mixedcos

Percentage Accurate: 66.3% → 98.6%

Time: 15.3s

Alternatives: 6

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double t_0 = c * (s_m * sqrt(x_m));
	return cos((2.0 * x_m)) / (t_0 * (x_m * t_0));
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c * (s_m * sqrt(x_m))
    code = cos((2.0d0 * x_m)) / (t_0 * (x_m * t_0))
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double t_0 = c * (s_m * Math.sqrt(x_m));
	return Math.cos((2.0 * x_m)) / (t_0 * (x_m * t_0));
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	t_0 = c * (s_m * math.sqrt(x_m))
	return math.cos((2.0 * x_m)) / (t_0 * (x_m * t_0))

Julia

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

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	t_0 = c * (s_m * sqrt(x_m));
	tmp = cos((2.0 * x_m)) / (t_0 * (x_m * t_0));
end

Wolfram

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

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in c around 0 66.4%

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

   1. unpow2 66.4%

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

   2. rem-square-sqrt 66.4%

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

   3. swap-sqr 72.1%

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

   4. unpow2 72.1%

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

   5. unpow2 72.1%

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

   6. unpow2 72.1%

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

   7. unswap-sqr 90.2%

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

   8. rem-sqrt-square 98.3%

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

5. Simplified 98.3%

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

   1. pow-prod-down 81.6%

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

   2. *-commutative 81.6%

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

   3. unpow2 81.6%

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

   4. add-sqr-sqrt 46.8%

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

   5. fabs-sqr 46.8%

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

   6. add-sqr-sqrt 57.1%

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

   7. *-commutative 57.1%

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

   8. associate-*l* 55.6%

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

   9. add-sqr-sqrt 37.4%

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

   10. fabs-sqr 37.4%

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

   11. add-sqr-sqrt 79.3%

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

   12. associate-*l* 74.1%

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

   13. unpow2 74.1%

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

   14. associate-*r* 75.2%

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

   15. *-commutative 75.2%

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

   16. add-sqr-sqrt 38.9%

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

   17. associate-*r* 39.0%

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

7. Applied egg-rr 49.4%

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

8. Final simplification 49.4%

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

Alternative 2: 95.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x_m \cdot s_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x_m\right)}{x_m \cdot \left(\left(c \cdot s_m\right) \cdot \left(s_m \cdot \left(x_m \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (if (<= x_m 2.1e-16)
   (pow (* c (* x_m s_m)) -2.0)
   (/ (cos (* 2.0 x_m)) (* x_m (* (* c s_m) (* s_m (* x_m c)))))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 2.1e-16) {
		tmp = pow((c * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((2.0 * x_m)) / (x_m * ((c * s_m) * (s_m * (x_m * c))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2.1d-16) then
        tmp = (c * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((2.0d0 * x_m)) / (x_m * ((c * s_m) * (s_m * (x_m * c))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 2.1e-16) {
		tmp = Math.pow((c * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((2.0 * x_m)) / (x_m * ((c * s_m) * (s_m * (x_m * c))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	tmp = 0
	if x_m <= 2.1e-16:
		tmp = math.pow((c * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((2.0 * x_m)) / (x_m * ((c * s_m) * (s_m * (x_m * c))))
	return tmp

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	tmp = 0.0
	if (x_m <= 2.1e-16)
		tmp = Float64(c * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(x_m * Float64(Float64(c * s_m) * Float64(s_m * Float64(x_m * c)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp_2 = code(x_m, c, s_m)
	tmp = 0.0;
	if (x_m <= 2.1e-16)
		tmp = (c * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((2.0 * x_m)) / (x_m * ((c * s_m) * (s_m * (x_m * c))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := If[LessEqual[x$95$m, 2.1e-16], N[Power[N[(c * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(N[(c * s$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1000000000000001e-16

   1. Initial program 71.9%

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

   3. Taylor expanded in x around 0 60.7%

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

      1. unpow2 60.7%

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

      2. rem-square-sqrt 60.7%

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

      3. swap-sqr 66.4%

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

      4. unpow2 66.4%

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

      5. unpow2 66.4%

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

      6. unpow2 66.4%

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

      7. unswap-sqr 81.0%

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

      8. rem-sqrt-square 84.5%

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

   5. Simplified 84.5%

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

      1. expm1-log1p-u 83.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)\right)} \]

      2. expm1-udef 77.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)} - 1} \]

      3. pow-flip 77.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot \left|s \cdot x\right|\right)}^{\left(-2\right)}}\right)} - 1 \]

      4. add-sqr-sqrt 49.2%

         \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \left|\color{blue}{\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}}\right|\right)}^{\left(-2\right)}\right)} - 1 \]

      5. fabs-sqr 49.2%

         \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

      6. add-sqr-sqrt 77.2%

         \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

      7. metadata-eval 77.2%

         \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \left(s \cdot x\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   7. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(c \cdot \left(s \cdot x\right)\right)}^{-2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 84.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(c \cdot \left(s \cdot x\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 84.7%

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

   9. Simplified 84.7%

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

   if 2.1000000000000001e-16 < x

   1. Initial program 79.5%

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

   3. Taylor expanded in c around 0 75.4%

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

      1. unpow2 75.4%

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

      2. rem-square-sqrt 75.4%

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

      3. swap-sqr 80.9%

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

      4. unpow2 80.9%

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

      5. unpow2 80.9%

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

      6. unpow2 80.9%

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

      7. unswap-sqr 88.3%

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

      8. rem-sqrt-square 98.5%

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

   5. Simplified 98.5%

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

      1. /-rgt-identity 98.5%

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

      2. unpow2 98.5%

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

      3. associate-/l* 98.5%

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

      4. add-sqr-sqrt 45.8%

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

      5. fabs-sqr 45.8%

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

      6. add-sqr-sqrt 73.6%

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

      7. add-sqr-sqrt 45.8%

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

      8. fabs-sqr 45.8%

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

      9. add-sqr-sqrt 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. associate-/l* 98.5%

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

      2. associate-*l* 96.0%

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

      3. /-rgt-identity 96.0%

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

      4. associate-*l* 98.5%

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

      5. associate-*r* 98.5%

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

      6. *-commutative 98.5%

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

      7. associate-*l* 96.9%

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

      8. associate-*r* 98.1%

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

      9. *-commutative 98.1%

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

      10. associate-*l* 96.9%

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

   9. Applied egg-rr 96.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double t_0 = c * (x_m * s_m);
	return (cos((2.0 * x_m)) / t_0) / t_0;
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c * (x_m * s_m)
    code = (cos((2.0d0 * x_m)) / t_0) / t_0
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double t_0 = c * (x_m * s_m);
	return (Math.cos((2.0 * x_m)) / t_0) / t_0;
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	t_0 = c * (x_m * s_m)
	return (math.cos((2.0 * x_m)) / t_0) / t_0

Julia

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

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	t_0 = c * (x_m * s_m);
	tmp = (cos((2.0 * x_m)) / t_0) / t_0;
end

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. cos-2 74.1%

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

   2. div-sub 42.1%

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

   3. pow2 42.1%

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

   4. *-commutative 42.1%

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

   5. associate-*l* 36.4%

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

   6. pow2 36.4%

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

   7. pow2 36.4%

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

   8. *-commutative 36.4%

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

   9. associate-*l* 36.0%

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

   10. pow2 36.0%

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

4. Applied egg-rr 36.0%

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

   1. unpow2 36.0%

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

   2. rem-square-sqrt 36.0%

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

   3. swap-sqr 36.8%

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

   4. unpow2 36.8%

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

   5. unpow2 36.8%

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

   6. unpow2 36.8%

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

   7. unswap-sqr 37.2%

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

   8. rem-sqrt-square 37.4%

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

   9. unpow2 37.4%

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

   10. rem-square-sqrt 37.4%

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

   11. swap-sqr 41.0%

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

   12. unpow2 41.0%

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

6. Simplified 62.2%

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

8. Applied egg-rr 98.6%

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

9. Final simplification 98.6%

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

Alternative 4: 78.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in x around 0 61.2%

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

   1. unpow2 61.2%

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

   2. rem-square-sqrt 61.2%

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

   3. swap-sqr 65.8%

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

   4. unpow2 65.8%

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

   5. unpow2 65.8%

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

   6. unpow2 65.8%

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

   7. unswap-sqr 76.5%

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

   8. rem-sqrt-square 79.4%

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

5. Simplified 79.4%

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

   1. expm1-log1p-u 78.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)\right)} \]

   2. expm1-udef 73.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)} - 1} \]

   3. pow-flip 73.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot \left|s \cdot x\right|\right)}^{\left(-2\right)}}\right)} - 1 \]

   4. add-sqr-sqrt 44.8%

      \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \left|\color{blue}{\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}}\right|\right)}^{\left(-2\right)}\right)} - 1 \]

   5. fabs-sqr 44.8%

      \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

   6. add-sqr-sqrt 73.7%

      \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

   7. metadata-eval 73.7%

      \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \left(s \cdot x\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

7. Applied egg-rr 73.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(c \cdot \left(s \cdot x\right)\right)}^{-2}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 79.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(c \cdot \left(s \cdot x\right)\right)}^{-2}\right)\right)} \]

   2. expm1-log1p 79.5%

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

9. Simplified 79.5%

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

10. Final simplification 79.5%

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

Alternative 5: 78.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := Block[{t$95$0 = N[(c * 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 74.1%

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

3. Taylor expanded in x around 0 61.2%

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

   1. unpow2 61.2%

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

   2. rem-square-sqrt 61.2%

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

   3. swap-sqr 65.8%

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

   4. unpow2 65.8%

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

   5. unpow2 65.8%

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

   6. unpow2 65.8%

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

   7. unswap-sqr 76.5%

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

   8. rem-sqrt-square 79.4%

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

5. Simplified 79.4%

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

   1. unpow2 79.4%

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

   2. add-sqr-sqrt 46.8%

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

   3. fabs-sqr 46.8%

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

   4. add-sqr-sqrt 59.6%

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

   5. add-sqr-sqrt 37.5%

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

   6. fabs-sqr 37.5%

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

   7. add-sqr-sqrt 79.4%

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

7. Applied egg-rr 79.4%

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

8. Final simplification 79.4%

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

Alternative 6: 78.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := Block[{t$95$0 = N[(c * 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 74.1%

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

   1. cos-2 74.1%

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

   2. div-sub 42.1%

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

   3. pow2 42.1%

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

   4. *-commutative 42.1%

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

   5. associate-*l* 36.4%

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

   6. pow2 36.4%

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

   7. pow2 36.4%

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

   8. *-commutative 36.4%

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

   9. associate-*l* 36.0%

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

   10. pow2 36.0%

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

4. Applied egg-rr 36.0%

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

   1. unpow2 36.0%

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

   2. rem-square-sqrt 36.0%

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

   3. swap-sqr 36.8%

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

   4. unpow2 36.8%

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

   5. unpow2 36.8%

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

   6. unpow2 36.8%

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

   7. unswap-sqr 37.2%

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

   8. rem-sqrt-square 37.4%

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

   9. unpow2 37.4%

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

   10. rem-square-sqrt 37.4%

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

   11. swap-sqr 41.0%

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

   12. unpow2 41.0%

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

6. Simplified 62.2%

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

8. Applied egg-rr 98.6%

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

9. Taylor expanded in x around 0 79.5%

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

10. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024014 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))