1%

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

   11. Taylor expanded in x around 0 70.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-16}:\\ \;\;\;\;\frac{-1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot \left(-s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(c \cdot \left(x \cdot s\right)\right)}}{x \cdot c}\\ \end{array} \]

Alternative 9: 79.1% accurate, 14.9× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot t_0}}{x \cdot c}\\ \end{array} \end{array} \]

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 3.6e+122)
     (/ (+ 1.0 (* -2.0 (* x x))) (* t_0 t_0))
     (/ (/ 1.0 (* s t_0)) (* x c)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 3.6e+122) {
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0);
	} else {
		tmp = (1.0 / (s * t_0)) / (x * c);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 3.6d+122) then
        tmp = (1.0d0 + ((-2.0d0) * (x * x))) / (t_0 * t_0)
    else
        tmp = (1.0d0 / (s * t_0)) / (x * c)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 3.6e+122) {
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0);
	} else {
		tmp = (1.0 / (s * t_0)) / (x * c);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 3.6e+122:
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0)
	else:
		tmp = (1.0 / (s * t_0)) / (x * c)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 3.6e+122)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x * x))) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(s * t_0)) / Float64(x * c));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 3.6e+122)
		tmp = (1.0 + (-2.0 * (x * x))) / (t_0 * t_0);
	else
		tmp = (1.0 / (s * t_0)) / (x * c);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.6e+122], N[(N[(1.0 + N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(s * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6000000000000003e122

   1. Initial program 65.2%

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

      1. associate-/r* 64.8%

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

      2. *-commutative 64.8%

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

      3. associate-*l* 59.5%

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

      4. unpow2 59.5%

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

      5. unpow2 59.5%

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

      6. associate-*r* 64.6%

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

      7. associate-/r* 66.3%

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

      8. associate-/l/ 66.3%

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

      9. associate-/l/ 66.3%

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

      10. *-commutative 66.3%

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

      11. associate-*l* 64.8%

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

      12. unpow2 64.8%

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

      13. associate-*l* 59.8%

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

      14. unpow2 59.8%

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

      15. unswap-sqr 76.2%

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

      16. *-commutative 76.2%

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

      17. *-commutative 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in x around 0 59.9%

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

      1. unpow2 59.9%

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

   6. Simplified 59.9%

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

      1. unswap-sqr 73.3%

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

      2. unpow2 73.3%

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

      3. *-commutative 73.3%

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

      4. associate-*r* 73.8%

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

      5. *-commutative 73.8%

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

   8. Applied egg-rr 73.8%

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

      1. unpow2 73.8%

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

   10. Applied egg-rr 73.8%

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

   if 3.6000000000000003e122 < x

   1. Initial program 63.8%

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

      1. associate-/r* 62.0%

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

      2. remove-double-neg 62.0%

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

      3. distribute-lft-neg-out 62.0%

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

      4. distribute-lft-neg-out 62.0%

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

      5. distribute-rgt-neg-out 62.0%

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

      6. associate-/l/ 63.8%

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

      7. distribute-rgt-neg-out 63.8%

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

      8. distribute-lft-neg-out 63.8%

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

      9. associate-*l* 65.7%

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

      10. distribute-lft-neg-in 65.7%

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

      11. distribute-lft-neg-out 65.7%

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

      12. remove-double-neg 65.7%

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

      13. associate-*r* 63.9%

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

      14. *-commutative 63.9%

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

      15. associate-*r* 61.9%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in x around 0 51.6%

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

      1. unpow2 51.6%

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

      2. unpow2 51.6%

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

      3. unpow2 51.6%

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

      4. swap-sqr 85.3%

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

      5. swap-sqr 99.5%

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

      6. unpow2 99.5%

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

      7. associate-*r* 97.3%

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

      8. *-commutative 97.3%

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

   6. Simplified 97.3%

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

      1. *-commutative 97.3%

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

      2. *-un-lft-identity 97.3%

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

      3. *-commutative 97.3%

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

      4. associate-*r* 99.5%

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

      5. pow2 99.5%

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

      6. frac-times 99.5%

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

      7. associate-*l/ 99.6%

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

      8. associate-*r* 97.4%

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

      9. *-commutative 97.4%

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

      10. associate-*r* 95.5%

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

      11. times-frac 93.2%

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

      12. *-commutative 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. associate-*l/ 93.2%

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

      2. *-lft-identity 93.2%

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

      3. associate-/l/ 93.1%

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

      4. *-commutative 93.1%

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

      5. *-commutative 93.1%

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

   10. Simplified 93.1%

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

   11. Taylor expanded in x around 0 70.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot \left(c \cdot \left(x \cdot s\right)\right)}}{x \cdot c}\\ \end{array} \]

Alternative 10: 79.8% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 64.5%

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

   1. associate-/r* 64.5%

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

   2. swap-sqr 85.1%

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

   3. associate-/r* 73.6%

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

   4. associate-/r* 73.3%

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

   5. associate-/r* 73.9%

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

   6. *-un-lft-identity 73.9%

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

   7. add-sqr-sqrt 73.9%

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

   8. times-frac 73.9%

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

5. Applied egg-rr 97.4%

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

6. Taylor expanded in x around 0 79.9%

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

   1. associate-/r* 80.0%

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

8. Simplified 80.0%

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

   1. associate-/r* 79.9%

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

   2. frac-2neg 79.9%

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

   3. metadata-eval 79.9%

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

   4. *-commutative 79.9%

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

   5. frac-times 80.0%

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

   6. metadata-eval 80.0%

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

   7. *-commutative 80.0%

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

10. Applied egg-rr 80.0%

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

11. Final simplification 80.0%

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

Alternative 11: 63.2% accurate, 24.1× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 70.4%

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

4. Taylor expanded in x around 0 63.5%

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

5. Final simplification 63.5%

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

Alternative 12: 69.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 70.4%

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

4. Taylor expanded in x around 0 63.5%

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

5. Taylor expanded in c around 0 63.5%

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

   1. unpow2 63.5%

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

   2. associate-*r* 70.8%

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

   3. *-commutative 70.8%

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

7. Simplified 70.8%

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

8. Final simplification 70.8%

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

Alternative 13: 79.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 64.5%

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

   1. associate-/r* 64.5%

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

   2. swap-sqr 85.1%

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

   3. associate-/r* 73.6%

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

   4. associate-/r* 73.3%

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

   5. associate-/r* 73.9%

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

   6. *-un-lft-identity 73.9%

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

   7. add-sqr-sqrt 73.9%

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

   8. times-frac 73.9%

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

5. Applied egg-rr 97.4%

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

6. Taylor expanded in x around 0 79.9%

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

   1. associate-/r* 80.0%

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

8. Simplified 80.0%

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

   1. frac-times 78.2%

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

   2. *-un-lft-identity 78.2%

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

   3. *-commutative 78.2%

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

   4. *-commutative 78.2%

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

10. Applied egg-rr 78.2%

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

11. Final simplification 78.2%

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

Alternative 14: 79.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. remove-double-neg 64.3%

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

   3. distribute-lft-neg-out 64.3%

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

   4. distribute-lft-neg-out 64.3%

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

   5. distribute-rgt-neg-out 64.3%

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

   6. associate-/l/ 65.0%

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

   7. distribute-rgt-neg-out 65.0%

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

   8. distribute-lft-neg-out 65.0%

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

   9. associate-*l* 67.4%

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

   10. distribute-lft-neg-in 67.4%

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

   11. distribute-lft-neg-out 67.4%

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

   12. remove-double-neg 67.4%

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

   13. associate-*r* 66.3%

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

   14. *-commutative 66.3%

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

   15. associate-*r* 64.5%

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

3. Simplified 64.5%

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

   1. associate-/r* 64.5%

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

   2. swap-sqr 85.1%

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

   3. associate-/r* 73.6%

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

   4. associate-/r* 73.3%

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

   5. associate-/r* 73.9%

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

   6. *-un-lft-identity 73.9%

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

   7. add-sqr-sqrt 73.9%

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

   8. times-frac 73.9%

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

5. Applied egg-rr 97.4%

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

   1. associate-*l/ 97.4%

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

   2. *-un-lft-identity 97.4%

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

   3. *-commutative 97.4%

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

   4. *-commutative 97.4%

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

7. Applied egg-rr 97.4%

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

8. Taylor expanded in x around 0 80.0%

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

9. Final simplification 80.0%

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

Alternative 15: 27.4% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. *-commutative 64.3%

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

   3. associate-*l* 58.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{{s}^{2} \cdot \left(x \cdot x\right)}} \]

   4. unpow2 58.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{\left(s \cdot s\right)} \cdot \left(x \cdot x\right)} \]

   5. unpow2 58.2%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(s \cdot s\right) \cdot \color{blue}{{x}^{2}}} \]

   6. associate-*r* 63.8%

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\color{blue}{s \cdot \left(s \cdot {x}^{2}\right)}} \]

   7. associate-/r* 65.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{s}}{s \cdot {x}^{2}}} \]

   8. associate-/l/ 65.2%

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{s \cdot {c}^{2}}}}{s \cdot {x}^{2}} \]

   9. associate-/l/ 65.2%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(s \cdot {x}^{2}\right) \cdot \left(s \cdot {c}^{2}\right)}} \]

   10. *-commutative 65.2%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({x}^{2} \cdot s\right)} \cdot \left(s \cdot {c}^{2}\right)} \]

   11. associate-*l* 63.6%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{x}^{2} \cdot \left(s \cdot \left(s \cdot {c}^{2}\right)\right)}} \]

   12. unpow2 63.6%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(s \cdot \left(s \cdot {c}^{2}\right)\right)} \]

   13. associate-*l* 58.1%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot {c}^{2}\right)}} \]

   14. unpow2 58.1%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)} \]

   15. unswap-sqr 73.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}} \]

   16. *-commutative 73.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(s \cdot c\right)\right)} \]

   17. *-commutative 73.9%

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

3. Simplified 73.9%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]

4. Taylor expanded in x around 0 51.1%

   \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 51.1%

      \[\leadsto \frac{1 + -2 \cdot \color{blue}{\left(x \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)} \]

6. Simplified 51.1%

   \[\leadsto \frac{\color{blue}{1 + -2 \cdot \left(x \cdot x\right)}}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)} \]

7. Taylor expanded in x around inf 26.9%

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 26.9%

      \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

   2. unpow2 26.9%

      \[\leadsto \frac{-2}{\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]

9. Simplified 26.9%

   \[\leadsto \color{blue}{\frac{-2}{\left(c \cdot c\right) \cdot \left(s \cdot s\right)}} \]

10. Final simplification 26.9%

    \[\leadsto \frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \]

Reproduce

herbie shell --seed 2023287 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))