mixedcos

Percentage Accurate: 66.8% → 99.5%

Time: 16.5s

Alternatives: 9

Speedup: 24.1×

• 31.2% 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: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x\_m}}{s\_m}}{c\_m}}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot -2\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (* s_m (* x_m c_m))))
   (if (<= x_m 4e-33)
     (/ (/ (/ (/ 1.0 x_m) s_m) c_m) (* c_m (* x_m s_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 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 4e-33) {
		tmp = (((1.0 / x_m) / s_m) / c_m) / (c_m * (x_m * s_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 = s_m * (x_m * c_m)
    if (x_m <= 4d-33) then
        tmp = (((1.0d0 / x_m) / s_m) / c_m) / (c_m * (x_m * s_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 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 4e-33) {
		tmp = (((1.0 / x_m) / s_m) / c_m) / (c_m * (x_m * s_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 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 4e-33:
		tmp = (((1.0 / x_m) / s_m) / c_m) / (c_m * (x_m * s_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(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 4e-33)
		tmp = Float64(Float64(Float64(Float64(1.0 / x_m) / s_m) / c_m) / Float64(c_m * Float64(x_m * s_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 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 4e-33)
		tmp = (((1.0 / x_m) / s_m) / c_m) / (c_m * (x_m * s_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[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4e-33], N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $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 < 4.0000000000000002e-33

   1. Initial program 62.1%

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

      1. add-cbrt-cube 58.5%

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

      2. pow3 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-*r* 54.3%

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

      5. unpow2 54.3%

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

      6. associate-*l* 54.2%

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

      7. pow-prod-down 65.0%

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

      8. pow-prod-down 72.3%

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

   4. Applied egg-rr 72.3%

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

      1. rem-cbrt-cube 96.1%

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

      2. associate-*l* 98.0%

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

      3. *-commutative 98.0%

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

      4. unpow2 98.0%

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

      5. associate-/r* 98.0%

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

      6. *-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-rgt-identity 98.0%

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

      2. times-frac 98.1%

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

      3. *-commutative 98.1%

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

   8. Applied egg-rr 98.1%

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

   9. Taylor expanded in x around 0 85.4%

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

       1. associate-*r* 82.4%

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

       2. associate-/l/ 82.3%

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

       3. associate-/l/ 85.5%

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

   11. Simplified 85.5%

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

   if 4.0000000000000002e-33 < x

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l* 63.1%

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

      3. associate-/r* 63.2%

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

      4. associate-/l/ 62.3%

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

      5. associate-/l/ 60.6%

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

      6. cos-neg 60.6%

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

      7. *-commutative 60.6%

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

      8. distribute-rgt-neg-in 60.6%

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

      9. metadata-eval 60.6%

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

      10. *-commutative 60.6%

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

      11. associate-*l* 57.9%

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

      12. unpow2 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in x around inf 60.2%

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

      1. *-commutative 60.2%

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

      2. *-commutative 60.2%

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

      3. associate-*r* 58.7%

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

      4. unpow2 58.7%

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

      5. unpow2 58.7%

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

      6. swap-sqr 72.7%

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

      7. unpow2 72.7%

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

      8. swap-sqr 98.2%

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

      9. unpow2 98.2%

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

      10. associate-*l* 99.5%

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

      11. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. unpow2 99.5%

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

      2. associate-*r* 98.3%

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

      3. associate-*r* 96.6%

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

   9. Applied egg-rr 96.6%

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

       1. *-un-lft-identity 96.6%

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

       2. associate-*l* 98.3%

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

       3. associate-*r* 99.5%

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

       4. times-frac 99.4%

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

       5. associate-*r* 98.3%

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

       6. *-commutative 98.3%

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

       7. associate-*r* 98.2%

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

       8. *-commutative 98.2%

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

   11. Applied egg-rr 98.2%

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

       1. associate-*l/ 98.3%

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

       2. *-lft-identity 98.3%

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

       3. associate-*r* 97.1%

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

       4. *-commutative 97.1%

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

       5. *-commutative 97.1%

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

       6. associate-*r* 98.2%

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

       7. *-commutative 98.2%

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

       8. *-commutative 98.2%

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

   13. Simplified 98.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot -2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 10^{-175}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot \left(t\_1 \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(x\_m \cdot s\_m\right) \cdot \left(c\_m \cdot t\_1\right)}\\ \end{array} \end{array} \]

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))) (t_1 (* c_m (* x_m s_m))))
   (if (<= c_m 1e-175)
     (/ t_0 (* x_m (* t_1 (* c_m s_m))))
     (/ t_0 (* (* x_m s_m) (* c_m t_1))))))

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 t_1 = c_m * (x_m * s_m);
	double tmp;
	if (c_m <= 1e-175) {
		tmp = t_0 / (x_m * (t_1 * (c_m * s_m)));
	} else {
		tmp = t_0 / ((x_m * s_m) * (c_m * t_1));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * (-2.0d0)))
    t_1 = c_m * (x_m * s_m)
    if (c_m <= 1d-175) then
        tmp = t_0 / (x_m * (t_1 * (c_m * s_m)))
    else
        tmp = t_0 / ((x_m * s_m) * (c_m * t_1))
    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 t_1 = c_m * (x_m * s_m);
	double tmp;
	if (c_m <= 1e-175) {
		tmp = t_0 / (x_m * (t_1 * (c_m * s_m)));
	} else {
		tmp = t_0 / ((x_m * s_m) * (c_m * t_1));
	}
	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))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if c_m <= 1e-175:
		tmp = t_0 / (x_m * (t_1 * (c_m * s_m)))
	else:
		tmp = t_0 / ((x_m * s_m) * (c_m * t_1))
	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))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (c_m <= 1e-175)
		tmp = Float64(t_0 / Float64(x_m * Float64(t_1 * Float64(c_m * s_m))));
	else
		tmp = Float64(t_0 / Float64(Float64(x_m * s_m) * Float64(c_m * t_1)));
	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));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (c_m <= 1e-175)
		tmp = t_0 / (x_m * (t_1 * (c_m * s_m)));
	else
		tmp = t_0 / ((x_m * s_m) * (c_m * t_1));
	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]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 1e-175], N[(t$95$0 / N[(x$95$m * N[(t$95$1 * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1e-175

   1. Initial program 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*l* 58.3%

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

      3. associate-/r* 58.3%

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

      4. associate-/l/ 57.2%

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

      5. associate-/l/ 55.1%

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

      6. cos-neg 55.1%

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

      7. *-commutative 55.1%

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

      8. distribute-rgt-neg-in 55.1%

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

      9. metadata-eval 55.1%

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

      10. *-commutative 55.1%

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

      11. associate-*l* 52.1%

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

      12. unpow2 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in x around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. *-commutative 53.9%

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

      3. associate-*r* 53.1%

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

      4. unpow2 53.1%

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

      5. unpow2 53.1%

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

      6. swap-sqr 70.9%

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

      7. unpow2 70.9%

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

      8. swap-sqr 95.2%

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

      9. unpow2 95.2%

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

      10. associate-*l* 98.8%

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

      11. *-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. unpow2 98.8%

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

      2. associate-*r* 94.1%

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

      3. associate-*r* 88.5%

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

   9. Applied egg-rr 88.5%

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

   if 1e-175 < c

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*l* 73.2%

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

      3. associate-/r* 73.2%

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

      4. associate-/l/ 73.2%

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

      5. associate-/l/ 71.4%

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

      6. cos-neg 71.4%

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

      7. *-commutative 71.4%

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

      8. distribute-rgt-neg-in 71.4%

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

      9. metadata-eval 71.4%

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

      10. *-commutative 71.4%

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

      11. associate-*l* 64.0%

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

      12. unpow2 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in x around inf 63.9%

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

      1. *-commutative 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*r* 64.0%

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

      4. unpow2 64.0%

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

      5. unpow2 64.0%

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

      6. swap-sqr 82.1%

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

      7. unpow2 82.1%

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

      8. swap-sqr 98.7%

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

      9. unpow2 98.7%

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

      10. associate-*l* 97.8%

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

      11. *-commutative 97.8%

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

   7. Simplified 97.8%

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

      1. unpow2 97.8%

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

      2. associate-*r* 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 10^{-175}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{x \cdot \left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 4 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x\_m}}{s\_m}}{c\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{s\_m \cdot \left(t\_0 \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

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 (* c_m (* x_m s_m))))
   (if (<= x_m 4e-32)
     (/ (/ (/ (/ 1.0 x_m) s_m) c_m) t_0)
     (/ (cos (* x_m -2.0)) (* s_m (* t_0 (* 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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4e-32) {
		tmp = (((1.0 / x_m) / s_m) / c_m) / t_0;
	} else {
		tmp = cos((x_m * -2.0)) / (s_m * (t_0 * (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 = c_m * (x_m * s_m)
    if (x_m <= 4d-32) then
        tmp = (((1.0d0 / x_m) / s_m) / c_m) / t_0
    else
        tmp = cos((x_m * (-2.0d0))) / (s_m * (t_0 * (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 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4e-32) {
		tmp = (((1.0 / x_m) / s_m) / c_m) / t_0;
	} else {
		tmp = Math.cos((x_m * -2.0)) / (s_m * (t_0 * (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 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 4e-32:
		tmp = (((1.0 / x_m) / s_m) / c_m) / t_0
	else:
		tmp = math.cos((x_m * -2.0)) / (s_m * (t_0 * (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 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 4e-32)
		tmp = Float64(Float64(Float64(Float64(1.0 / x_m) / s_m) / c_m) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(s_m * Float64(t_0 * 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 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 4e-32)
		tmp = (((1.0 / x_m) / s_m) / c_m) / t_0;
	else
		tmp = cos((x_m * -2.0)) / (s_m * (t_0 * (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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4e-32], N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(t$95$0 * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000022e-32

   1. Initial program 62.1%

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

      1. add-cbrt-cube 58.5%

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

      2. pow3 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-*r* 54.3%

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

      5. unpow2 54.3%

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

      6. associate-*l* 54.2%

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

      7. pow-prod-down 65.0%

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

      8. pow-prod-down 72.3%

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

   4. Applied egg-rr 72.3%

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

      1. rem-cbrt-cube 96.1%

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

      2. associate-*l* 98.0%

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

      3. *-commutative 98.0%

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

      4. unpow2 98.0%

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

      5. associate-/r* 98.0%

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

      6. *-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-rgt-identity 98.0%

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

      2. times-frac 98.1%

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

      3. *-commutative 98.1%

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

   8. Applied egg-rr 98.1%

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

   9. Taylor expanded in x around 0 85.4%

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

       1. associate-*r* 82.4%

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

       2. associate-/l/ 82.3%

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

       3. associate-/l/ 85.5%

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

   11. Simplified 85.5%

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

   if 4.00000000000000022e-32 < x

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l* 63.1%

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

      3. associate-/r* 63.2%

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

      4. associate-/l/ 62.3%

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

      5. associate-/l/ 60.6%

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

      6. cos-neg 60.6%

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

      7. *-commutative 60.6%

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

      8. distribute-rgt-neg-in 60.6%

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

      9. metadata-eval 60.6%

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

      10. *-commutative 60.6%

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

      11. associate-*l* 57.9%

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

      12. unpow2 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in x around inf 60.2%

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

      1. *-commutative 60.2%

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

      2. *-commutative 60.2%

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

      3. associate-*r* 58.7%

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

      4. unpow2 58.7%

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

      5. unpow2 58.7%

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

      6. swap-sqr 72.7%

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

      7. unpow2 72.7%

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

      8. swap-sqr 98.2%

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

      9. unpow2 98.2%

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

      10. associate-*l* 99.5%

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

      11. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. unpow2 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 98.2%

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

      4. associate-*r* 91.6%

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 2.7× speedup

Math

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

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
 (/ (cos (* x_m -2.0)) (* (* x_m s_m) (* c_m (* c_m (* x_m s_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 * (c_m * (x_m * s_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 * (c_m * (x_m * s_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 * (c_m * (x_m * s_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 * (c_m * (x_m * s_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(cos(Float64(x_m * -2.0)) / Float64(Float64(x_m * s_m) * Float64(c_m * Float64(c_m * Float64(x_m * s_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 * (c_m * (x_m * s_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[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.6%

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

   1. *-commutative 61.6%

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

   2. associate-*l* 64.4%

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

   3. associate-/r* 64.4%

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

   4. associate-/l/ 63.8%

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

   5. associate-/l/ 61.9%

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

   6. cos-neg 61.9%

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

   7. *-commutative 61.9%

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

   8. distribute-rgt-neg-in 61.9%

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

   9. metadata-eval 61.9%

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

   10. *-commutative 61.9%

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

   11. associate-*l* 57.0%

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

   12. unpow2 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in x around inf 58.0%

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

   1. *-commutative 58.0%

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

   2. *-commutative 58.0%

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

   3. associate-*r* 57.6%

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

   4. unpow2 57.6%

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

   5. unpow2 57.6%

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

   6. swap-sqr 75.5%

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

   7. unpow2 75.5%

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

   8. swap-sqr 96.6%

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

   9. unpow2 96.6%

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

   10. associate-*l* 98.4%

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

   11. *-commutative 98.4%

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

7. Simplified 98.4%

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

   1. unpow2 98.4%

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

   2. associate-*r* 94.1%

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

9. Applied egg-rr 94.1%

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

10. Final simplification 94.1%

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

Alternative 5: 97.1% accurate, 2.7× speedup

Math

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

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)) (* c_m (* x_m s_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)) / (c_m * (x_m * s_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)) / (c_m * (x_m * s_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)) / (c_m * (x_m * s_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)) / (c_m * (x_m * s_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)) / c_m) / Float64(x_m * s_m)) / Float64(c_m * Float64(x_m * s_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)) / (c_m * (x_m * s_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] / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.6%

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

   1. add-cbrt-cube 58.1%

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

   2. pow3 58.1%

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

   3. *-commutative 58.1%

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

   4. associate-*r* 55.0%

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

   5. unpow2 55.0%

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

   6. associate-*l* 54.5%

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

   7. pow-prod-down 64.7%

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

   8. pow-prod-down 72.8%

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

4. Applied egg-rr 72.8%

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

   1. rem-cbrt-cube 96.6%

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

   2. associate-*l* 98.4%

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

   3. *-commutative 98.4%

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

   4. unpow2 98.4%

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

   5. associate-/r* 98.4%

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

   6. *-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

   1. *-rgt-identity 98.4%

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

   2. times-frac 98.4%

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

   3. *-commutative 98.4%

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

8. Applied egg-rr 98.4%

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

   1. un-div-inv 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 6: 76.2% accurate, 24.1× speedup

Math

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

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 (* (* c_m s_m) (* x_m (* c_m (* x_m s_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 * s_m) * (x_m * (c_m * (x_m * s_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 * s_m) * (x_m * (c_m * (x_m * s_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 * s_m) * (x_m * (c_m * (x_m * s_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 * s_m) * (x_m * (c_m * (x_m * s_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(c_m * s_m) * Float64(x_m * Float64(c_m * Float64(x_m * s_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 * s_m) * (x_m * (c_m * (x_m * s_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[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.6%

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

3. Taylor expanded in x around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. associate-*r* 52.4%

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

   3. unpow2 52.4%

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

   4. unpow2 52.4%

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

   5. swap-sqr 66.2%

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

   6. unpow2 66.2%

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

   7. swap-sqr 79.3%

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

   8. unpow2 79.3%

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

   9. associate-*l* 80.4%

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

   10. *-commutative 80.4%

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

5. Simplified 80.4%

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

   1. unpow2 80.4%

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

   2. associate-*r* 78.1%

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

   3. associate-*l* 75.6%

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

7. Applied egg-rr 75.6%

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

8. Final simplification 75.6%

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

Alternative 7: 79.5% accurate, 24.1× speedup

Math

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

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 (* c_m (* x_m s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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(c_m * Float64(x_m * s_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 = c_m * (x_m * s_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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.6%

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

3. Taylor expanded in x around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. associate-*r* 52.4%

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

   3. unpow2 52.4%

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

   4. unpow2 52.4%

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

   5. swap-sqr 66.2%

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

   6. unpow2 66.2%

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

   7. swap-sqr 79.3%

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

   8. unpow2 79.3%

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

   9. associate-*l* 80.4%

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

   10. *-commutative 80.4%

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

5. Simplified 80.4%

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

   1. unpow2 80.4%

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

7. Applied egg-rr 80.4%

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

8. Final simplification 80.4%

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

Alternative 8: 79.6% accurate, 24.1× speedup

Math

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

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 (* c_m (* x_m s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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 = c_m * (x_m * s_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(c_m * Float64(x_m * s_m))
	return Float64(Float64(1.0 / 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 = c_m * (x_m * s_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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 61.6%

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

   1. add-cbrt-cube 58.1%

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

   2. pow3 58.1%

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

   3. *-commutative 58.1%

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

   4. associate-*r* 55.0%

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

   5. unpow2 55.0%

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

   6. associate-*l* 54.5%

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

   7. pow-prod-down 64.7%

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

   8. pow-prod-down 72.8%

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

4. Applied egg-rr 72.8%

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

   1. rem-cbrt-cube 96.6%

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

   2. associate-*l* 98.4%

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

   3. *-commutative 98.4%

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

   4. unpow2 98.4%

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

   5. associate-/r* 98.4%

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

   6. *-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in x around 0 80.4%

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

8. Final simplification 80.4%

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

Alternative 9: 79.7% accurate, 24.1× speedup

Math

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

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 x_m) s_m) c_m) (* c_m (* x_m s_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) / s_m) / c_m) / (c_m * (x_m * s_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) / s_m) / c_m) / (c_m * (x_m * s_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) / s_m) / c_m) / (c_m * (x_m * s_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) / s_m) / c_m) / (c_m * (x_m * s_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 / x_m) / s_m) / c_m) / Float64(c_m * Float64(x_m * s_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) / s_m) / c_m) / (c_m * (x_m * s_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 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.6%

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

   1. add-cbrt-cube 58.1%

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

   2. pow3 58.1%

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

   3. *-commutative 58.1%

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

   4. associate-*r* 55.0%

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

   5. unpow2 55.0%

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

   6. associate-*l* 54.5%

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

   7. pow-prod-down 64.7%

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

   8. pow-prod-down 72.8%

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

4. Applied egg-rr 72.8%

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

   1. rem-cbrt-cube 96.6%

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

   2. associate-*l* 98.4%

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

   3. *-commutative 98.4%

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

   4. unpow2 98.4%

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

   5. associate-/r* 98.4%

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

   6. *-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

   1. *-rgt-identity 98.4%

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

   2. times-frac 98.4%

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

   3. *-commutative 98.4%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in x around 0 80.4%

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

    1. associate-*r* 78.1%

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

    2. associate-/l/ 78.1%

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

    3. associate-/l/ 80.5%

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

11. Simplified 80.5%

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

12. Final simplification 80.5%

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

Reproduce

herbie shell --seed 2024040 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))