mixedcos

Percentage Accurate: 66.7% → 98.4%

Time: 15.4s

Alternatives: 8

Speedup: 24.1×

• 32.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{s\_m}}{x\_m}}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* x_m (* s_m c_m))))
   (if (<= x_m 1e-13)
     (/ (/ (/ (/ 1.0 s_m) x_m) c_m) (* (* x_m s_m) c_m))
     (/ (/ (cos (* x_m 2.0)) t_0) t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 1e-13) {
		tmp = (((1.0 / s_m) / x_m) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (cos((x_m * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (s_m * c_m)
    if (x_m <= 1d-13) then
        tmp = (((1.0d0 / s_m) / x_m) / c_m) / ((x_m * s_m) * c_m)
    else
        tmp = (cos((x_m * 2.0d0)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 1e-13) {
		tmp = (((1.0 / s_m) / x_m) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = x_m * (s_m * c_m)
	tmp = 0
	if x_m <= 1e-13:
		tmp = (((1.0 / s_m) / x_m) / c_m) / ((x_m * s_m) * c_m)
	else:
		tmp = (math.cos((x_m * 2.0)) / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(x_m * Float64(s_m * c_m))
	tmp = 0.0
	if (x_m <= 1e-13)
		tmp = Float64(Float64(Float64(Float64(1.0 / s_m) / x_m) / c_m) / Float64(Float64(x_m * s_m) * c_m));
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = x_m * (s_m * c_m);
	tmp = 0.0;
	if (x_m <= 1e-13)
		tmp = (((1.0 / s_m) / x_m) / c_m) / ((x_m * s_m) * c_m);
	else
		tmp = (cos((x_m * 2.0)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1e-13], N[(N[(N[(N[(1.0 / s$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-13

   1. Initial program 66.0%

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

      1. associate-/r* 66.1%

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

      2. *-commutative 66.1%

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

      3. unpow2 66.1%

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

      4. sqr-neg 66.1%

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

      5. unpow2 66.1%

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

      6. cos-neg 66.1%

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

      7. *-commutative 66.1%

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

      8. distribute-rgt-neg-in 66.1%

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

      9. metadata-eval 66.1%

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

      10. unpow2 66.1%

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

      11. sqr-neg 66.1%

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

      12. unpow2 66.1%

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

      13. associate-*r* 61.4%

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

      14. unpow2 61.4%

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

      15. *-commutative 61.4%

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

   3. Simplified 61.4%

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

      1. associate-/l/ 61.3%

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

      2. associate-/r* 61.4%

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

      3. associate-/l/ 61.5%

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

      4. add-cube-cbrt 61.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}}{{c}^{2}} \]

      5. associate-/l* 61.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}{{c}^{2}}} \]

   6. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 80.6%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]

      2. unpow2 80.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}} \]

      3. rem-3cbrt-lft 80.8%

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

      4. *-commutative 80.8%

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

      5. associate-/l* 80.5%

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

      6. *-commutative 80.5%

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

   8. Simplified 80.5%

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

      1. *-commutative 80.5%

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

      2. div-inv 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*l* 80.5%

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

      5. pow-flip 80.8%

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

      6. metadata-eval 80.8%

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

      7. unpow-prod-down 98.6%

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

      8. metadata-eval 98.6%

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

      9. pow-flip 97.8%

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

      10. div-inv 97.8%

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

      11. unpow2 97.8%

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

      12. associate-/r* 98.6%

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

      13. *-commutative 98.6%

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

      14. associate-/r* 98.7%

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

      15. *-commutative 98.7%

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

   10. Applied egg-rr 98.7%

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

   11. Taylor expanded in x around 0 86.7%

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

       1. associate-/r* 86.8%

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

   13. Simplified 86.8%

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

   if 1e-13 < x

   1. Initial program 78.4%

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

      1. associate-/r* 78.3%

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

      2. *-commutative 78.3%

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

      3. unpow2 78.3%

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

      4. sqr-neg 78.3%

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

      5. unpow2 78.3%

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

      6. cos-neg 78.3%

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

      7. *-commutative 78.3%

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

      8. distribute-rgt-neg-in 78.3%

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

      9. metadata-eval 78.3%

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

      10. unpow2 78.3%

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

      11. sqr-neg 78.3%

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

      12. unpow2 78.3%

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

      13. associate-*r* 70.3%

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

      14. unpow2 70.3%

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

      15. *-commutative 70.3%

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

   3. Simplified 70.3%

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

   6. Applied egg-rr 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. associate-*r* 98.2%

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

      3. times-frac 98.2%

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

      4. *-commutative 98.2%

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

   8. Applied egg-rr 98.2%

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

      1. associate-*l/ 98.2%

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

      2. *-un-lft-identity 98.2%

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

      3. frac-times 98.3%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. associate-*l* 96.8%

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

      8. *-commutative 96.8%

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

      9. associate-*l* 96.7%

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

   10. Applied egg-rr 96.7%

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

4. Final simplification 89.2%

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

Alternative 2: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{t\_0}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot \left(\left(x\_m \cdot s\_m\right) \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{s\_m}}{\left(x\_m \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(x\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m 2.0))))
   (if (<= x_m 4.4e+61)
     (/ (/ t_0 c_m) (* (* x_m s_m) (* (* x_m s_m) c_m)))
     (/ (/ t_0 s_m) (* (* x_m (* s_m c_m)) (* x_m c_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 4.4e+61) {
		tmp = (t_0 / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m));
	} else {
		tmp = (t_0 / s_m) / ((x_m * (s_m * c_m)) * (x_m * c_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    if (x_m <= 4.4d+61) then
        tmp = (t_0 / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m))
    else
        tmp = (t_0 / s_m) / ((x_m * (s_m * c_m)) * (x_m * c_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 4.4e+61) {
		tmp = (t_0 / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m));
	} else {
		tmp = (t_0 / s_m) / ((x_m * (s_m * c_m)) * (x_m * c_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 4.4e+61:
		tmp = (t_0 / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m))
	else:
		tmp = (t_0 / s_m) / ((x_m * (s_m * c_m)) * (x_m * c_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 4.4e+61)
		tmp = Float64(Float64(t_0 / c_m) / Float64(Float64(x_m * s_m) * Float64(Float64(x_m * s_m) * c_m)));
	else
		tmp = Float64(Float64(t_0 / s_m) / Float64(Float64(x_m * Float64(s_m * c_m)) * Float64(x_m * c_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 4.4e+61)
		tmp = (t_0 / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m));
	else
		tmp = (t_0 / s_m) / ((x_m * (s_m * c_m)) * (x_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 4.4e+61], N[(N[(t$95$0 / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4000000000000001e61

   1. Initial program 66.6%

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

      1. associate-/r* 66.6%

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

      2. *-commutative 66.6%

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

      3. unpow2 66.6%

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

      4. sqr-neg 66.6%

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

      5. unpow2 66.6%

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

      6. cos-neg 66.6%

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

      7. *-commutative 66.6%

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

      8. distribute-rgt-neg-in 66.6%

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

      9. metadata-eval 66.6%

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

      10. unpow2 66.6%

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

      11. sqr-neg 66.6%

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

      12. unpow2 66.6%

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

      13. associate-*r* 62.3%

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

      14. unpow2 62.3%

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

      15. *-commutative 62.3%

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

   3. Simplified 62.3%

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

      1. associate-/l/ 62.3%

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

      2. associate-/r* 62.3%

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

      3. associate-/l/ 62.4%

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

      4. add-cube-cbrt 62.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}}{{c}^{2}} \]

      5. associate-/l* 62.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}{{c}^{2}}} \]

   6. Applied egg-rr 80.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]

      2. unpow2 80.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}} \]

      3. rem-3cbrt-lft 80.3%

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

      4. *-commutative 80.3%

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

      5. associate-/l* 80.1%

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

      6. *-commutative 80.1%

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

   8. Simplified 80.1%

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

      1. *-commutative 80.1%

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

      2. div-inv 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*l* 80.1%

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

      5. pow-flip 80.3%

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

      6. metadata-eval 80.3%

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

      7. unpow-prod-down 98.7%

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

      8. metadata-eval 98.7%

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

      9. pow-flip 97.9%

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

      10. div-inv 97.9%

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

      11. unpow2 97.9%

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

      12. associate-/r* 98.6%

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

      13. associate-/r* 98.7%

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

      14. associate-/l/ 95.4%

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

      15. *-commutative 95.4%

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

   10. Applied egg-rr 95.4%

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

   if 4.4000000000000001e61 < x

   1. Initial program 79.7%

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

      1. associate-/r* 79.7%

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

      2. *-commutative 79.7%

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

      3. unpow2 79.7%

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

      4. sqr-neg 79.7%

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

      5. unpow2 79.7%

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

      6. cos-neg 79.7%

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

      7. *-commutative 79.7%

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

      8. distribute-rgt-neg-in 79.7%

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

      9. metadata-eval 79.7%

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

      10. unpow2 79.7%

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

      11. sqr-neg 79.7%

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

      12. unpow2 79.7%

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

      13. associate-*r* 69.1%

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

      14. unpow2 69.1%

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

      15. *-commutative 69.1%

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

   3. Simplified 69.1%

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

   6. Applied egg-rr 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-*r* 97.9%

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

      3. times-frac 97.8%

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

      4. *-commutative 97.8%

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

   8. Applied egg-rr 97.8%

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

      1. associate-*l/ 97.8%

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

      2. *-un-lft-identity 97.8%

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

      3. frac-times 95.9%

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

      4. *-un-lft-identity 95.9%

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

      5. *-commutative 95.9%

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

      6. associate-*l* 92.2%

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

      7. *-commutative 92.2%

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

   10. Applied egg-rr 92.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{\left(x \cdot s\right) \cdot \left(\left(x \cdot s\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 2.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (/ (/ (cos (* x_m 2.0)) (* x_m s_m)) c_m) (* (* x_m s_m) c_m)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return ((cos((x_m * 2.0)) / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return ((math.cos((x_m * 2.0)) / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. associate-/r* 69.1%

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

   2. *-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. sqr-neg 69.1%

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

   5. unpow2 69.1%

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

   6. cos-neg 69.1%

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

   7. *-commutative 69.1%

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

   8. distribute-rgt-neg-in 69.1%

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

   9. metadata-eval 69.1%

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

   10. unpow2 69.1%

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

   11. sqr-neg 69.1%

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

   12. unpow2 69.1%

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

   13. associate-*r* 63.6%

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

   14. unpow2 63.6%

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

   15. *-commutative 63.6%

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

3. Simplified 63.6%

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

   1. associate-/l/ 63.6%

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

   2. associate-/r* 63.6%

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

   3. associate-/l/ 63.6%

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

   4. add-cube-cbrt 63.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}}{{c}^{2}} \]

   5. associate-/l* 63.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}{{c}^{2}}} \]

6. Applied egg-rr 81.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 81.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]

   2. unpow2 81.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}} \]

   3. rem-3cbrt-lft 81.8%

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

   4. *-commutative 81.8%

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

   5. associate-/l* 81.5%

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

   6. *-commutative 81.5%

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

8. Simplified 81.5%

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

   1. *-commutative 81.5%

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

   2. div-inv 81.5%

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

   3. *-commutative 81.5%

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

   4. associate-*l* 81.5%

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

   5. pow-flip 81.7%

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

   6. metadata-eval 81.7%

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

   7. unpow-prod-down 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow-flip 98.2%

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

   10. div-inv 98.2%

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

   11. unpow2 98.2%

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

   12. associate-/r* 98.8%

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

   13. *-commutative 98.8%

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

   14. associate-/r* 98.9%

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

   15. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 4: 94.1% accurate, 2.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (/ (cos (* x_m 2.0)) c_m) (* (* x_m s_m) (* (* x_m s_m) c_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (cos((x_m * 2.0)) / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m));
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return (math.cos((x_m * 2.0)) / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. associate-/r* 69.1%

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

   2. *-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. sqr-neg 69.1%

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

   5. unpow2 69.1%

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

   6. cos-neg 69.1%

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

   7. *-commutative 69.1%

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

   8. distribute-rgt-neg-in 69.1%

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

   9. metadata-eval 69.1%

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

   10. unpow2 69.1%

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

   11. sqr-neg 69.1%

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

   12. unpow2 69.1%

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

   13. associate-*r* 63.6%

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

   14. unpow2 63.6%

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

   15. *-commutative 63.6%

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

3. Simplified 63.6%

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

   1. associate-/l/ 63.6%

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

   2. associate-/r* 63.6%

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

   3. associate-/l/ 63.6%

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

   4. add-cube-cbrt 63.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}}{{c}^{2}} \]

   5. associate-/l* 63.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}{{c}^{2}}} \]

6. Applied egg-rr 81.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 81.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]

   2. unpow2 81.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}} \]

   3. rem-3cbrt-lft 81.8%

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

   4. *-commutative 81.8%

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

   5. associate-/l* 81.5%

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

   6. *-commutative 81.5%

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

8. Simplified 81.5%

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

   1. *-commutative 81.5%

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

   2. div-inv 81.5%

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

   3. *-commutative 81.5%

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

   4. associate-*l* 81.5%

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

   5. pow-flip 81.7%

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

   6. metadata-eval 81.7%

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

   7. unpow-prod-down 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow-flip 98.2%

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

   10. div-inv 98.2%

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

   11. unpow2 98.2%

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

   12. associate-/r* 98.8%

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

   13. associate-/r* 98.9%

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

   14. associate-/l/ 95.9%

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

   15. *-commutative 95.9%

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

10. Applied egg-rr 95.9%

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

11. Final simplification 95.9%

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

Alternative 5: 80.0% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (/ (/ (/ 1.0 s_m) x_m) c_m) (* (* x_m s_m) c_m)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (((1.0 / s_m) / x_m) / c_m) / ((x_m * s_m) * c_m);
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return (((1.0 / s_m) / x_m) / c_m) / ((x_m * s_m) * c_m)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[(N[(N[(1.0 / s$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. associate-/r* 69.1%

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

   2. *-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. sqr-neg 69.1%

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

   5. unpow2 69.1%

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

   6. cos-neg 69.1%

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

   7. *-commutative 69.1%

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

   8. distribute-rgt-neg-in 69.1%

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

   9. metadata-eval 69.1%

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

   10. unpow2 69.1%

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

   11. sqr-neg 69.1%

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

   12. unpow2 69.1%

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

   13. associate-*r* 63.6%

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

   14. unpow2 63.6%

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

   15. *-commutative 63.6%

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

3. Simplified 63.6%

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

   1. associate-/l/ 63.6%

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

   2. associate-/r* 63.6%

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

   3. associate-/l/ 63.6%

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

   4. add-cube-cbrt 63.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}}{{c}^{2}} \]

   5. associate-/l* 63.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}{{c}^{2}}} \]

6. Applied egg-rr 81.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 81.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]

   2. unpow2 81.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}} \]

   3. rem-3cbrt-lft 81.8%

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

   4. *-commutative 81.8%

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

   5. associate-/l* 81.5%

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

   6. *-commutative 81.5%

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

8. Simplified 81.5%

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

   1. *-commutative 81.5%

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

   2. div-inv 81.5%

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

   3. *-commutative 81.5%

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

   4. associate-*l* 81.5%

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

   5. pow-flip 81.7%

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

   6. metadata-eval 81.7%

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

   7. unpow-prod-down 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow-flip 98.2%

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

   10. div-inv 98.2%

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

   11. unpow2 98.2%

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

   12. associate-/r* 98.8%

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

   13. *-commutative 98.8%

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

   14. associate-/r* 98.9%

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

   15. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

11. Taylor expanded in x around 0 81.2%

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

    1. associate-/r* 81.3%

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

13. Simplified 81.3%

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

14. Final simplification 81.3%

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

Alternative 6: 79.3% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (/ 1.0 c_m) (* (* x_m s_m) (* (* x_m s_m) c_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (1.0 / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m));
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return (1.0 / c_m) / ((x_m * s_m) * ((x_m * s_m) * c_m))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. associate-/r* 69.1%

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

   2. *-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. sqr-neg 69.1%

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

   5. unpow2 69.1%

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

   6. cos-neg 69.1%

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

   7. *-commutative 69.1%

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

   8. distribute-rgt-neg-in 69.1%

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

   9. metadata-eval 69.1%

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

   10. unpow2 69.1%

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

   11. sqr-neg 69.1%

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

   12. unpow2 69.1%

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

   13. associate-*r* 63.6%

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

   14. unpow2 63.6%

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

   15. *-commutative 63.6%

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

3. Simplified 63.6%

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

   1. associate-/l/ 63.6%

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

   2. associate-/r* 63.6%

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

   3. associate-/l/ 63.6%

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

   4. add-cube-cbrt 63.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}}{{c}^{2}} \]

   5. associate-/l* 63.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{\cos \left(x \cdot -2\right)}{{x}^{2}}}{{s}^{2}}}}{{c}^{2}}} \]

6. Applied egg-rr 81.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 81.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)}^{2} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}}} \]

   2. unpow2 81.6%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right) \cdot {\left(x \cdot s\right)}^{-2}}}{{c}^{2}} \]

   3. rem-3cbrt-lft 81.8%

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

   4. *-commutative 81.8%

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

   5. associate-/l* 81.5%

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

   6. *-commutative 81.5%

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

8. Simplified 81.5%

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

   1. *-commutative 81.5%

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

   2. div-inv 81.5%

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

   3. *-commutative 81.5%

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

   4. associate-*l* 81.5%

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

   5. pow-flip 81.7%

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

   6. metadata-eval 81.7%

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

   7. unpow-prod-down 98.9%

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

   8. metadata-eval 98.9%

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

   9. pow-flip 98.2%

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

   10. div-inv 98.2%

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

   11. unpow2 98.2%

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

   12. associate-/r* 98.8%

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

   13. associate-/r* 98.9%

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

   14. associate-/l/ 95.9%

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

   15. *-commutative 95.9%

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

10. Applied egg-rr 95.9%

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

11. Taylor expanded in x around 0 80.1%

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

12. Final simplification 80.1%

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

Alternative 7: 79.9% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(x\_m \cdot s\_m\right) \cdot c\_m\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* x_m s_m) c_m))) (/ 1.0 (* t_0 t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (x_m * s_m) * c_m;
	return 1.0 / (t_0 * t_0);
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (x_m * s_m) * c_m
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 69.0%

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

   1. associate-/r* 69.1%

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

   2. *-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. sqr-neg 69.1%

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

   5. unpow2 69.1%

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

   6. cos-neg 69.1%

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

   7. *-commutative 69.1%

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

   8. distribute-rgt-neg-in 69.1%

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

   9. metadata-eval 69.1%

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

   10. unpow2 69.1%

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

   11. sqr-neg 69.1%

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

   12. unpow2 69.1%

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

   13. associate-*r* 63.6%

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

   14. unpow2 63.6%

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

   15. *-commutative 63.6%

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

3. Simplified 63.6%

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

5. Taylor expanded in x around 0 57.6%

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

   1. associate-/r* 57.7%

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

   2. *-commutative 57.7%

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

   3. unpow2 57.7%

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

   4. unpow2 57.7%

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

   5. swap-sqr 70.3%

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

   6. unpow2 70.3%

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

   7. associate-/r* 70.5%

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

   8. unpow2 70.5%

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

   9. unpow2 70.5%

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

   10. swap-sqr 80.9%

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

   11. unpow2 80.9%

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

7. Simplified 80.9%

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

   1. unpow2 80.9%

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

9. Applied egg-rr 80.9%

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

10. Final simplification 80.9%

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

Alternative 8: 74.9% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ 1.0 (* (* x_m c_m) (* s_m (* (* x_m s_m) c_m)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 1.0 / ((x_m * c_m) * (s_m * ((x_m * s_m) * c_m)));
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 1.0 / ((x_m * c_m) * (s_m * ((x_m * s_m) * c_m)))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. associate-/r* 69.1%

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

   2. *-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. sqr-neg 69.1%

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

   5. unpow2 69.1%

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

   6. cos-neg 69.1%

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

   7. *-commutative 69.1%

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

   8. distribute-rgt-neg-in 69.1%

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

   9. metadata-eval 69.1%

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

   10. unpow2 69.1%

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

   11. sqr-neg 69.1%

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

   12. unpow2 69.1%

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

   13. associate-*r* 63.6%

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

   14. unpow2 63.6%

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

   15. *-commutative 63.6%

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

3. Simplified 63.6%

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

5. Taylor expanded in x around 0 57.6%

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

   1. associate-/r* 57.7%

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

   2. *-commutative 57.7%

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

   3. unpow2 57.7%

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

   4. unpow2 57.7%

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

   5. swap-sqr 70.3%

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

   6. unpow2 70.3%

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

   7. associate-/r* 70.5%

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

   8. unpow2 70.5%

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

   9. unpow2 70.5%

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

   10. swap-sqr 80.9%

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

   11. unpow2 80.9%

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

7. Simplified 80.9%

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

   1. unpow2 80.9%

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

   2. associate-*r* 79.6%

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

   3. associate-*l* 77.5%

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

9. Applied egg-rr 77.5%

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

10. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))