mixedcos

Percentage Accurate: 66.4% → 99.5%

Time: 13.2s

Alternatives: 9

Speedup: 24.1×

• 30.9% 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: 66.4% 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, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{s\_m \cdot \left(x\_m \cdot c\_m\right)}\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x\_m \cdot s\_m}}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

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 (/ 1.0 (* s_m (* x_m c_m)))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 9e+36)
     (/ (/ (/ t_1 (* x_m s_m)) c_m) (* (* x_m s_m) c_m))
     (* t_1 (* 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 = 1.0 / (s_m * (x_m * c_m));
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 9e+36) {
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = t_1 * (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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (s_m * (x_m * c_m))
    t_1 = cos((x_m * 2.0d0))
    if (x_m <= 9d+36) then
        tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
    else
        tmp = t_1 * (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 = 1.0 / (s_m * (x_m * c_m));
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 9e+36) {
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = t_1 * (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 = 1.0 / (s_m * (x_m * c_m))
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 9e+36:
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
	else:
		tmp = t_1 * (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(1.0 / Float64(s_m * Float64(x_m * c_m)))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 9e+36)
		tmp = Float64(Float64(Float64(t_1 / Float64(x_m * s_m)) / c_m) / Float64(Float64(x_m * s_m) * c_m));
	else
		tmp = Float64(t_1 * Float64(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 = 1.0 / (s_m * (x_m * c_m));
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 9e+36)
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	else
		tmp = t_1 * (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[(1.0 / N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 9e+36], N[(N[(N[(t$95$1 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.99999999999999994e36

   1. Initial program 67.8%

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

      1. *-un-lft-identity 67.8%

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

      2. add-sqr-sqrt 67.8%

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

      3. times-frac 67.8%

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

      4. sqrt-prod 67.8%

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

      5. sqrt-pow1 44.6%

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

      6. metadata-eval 44.6%

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

      7. pow1 44.6%

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

      8. *-commutative 44.6%

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

      9. associate-*r* 40.4%

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

      10. unpow2 40.4%

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

      11. pow-prod-down 44.6%

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

      12. sqrt-pow1 44.5%

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

      13. metadata-eval 44.5%

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

      14. pow1 44.5%

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

      15. *-commutative 44.5%

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

   4. Applied egg-rr 97.4%

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

      1. associate-*l/ 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-commutative 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. *-un-lft-identity 97.5%

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

      2. times-frac 97.0%

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

   8. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. *-lft-identity 97.0%

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

      3. *-commutative 97.0%

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

   10. Simplified 97.0%

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

   if 8.99999999999999994e36 < x

   1. Initial program 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. add-sqr-sqrt 62.0%

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

      3. times-frac 61.9%

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

      4. sqrt-prod 61.9%

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

      5. sqrt-pow1 49.8%

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

      6. metadata-eval 49.8%

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

      7. pow1 49.8%

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

      8. *-commutative 49.8%

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

      9. associate-*r* 47.9%

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

      10. unpow2 47.9%

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

      11. pow-prod-down 49.8%

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

      12. sqrt-pow1 49.1%

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

      13. metadata-eval 49.1%

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

      14. pow1 49.1%

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

      15. *-commutative 49.1%

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

   4. Applied egg-rr 96.3%

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

      1. *-commutative 96.3%

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

      2. clear-num 96.3%

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

      3. associate-/r* 96.4%

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

      4. frac-times 93.1%

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

      5. *-un-lft-identity 93.1%

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

      6. *-commutative 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-*r/ 93.1%

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

      3. associate-/r/ 93.2%

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

      4. associate-/r* 93.2%

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

      5. associate-*l* 96.4%

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

      6. pow2 96.4%

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

      7. pow-flip 96.4%

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

      8. associate-*r* 96.5%

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

      9. *-commutative 96.5%

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

      10. associate-*l* 96.2%

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

      11. metadata-eval 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. metadata-eval 96.2%

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

      2. pow-prod-up 96.2%

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

      3. unpow-1 96.2%

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

      4. *-commutative 96.2%

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

      5. associate-*r* 93.1%

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

      6. *-commutative 93.1%

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

      7. associate-*l* 93.2%

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

      8. unpow-1 93.2%

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

      9. *-commutative 93.2%

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

      10. associate-*r* 93.3%

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

      11. *-commutative 93.3%

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

      12. associate-*l* 96.6%

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

   10. Applied egg-rr 96.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot s}}{c}}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(x \cdot 2\right) \cdot \left(\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x\_m \cdot s\_m}}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (* s_m (* x_m c_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 3.3e+35)
     (/ (/ (/ t_1 (* x_m s_m)) c_m) (* (* x_m s_m) c_m))
     (* t_1 (/ (/ 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 = s_m * (x_m * c_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 3.3e+35) {
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = t_1 * ((1.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = cos((x_m * 2.0d0))
    if (x_m <= 3.3d+35) then
        tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
    else
        tmp = t_1 * ((1.0d0 / 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 t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 3.3e+35) {
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = t_1 * ((1.0 / 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)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 3.3e+35:
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
	else:
		tmp = t_1 * ((1.0 / 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))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 3.3e+35)
		tmp = Float64(Float64(Float64(t_1 / Float64(x_m * s_m)) / c_m) / Float64(Float64(x_m * s_m) * c_m));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 / 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);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 3.3e+35)
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	else
		tmp = t_1 * ((1.0 / 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]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 3.3e+35], N[(N[(N[(t$95$1 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3000000000000002e35

   1. Initial program 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. times-frac 67.6%

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

      4. sqrt-prod 67.6%

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

      5. sqrt-pow1 44.4%

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

      6. metadata-eval 44.4%

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

      7. pow1 44.4%

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

      8. *-commutative 44.4%

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

      9. associate-*r* 40.1%

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

      10. unpow2 40.1%

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

      11. pow-prod-down 44.4%

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

      12. sqrt-pow1 44.7%

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

      13. metadata-eval 44.7%

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

      14. pow1 44.7%

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

      15. *-commutative 44.7%

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

   4. Applied egg-rr 97.4%

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

      1. associate-*l/ 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-commutative 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. *-un-lft-identity 97.5%

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

      2. times-frac 97.0%

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

   8. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. *-lft-identity 97.0%

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

      3. *-commutative 97.0%

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

   10. Simplified 97.0%

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

   if 3.3000000000000002e35 < x

   1. Initial program 62.6%

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

      1. *-un-lft-identity 62.6%

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

      2. add-sqr-sqrt 62.6%

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

      3. times-frac 62.5%

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

      4. sqrt-prod 62.5%

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

      5. sqrt-pow1 50.6%

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

      6. metadata-eval 50.6%

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

      7. pow1 50.6%

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

      8. *-commutative 50.6%

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

      9. associate-*r* 48.7%

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

      10. unpow2 48.7%

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

      11. pow-prod-down 50.6%

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

      12. sqrt-pow1 48.3%

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

      13. metadata-eval 48.3%

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

      14. pow1 48.3%

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

      15. *-commutative 48.3%

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

   4. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

      2. clear-num 96.4%

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

      3. associate-/r* 96.4%

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

      4. frac-times 93.2%

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

      5. *-un-lft-identity 93.2%

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

      6. *-commutative 93.2%

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

   6. Applied egg-rr 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*r/ 93.2%

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

      3. associate-/r/ 93.3%

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

      4. associate-/r* 93.3%

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

      5. associate-*l* 96.5%

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

      6. pow2 96.5%

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

      7. pow-flip 96.5%

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

      8. associate-*r* 96.5%

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

      9. *-commutative 96.5%

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

      10. associate-*l* 96.2%

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

      11. metadata-eval 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. sqr-pow 96.3%

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

      2. pow2 96.3%

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

      3. metadata-eval 96.3%

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

      4. unpow-1 96.3%

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

      5. *-commutative 96.3%

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

      6. associate-*r* 96.5%

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

      7. *-commutative 96.5%

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

      8. metadata-eval 96.5%

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

      9. *-commutative 96.5%

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

      10. frac-times 96.3%

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

      11. associate-/l/ 96.3%

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

      12. div-inv 96.4%

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

      13. pow2 96.4%

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

      14. clear-num 96.4%

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

      15. un-div-inv 96.3%

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

      16. div-inv 96.2%

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

      17. clear-num 96.4%

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

      18. div-inv 96.3%

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

      19. clear-num 96.4%

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

      20. /-rgt-identity 96.4%

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

   10. Applied egg-rr 96.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot s}}{c}}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(x \cdot 2\right) \cdot \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x\_m \cdot s\_m}}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (* s_m (* x_m c_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 1e+29)
     (/ (/ (/ t_1 (* x_m s_m)) c_m) (* (* x_m s_m) c_m))
     (/ (/ t_1 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 t_1 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1e+29) {
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (t_1 / 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) :: t_1
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = cos((x_m * 2.0d0))
    if (x_m <= 1d+29) then
        tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
    else
        tmp = (t_1 / 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 t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 1e+29) {
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	} else {
		tmp = (t_1 / 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)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 1e+29:
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m)
	else:
		tmp = (t_1 / 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))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 1e+29)
		tmp = Float64(Float64(Float64(t_1 / Float64(x_m * s_m)) / c_m) / Float64(Float64(x_m * s_m) * c_m));
	else
		tmp = Float64(Float64(t_1 / 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);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 1e+29)
		tmp = ((t_1 / (x_m * s_m)) / c_m) / ((x_m * s_m) * c_m);
	else
		tmp = (t_1 / 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]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 1e+29], N[(N[(N[(t$95$1 / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999914e28

   1. Initial program 68.0%

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

      1. *-un-lft-identity 68.0%

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

      2. add-sqr-sqrt 68.0%

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

      3. times-frac 68.0%

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

      4. sqrt-prod 68.0%

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

      5. sqrt-pow1 44.6%

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

      6. metadata-eval 44.6%

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

      7. pow1 44.6%

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

      8. *-commutative 44.6%

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

      9. associate-*r* 40.3%

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

      10. unpow2 40.3%

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

      11. pow-prod-down 44.6%

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

      12. sqrt-pow1 44.9%

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

      13. metadata-eval 44.9%

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

      14. pow1 44.9%

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

      15. *-commutative 44.9%

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

   4. Applied egg-rr 97.4%

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

      1. associate-*l/ 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-commutative 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. *-un-lft-identity 97.5%

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

      2. times-frac 97.0%

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

   8. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. *-lft-identity 97.0%

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

      3. *-commutative 97.0%

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

   10. Simplified 97.0%

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

   if 9.99999999999999914e28 < x

   1. Initial program 61.5%

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

      1. *-un-lft-identity 61.5%

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

      2. add-sqr-sqrt 61.5%

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

      3. times-frac 61.5%

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

      4. sqrt-prod 61.5%

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

      5. sqrt-pow1 49.7%

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

      6. metadata-eval 49.7%

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

      7. pow1 49.7%

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

      8. *-commutative 49.7%

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

      9. associate-*r* 47.9%

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

      10. unpow2 47.9%

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

      11. pow-prod-down 49.8%

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

      12. sqrt-pow1 47.5%

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

      13. metadata-eval 47.5%

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

      14. pow1 47.5%

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

      15. *-commutative 47.5%

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

   4. Applied egg-rr 96.5%

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

      1. *-commutative 96.5%

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

      2. clear-num 96.4%

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

      3. associate-/r* 96.4%

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

      4. frac-times 91.7%

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

      5. *-un-lft-identity 91.7%

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

      6. *-commutative 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. *-commutative 91.7%

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

      2. associate-*r/ 91.8%

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

      3. associate-/r/ 91.8%

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

      4. associate-/r* 91.8%

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

      5. associate-*l* 96.5%

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

      6. pow2 96.5%

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

      7. pow-flip 96.5%

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

      8. associate-*r* 96.6%

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

      9. *-commutative 96.6%

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

      10. associate-*l* 96.3%

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

      11. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. *-commutative 96.3%

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

      2. sqr-pow 96.3%

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

      3. associate-*r* 96.2%

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

      4. metadata-eval 96.2%

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

      5. unpow-1 96.2%

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

      6. *-commutative 96.2%

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

      7. associate-*r* 93.3%

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

      8. *-commutative 93.3%

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

      9. div-inv 93.3%

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

      10. metadata-eval 93.3%

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

      11. unpow-1 93.3%

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

      12. *-commutative 93.3%

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

      13. associate-*r* 96.5%

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

      14. *-commutative 96.5%

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

      15. div-inv 96.5%

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

   10. Applied egg-rr 96.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot s}}{c}}{\left(x \cdot s\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot s\_m\right) \cdot c\_m} \cdot \frac{\frac{1}{c\_m}}{x\_m \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))))
   (if (<= x_m 2.4e-9)
     (* (/ 1.0 (* (* x_m s_m) c_m)) (/ (/ 1.0 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 <= 2.4e-9) {
		tmp = (1.0 / ((x_m * s_m) * c_m)) * ((1.0 / 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 <= 2.4d-9) then
        tmp = (1.0d0 / ((x_m * s_m) * c_m)) * ((1.0d0 / 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 <= 2.4e-9) {
		tmp = (1.0 / ((x_m * s_m) * c_m)) * ((1.0 / 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 <= 2.4e-9:
		tmp = (1.0 / ((x_m * s_m) * c_m)) * ((1.0 / 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 <= 2.4e-9)
		tmp = Float64(Float64(1.0 / Float64(Float64(x_m * s_m) * c_m)) * Float64(Float64(1.0 / 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 <= 2.4e-9)
		tmp = (1.0 / ((x_m * s_m) * c_m)) * ((1.0 / 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, 2.4e-9], N[(N[(1.0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] / 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 < 2.4e-9

   1. Initial program 68.4%

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

      1. *-un-lft-identity 68.4%

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

      2. add-sqr-sqrt 68.4%

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

      3. times-frac 68.3%

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

      4. sqrt-prod 68.3%

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

      5. sqrt-pow1 45.6%

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

      6. metadata-eval 45.6%

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

      7. pow1 45.6%

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

      8. *-commutative 45.6%

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

      9. associate-*r* 41.2%

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

      10. unpow2 41.2%

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

      11. pow-prod-down 45.6%

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

      12. sqrt-pow1 45.5%

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

      13. metadata-eval 45.5%

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

      14. pow1 45.5%

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

      15. *-commutative 45.5%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around 0 86.1%

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

      1. associate-/r* 86.1%

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

   7. Simplified 86.1%

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

   if 2.4e-9 < x

   1. Initial program 61.2%

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

      1. *-un-lft-identity 61.2%

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

      2. add-sqr-sqrt 61.2%

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

      3. times-frac 61.2%

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

      4. sqrt-prod 61.2%

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

      5. sqrt-pow1 46.2%

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

      6. metadata-eval 46.2%

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

      7. pow1 46.2%

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

      8. *-commutative 46.2%

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

      9. associate-*r* 44.6%

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

      10. unpow2 44.6%

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

      11. pow-prod-down 46.2%

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

      12. sqrt-pow1 45.7%

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

      13. metadata-eval 45.7%

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

      14. pow1 45.7%

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

      15. *-commutative 45.7%

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

   4. Applied egg-rr 96.8%

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

      1. *-commutative 96.8%

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

      2. clear-num 96.7%

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

      3. associate-/r* 96.7%

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

      4. frac-times 92.5%

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

      5. *-un-lft-identity 92.5%

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

      6. *-commutative 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-*r/ 92.6%

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

      3. associate-/r/ 92.7%

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

      4. associate-/r* 92.6%

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

      5. associate-*l* 96.8%

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

      6. pow2 96.8%

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

      7. pow-flip 96.8%

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

      8. associate-*r* 96.9%

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

      9. *-commutative 96.9%

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

      10. associate-*l* 96.6%

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

      11. metadata-eval 96.6%

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

   8. Applied egg-rr 96.6%

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

      1. *-commutative 96.6%

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

      2. sqr-pow 96.6%

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

      3. associate-*r* 96.5%

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

      4. metadata-eval 96.5%

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

      5. unpow-1 96.5%

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

      6. *-commutative 96.5%

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

      7. associate-*r* 93.9%

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

      8. *-commutative 93.9%

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

      9. div-inv 94.0%

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

      10. metadata-eval 94.0%

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

      11. unpow-1 94.0%

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

      12. *-commutative 94.0%

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

      13. associate-*r* 96.8%

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

      14. *-commutative 96.8%

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

      15. div-inv 96.9%

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

   10. Applied egg-rr 96.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot c} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(x\_m \cdot s\_m\right) \cdot c\_m\\ \frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.5%

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

   4. sqrt-prod 66.5%

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

   5. sqrt-pow1 45.8%

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

   6. metadata-eval 45.8%

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

   7. pow1 45.8%

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

   8. *-commutative 45.8%

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

   9. associate-*r* 42.1%

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

   10. unpow2 42.1%

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

   11. pow-prod-down 45.8%

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

   12. sqrt-pow1 45.5%

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

   13. metadata-eval 45.5%

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

   14. pow1 45.5%

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

   15. *-commutative 45.5%

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

4. Applied egg-rr 97.2%

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

   1. associate-*l/ 97.2%

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

   2. *-un-lft-identity 97.2%

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

   3. *-commutative 97.2%

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

6. Applied egg-rr 97.2%

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

7. Final simplification 97.2%

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

Alternative 6: 93.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.5%

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

   4. sqrt-prod 66.5%

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

   5. sqrt-pow1 45.8%

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

   6. metadata-eval 45.8%

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

   7. pow1 45.8%

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

   8. *-commutative 45.8%

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

   9. associate-*r* 42.1%

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

   10. unpow2 42.1%

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

   11. pow-prod-down 45.8%

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

   12. sqrt-pow1 45.5%

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

   13. metadata-eval 45.5%

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

   14. pow1 45.5%

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

   15. *-commutative 45.5%

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

4. Applied egg-rr 97.2%

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

   1. associate-/r* 97.2%

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

   2. frac-times 94.6%

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

   3. metadata-eval 94.6%

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

   4. times-frac 94.6%

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

   5. *-un-lft-identity 94.6%

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

   6. *-un-lft-identity 94.6%

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

   7. *-commutative 94.6%

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

6. Applied egg-rr 94.6%

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

7. Final simplification 94.6%

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

Alternative 7: 79.5% accurate, 20.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (* (/ 1.0 (* (* x_m s_m) c_m)) (/ (/ 1.0 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)) * ((1.0 / 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)) * ((1.0d0 / 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)) * ((1.0 / 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)) * ((1.0 / 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(1.0 / Float64(Float64(x_m * s_m) * c_m)) * Float64(Float64(1.0 / 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)) * ((1.0 / 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[(1.0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.5%

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

   4. sqrt-prod 66.5%

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

   5. sqrt-pow1 45.8%

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

   6. metadata-eval 45.8%

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

   7. pow1 45.8%

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

   8. *-commutative 45.8%

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

   9. associate-*r* 42.1%

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

   10. unpow2 42.1%

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

   11. pow-prod-down 45.8%

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

   12. sqrt-pow1 45.5%

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

   13. metadata-eval 45.5%

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

   14. pow1 45.5%

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

   15. *-commutative 45.5%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in x around 0 76.0%

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

   1. associate-/r* 76.1%

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

7. Simplified 76.1%

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

8. Final simplification 76.1%

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

Alternative 8: 79.5% accurate, 20.9× speedup

Math

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

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 (/ 1.0 (* (* x_m s_m) c_m)))) (* 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 = 1.0 / ((x_m * s_m) * c_m);
	return 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 = 1.0d0 / ((x_m * s_m) * c_m)
    code = 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 = 1.0 / ((x_m * s_m) * c_m);
	return 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 = 1.0 / ((x_m * s_m) * c_m)
	return 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(1.0 / Float64(Float64(x_m * s_m) * c_m))
	return 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 = 1.0 / ((x_m * s_m) * c_m);
	tmp = 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[(1.0 / N[(N[(x$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 66.5%

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

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.5%

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

   4. sqrt-prod 66.5%

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

   5. sqrt-pow1 45.8%

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

   6. metadata-eval 45.8%

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

   7. pow1 45.8%

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

   8. *-commutative 45.8%

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

   9. associate-*r* 42.1%

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

   10. unpow2 42.1%

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

   11. pow-prod-down 45.8%

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

   12. sqrt-pow1 45.5%

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

   13. metadata-eval 45.5%

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

   14. pow1 45.5%

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

   15. *-commutative 45.5%

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

4. Applied egg-rr 97.2%

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

5. Taylor expanded in x around 0 76.0%

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

6. Final simplification 76.0%

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

Alternative 9: 79.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

3. Taylor expanded in x around 0 53.2%

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

   1. associate-/r* 53.2%

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

   2. *-commutative 53.2%

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

   3. unpow2 53.2%

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

   4. unpow2 53.2%

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

   5. swap-sqr 62.6%

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

   6. unpow2 62.6%

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

   7. associate-/r* 62.6%

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

   8. unpow2 62.6%

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

   9. unpow2 62.6%

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

   10. swap-sqr 76.0%

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

   11. unpow2 76.0%

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

   12. *-commutative 76.0%

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

5. Simplified 76.0%

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

   1. *-commutative 76.0%

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

   2. pow2 76.0%

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

7. Applied egg-rr 76.0%

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

8. Final simplification 76.0%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))