mixedcos

Percentage Accurate: 67.2% → 98.8%

Time: 16.5s

Alternatives: 12

Speedup: 24.1×

• 32.6% 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: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.24:\\ \;\;\;\;{\left(c_m \cdot \left(x_m \cdot s_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{s_m \cdot \left(\left(x_m \cdot c_m\right) \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 0.24)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (cos (* x_m -2.0)) (* s_m (* (* x_m c_m) (* s_m (* x_m c_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 0.24) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 0.24d0) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * (-2.0d0))) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 0.24) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 0.24:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 0.24)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * c_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 0.24)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((x_m * -2.0)) / (s_m * ((x_m * c_m) * (s_m * (x_m * c_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 0.24], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.23999999999999999

   1. Initial program 73.6%

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

      1. associate-/r* 73.6%

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

      2. associate-*l* 73.6%

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

      3. unpow2 73.6%

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

      4. sqr-neg 73.6%

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

      5. unpow2 73.6%

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

      6. *-commutative 73.6%

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

      7. *-commutative 73.6%

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

      8. associate-/r* 73.6%

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

      9. cos-neg 73.6%

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

      10. *-commutative 73.6%

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

      11. distribute-rgt-neg-in 73.6%

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

      12. metadata-eval 73.6%

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

      13. associate-*r* 75.1%

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

      14. *-commutative 75.1%

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

      15. unpow2 75.1%

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

      16. sqr-neg 75.1%

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

      17. associate-*l* 81.2%

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

      18. associate-*r* 81.9%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in x around inf 69.7%

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

      1. associate-/r* 69.7%

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

      2. *-commutative 69.7%

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

      3. unpow2 69.7%

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

      4. unpow2 69.7%

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

      5. swap-sqr 82.6%

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

      6. unpow2 82.6%

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

      7. associate-/r* 82.6%

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

      8. *-commutative 82.6%

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

      9. unpow2 82.6%

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

      10. unpow2 82.6%

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

      11. swap-sqr 97.6%

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

      12. unpow2 97.6%

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

      13. associate-*r* 97.6%

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

      14. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around 0 64.5%

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

      1. associate-*r* 65.1%

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

      2. *-commutative 65.1%

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

      3. unpow2 65.1%

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

      4. unpow2 65.1%

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

      5. swap-sqr 76.5%

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

      6. unpow2 76.5%

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

      7. swap-sqr 88.7%

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

      8. associate-*r* 87.2%

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

      9. associate-*r* 87.2%

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

      10. unpow2 87.2%

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

      11. /-rgt-identity 87.2%

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

      12. unpow2 87.2%

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

      13. associate-/l* 87.2%

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

      14. associate-/l* 87.1%

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

      15. associate-*l/ 87.1%

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

      16. unpow-1 87.1%

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

      17. unpow-1 87.1%

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

      18. pow-sqr 87.2%

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

      19. metadata-eval 87.2%

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

   10. Simplified 88.0%

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

   if 0.23999999999999999 < x

   1. Initial program 66.7%

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

      1. associate-/r* 64.6%

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

      2. associate-*l* 64.6%

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

      3. unpow2 64.6%

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

      4. sqr-neg 64.6%

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

      5. unpow2 64.6%

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

      6. *-commutative 64.6%

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

      7. *-commutative 64.6%

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

      8. associate-/r* 66.7%

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

      9. cos-neg 66.7%

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

      10. *-commutative 66.7%

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

      11. distribute-rgt-neg-in 66.7%

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

      12. metadata-eval 66.7%

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

      13. associate-*r* 66.8%

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

      14. *-commutative 66.8%

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

      15. unpow2 66.8%

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

      16. sqr-neg 66.8%

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

      17. associate-*l* 72.8%

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

      18. associate-*r* 75.2%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in x around inf 58.6%

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

      1. associate-/r* 56.5%

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

      2. *-commutative 56.5%

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

      3. unpow2 56.5%

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

      4. unpow2 56.5%

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

      5. swap-sqr 76.6%

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

      6. unpow2 76.6%

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

      7. associate-/r* 78.6%

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

      8. *-commutative 78.6%

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

      9. unpow2 78.6%

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

      10. unpow2 78.6%

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

      11. swap-sqr 95.8%

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

      12. unpow2 95.8%

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

      13. associate-*r* 97.3%

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

      14. *-commutative 97.3%

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

   7. Simplified 97.3%

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

      1. unpow2 97.3%

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

      2. associate-*r* 93.2%

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

      3. *-commutative 93.2%

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

      4. associate-*r* 94.5%

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

      5. *-commutative 94.5%

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

      6. associate-*r* 91.7%

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

      7. associate-*r* 90.3%

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

      8. *-commutative 90.3%

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

      9. associate-*r* 94.6%

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.24:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{1}{c_m}}{\left(x_m \cdot \left(s_m \cdot c_m\right)\right) \cdot \left(x_m \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{\left(s_m \cdot c_m\right) \cdot \left(x_m \cdot \left(c_m \cdot \left(x_m \cdot s_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 1.25e-204)
   (/ (/ 1.0 c_m) (* (* x_m (* s_m c_m)) (* x_m s_m)))
   (/ (cos (* x_m -2.0)) (* (* s_m c_m) (* x_m (* c_m (* x_m s_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.25e-204) {
		tmp = (1.0 / c_m) / ((x_m * (s_m * c_m)) * (x_m * s_m));
	} else {
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (c_m * (x_m * s_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 1.25d-204) then
        tmp = (1.0d0 / c_m) / ((x_m * (s_m * c_m)) * (x_m * s_m))
    else
        tmp = cos((x_m * (-2.0d0))) / ((s_m * c_m) * (x_m * (c_m * (x_m * s_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.25e-204) {
		tmp = (1.0 / c_m) / ((x_m * (s_m * c_m)) * (x_m * s_m));
	} else {
		tmp = Math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (c_m * (x_m * s_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 1.25e-204:
		tmp = (1.0 / c_m) / ((x_m * (s_m * c_m)) * (x_m * s_m))
	else:
		tmp = math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (c_m * (x_m * s_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 1.25e-204)
		tmp = Float64(Float64(1.0 / c_m) / Float64(Float64(x_m * Float64(s_m * c_m)) * Float64(x_m * s_m)));
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(s_m * c_m) * Float64(x_m * Float64(c_m * Float64(x_m * s_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 1.25e-204)
		tmp = (1.0 / c_m) / ((x_m * (s_m * c_m)) * (x_m * s_m));
	else
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (c_m * (x_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 1.25e-204], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e-204

   1. Initial program 75.1%

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

      1. associate-/r* 75.1%

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

      2. associate-*l* 75.1%

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

      3. unpow2 75.1%

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

      4. sqr-neg 75.1%

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

      5. unpow2 75.1%

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

      6. *-commutative 75.1%

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

      7. *-commutative 75.1%

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

      8. associate-/r* 75.1%

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

      9. cos-neg 75.1%

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

      10. *-commutative 75.1%

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

      11. distribute-rgt-neg-in 75.1%

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

      12. metadata-eval 75.1%

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

      13. associate-*r* 77.0%

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

      14. *-commutative 77.0%

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

      15. unpow2 77.0%

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

      16. sqr-neg 77.0%

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

      17. associate-*l* 82.8%

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

      18. associate-*r* 84.2%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around inf 71.6%

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

      1. associate-/r* 71.6%

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

      2. *-commutative 71.6%

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

      3. unpow2 71.6%

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

      4. unpow2 71.6%

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

      5. swap-sqr 84.1%

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

      6. unpow2 84.1%

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

      7. associate-/r* 84.1%

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

      8. *-commutative 84.1%

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

      9. unpow2 84.1%

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

      10. unpow2 84.1%

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

      11. swap-sqr 97.7%

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

      12. unpow2 97.7%

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

      13. associate-*r* 97.1%

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

      14. *-commutative 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around 0 65.0%

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

      1. associate-*r* 65.8%

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

      2. *-commutative 65.8%

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

      3. unpow2 65.8%

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

      4. unpow2 65.8%

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

      5. swap-sqr 75.0%

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

      6. unpow2 75.0%

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

      7. swap-sqr 85.8%

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

      8. associate-*r* 83.9%

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

      9. associate-*r* 83.9%

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

      10. unpow2 83.9%

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

      11. /-rgt-identity 83.9%

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

      12. unpow2 83.9%

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

      13. associate-/l* 83.9%

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

      14. associate-/l* 83.8%

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

      15. associate-*l/ 83.8%

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

      16. unpow-1 83.8%

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

      17. unpow-1 83.8%

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

      18. pow-sqr 83.9%

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

      19. metadata-eval 83.9%

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

   10. Simplified 85.6%

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

       1. metadata-eval 85.6%

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

       2. pow-sqr 85.5%

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

       3. inv-pow 85.5%

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

       4. inv-pow 85.5%

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

       5. associate-/r* 85.5%

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

       6. frac-times 84.2%

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

       7. *-un-lft-identity 84.2%

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

       8. *-commutative 84.2%

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

       9. associate-*r* 82.2%

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

       10. *-commutative 82.2%

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

       11. associate-*l* 84.2%

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

       12. *-commutative 84.2%

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

   12. Applied egg-rr 84.2%

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

   if 1.25e-204 < x

   1. Initial program 67.1%

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

      1. associate-/r* 65.8%

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

      2. associate-*l* 65.8%

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

      3. unpow2 65.8%

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

      4. sqr-neg 65.8%

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

      5. unpow2 65.8%

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

      6. *-commutative 65.8%

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

      7. *-commutative 65.8%

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

      8. associate-/r* 67.1%

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

      9. cos-neg 67.1%

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

      10. *-commutative 67.1%

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

      11. distribute-rgt-neg-in 67.1%

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

      12. metadata-eval 67.1%

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

      13. associate-*r* 67.3%

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

      14. *-commutative 67.3%

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

      15. unpow2 67.3%

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

      16. sqr-neg 67.3%

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

      17. associate-*l* 73.7%

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

      18. associate-*r* 74.4%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in x around inf 60.1%

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

      1. associate-/r* 58.8%

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

      2. *-commutative 58.8%

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

      3. unpow2 58.8%

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

      4. unpow2 58.8%

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

      5. swap-sqr 76.8%

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

      6. unpow2 76.8%

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

      7. associate-/r* 78.1%

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

      8. *-commutative 78.1%

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

      9. unpow2 78.1%

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

      10. unpow2 78.1%

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

      11. swap-sqr 96.4%

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

      12. unpow2 96.4%

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

      13. associate-*r* 98.2%

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

      14. *-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. associate-*r* 96.4%

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

      2. *-commutative 96.4%

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

      3. associate-*r* 96.4%

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

      4. unpow2 96.4%

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

      5. associate-*r* 93.8%

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

      6. associate-*l* 93.8%

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

      7. *-commutative 93.8%

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

      8. associate-*r* 94.7%

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

      9. *-commutative 94.7%

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

      10. associate-*r* 93.9%

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

   9. Applied egg-rr 93.9%

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

   10. Taylor expanded in s around 0 93.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(s \cdot c\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x_m \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;{\left(c_m \cdot \left(x_m \cdot s_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x_m \cdot -2\right)}{\left(s_m \cdot c_m\right) \cdot \left(x_m \cdot \left(s_m \cdot \left(x_m \cdot c_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 8.5e-9)
   (pow (* c_m (* x_m s_m)) -2.0)
   (/ (cos (* x_m -2.0)) (* (* s_m c_m) (* x_m (* s_m (* x_m c_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 8.5e-9) {
		tmp = pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 8.5d-9) then
        tmp = (c_m * (x_m * s_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * (-2.0d0))) / ((s_m * c_m) * (x_m * (s_m * (x_m * c_m))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 8.5e-9) {
		tmp = Math.pow((c_m * (x_m * s_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 8.5e-9:
		tmp = math.pow((c_m * (x_m * s_m)), -2.0)
	else:
		tmp = math.cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (s_m * (x_m * c_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 8.5e-9)
		tmp = Float64(c_m * Float64(x_m * s_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(Float64(s_m * c_m) * Float64(x_m * Float64(s_m * Float64(x_m * c_m)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 8.5e-9)
		tmp = (c_m * (x_m * s_m)) ^ -2.0;
	else
		tmp = cos((x_m * -2.0)) / ((s_m * c_m) * (x_m * (s_m * (x_m * c_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 8.5e-9], N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.5e-9

   1. Initial program 73.4%

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

      1. associate-/r* 73.4%

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

      2. associate-*l* 73.4%

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

      3. unpow2 73.4%

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

      4. sqr-neg 73.4%

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

      5. unpow2 73.4%

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

      6. *-commutative 73.4%

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

      7. *-commutative 73.4%

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

      8. associate-/r* 73.4%

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

      9. cos-neg 73.4%

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

      10. *-commutative 73.4%

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

      11. distribute-rgt-neg-in 73.4%

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

      12. metadata-eval 73.4%

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

      13. associate-*r* 75.0%

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

      14. *-commutative 75.0%

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

      15. unpow2 75.0%

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

      16. sqr-neg 75.0%

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

      17. associate-*l* 81.1%

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

      18. associate-*r* 81.8%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around inf 69.5%

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

      1. associate-/r* 69.5%

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

      2. *-commutative 69.5%

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

      3. unpow2 69.5%

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

      4. unpow2 69.5%

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

      5. swap-sqr 82.5%

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

      6. unpow2 82.5%

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

      7. associate-/r* 82.5%

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

      8. *-commutative 82.5%

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

      9. unpow2 82.5%

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

      10. unpow2 82.5%

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

      11. swap-sqr 97.6%

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

      12. unpow2 97.6%

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

      13. associate-*r* 97.6%

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

      14. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around 0 64.3%

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

      1. associate-*r* 64.9%

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

      2. *-commutative 64.9%

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

      3. unpow2 64.9%

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

      4. unpow2 64.9%

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

      5. swap-sqr 76.4%

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

      6. unpow2 76.4%

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

      7. swap-sqr 88.6%

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

      8. associate-*r* 87.1%

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

      9. associate-*r* 87.1%

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

      10. unpow2 87.1%

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

      11. /-rgt-identity 87.1%

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

      12. unpow2 87.1%

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

      13. associate-/l* 87.1%

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

      14. associate-/l* 87.0%

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

      15. associate-*l/ 87.0%

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

      16. unpow-1 87.0%

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

      17. unpow-1 87.0%

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

      18. pow-sqr 87.1%

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

      19. metadata-eval 87.1%

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

   10. Simplified 88.0%

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

   if 8.5e-9 < x

   1. Initial program 67.2%

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

      1. associate-/r* 65.1%

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

      2. associate-*l* 65.1%

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

      3. unpow2 65.1%

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

      4. sqr-neg 65.1%

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

      5. unpow2 65.1%

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

      6. *-commutative 65.1%

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

      7. *-commutative 65.1%

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

      8. associate-/r* 67.2%

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

      9. cos-neg 67.2%

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

      10. *-commutative 67.2%

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

      11. distribute-rgt-neg-in 67.2%

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

      12. metadata-eval 67.2%

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

      13. associate-*r* 67.3%

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

      14. *-commutative 67.3%

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

      15. unpow2 67.3%

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

      16. sqr-neg 67.3%

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

      17. associate-*l* 73.2%

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

      18. associate-*r* 75.6%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in x around inf 59.3%

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

      1. associate-/r* 57.2%

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

      2. *-commutative 57.2%

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

      3. unpow2 57.2%

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

      4. unpow2 57.2%

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

      5. swap-sqr 76.9%

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

      6. unpow2 76.9%

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

      7. associate-/r* 79.0%

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

      8. *-commutative 79.0%

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

      9. unpow2 79.0%

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

      10. unpow2 79.0%

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

      11. swap-sqr 95.9%

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

      12. unpow2 95.9%

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

      13. associate-*r* 97.3%

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

      14. *-commutative 97.3%

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

   7. Simplified 97.3%

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

      1. associate-*r* 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*r* 95.9%

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

      4. unpow2 95.9%

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

      5. associate-*r* 91.8%

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

      6. associate-*l* 91.8%

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

      7. *-commutative 91.8%

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

      8. associate-*r* 91.7%

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

      9. *-commutative 91.7%

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

      10. associate-*r* 90.5%

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

   9. Applied egg-rr 90.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{\cos \left(x_m \cdot -2\right)}{s_m} \cdot \frac{1}{x_m \cdot c_m}}{s_m \cdot \left(x_m \cdot c_m\right)} \end{array} \]

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)) s_m) (/ 1.0 (* x_m c_m))) (* s_m (* x_m c_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return ((cos((x_m * -2.0)) / s_m) * (1.0 / (x_m * c_m))) / (s_m * (x_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))) / s_m) * (1.0d0 / (x_m * c_m))) / (s_m * (x_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)) / s_m) * (1.0 / (x_m * c_m))) / (s_m * (x_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)) / s_m) * (1.0 / (x_m * c_m))) / (s_m * (x_m * c_m))

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(Float64(Float64(cos(Float64(x_m * -2.0)) / s_m) * Float64(1.0 / Float64(x_m * c_m))) / Float64(s_m * Float64(x_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)) / s_m) * (1.0 / (x_m * c_m))) / (s_m * (x_m * c_m));
end

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. associate-/r* 71.3%

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

   2. *-commutative 71.3%

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

   3. associate-*r* 66.4%

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

   4. unpow2 66.4%

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

   5. associate-/r* 66.9%

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

   6. add-sqr-sqrt 28.7%

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

   7. sqrt-unprod 50.5%

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

   8. swap-sqr 50.5%

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

   9. metadata-eval 50.5%

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

   10. metadata-eval 50.5%

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

   11. swap-sqr 50.5%

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

   12. *-commutative 50.5%

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

   13. *-commutative 50.5%

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

   14. sqrt-unprod 35.6%

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

   15. add-sqr-sqrt 66.9%

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

   16. *-un-lft-identity 66.9%

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

   17. associate-*r* 67.5%

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

4. Applied egg-rr 82.1%

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

   1. frac-times 82.3%

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

   2. *-un-lft-identity 82.3%

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

   3. pow-prod-down 98.3%

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

   4. associate-*r* 97.1%

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

   5. unpow2 97.1%

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

   6. associate-/r* 97.4%

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

6. Applied egg-rr 97.8%

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

   1. associate-/r* 97.8%

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

   2. div-inv 97.9%

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

   3. *-commutative 97.9%

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

8. Applied egg-rr 97.9%

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

9. Final simplification 97.9%

   \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{s} \cdot \frac{1}{x \cdot c}}{s \cdot \left(x \cdot c\right)} \]
10. Add Preprocessing

Alternative 5: 97.4% accurate, 2.7× speedup

Math

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

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)))) (/ (/ (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);
	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 = s_m * (x_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 = s_m * (x_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 = s_m * (x_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(s_m * Float64(x_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 = s_m * (x_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[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 71.8%

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

   1. associate-/r* 71.3%

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

   2. *-commutative 71.3%

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

   3. associate-*r* 66.4%

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

   4. unpow2 66.4%

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

   5. associate-/r* 66.9%

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

   6. add-sqr-sqrt 28.7%

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

   7. sqrt-unprod 50.5%

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

   8. swap-sqr 50.5%

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

   9. metadata-eval 50.5%

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

   10. metadata-eval 50.5%

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

   11. swap-sqr 50.5%

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

   12. *-commutative 50.5%

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

   13. *-commutative 50.5%

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

   14. sqrt-unprod 35.6%

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

   15. add-sqr-sqrt 66.9%

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

   16. *-un-lft-identity 66.9%

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

   17. associate-*r* 67.5%

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

4. Applied egg-rr 82.1%

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

   1. frac-times 82.3%

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

   2. *-un-lft-identity 82.3%

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

   3. pow-prod-down 98.3%

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

   4. associate-*r* 97.1%

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

   5. unpow2 97.1%

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

   6. associate-/r* 97.4%

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

6. Applied egg-rr 97.8%

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

7. Final simplification 97.8%

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

Alternative 6: 97.5% accurate, 2.7× speedup

Math

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

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)) s_m) (* x_m c_m)) (* s_m (* x_m c_m))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. associate-/r* 71.3%

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

   2. *-commutative 71.3%

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

   3. associate-*r* 66.4%

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

   4. unpow2 66.4%

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

   5. associate-/r* 66.9%

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

   6. add-sqr-sqrt 28.7%

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

   7. sqrt-unprod 50.5%

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

   8. swap-sqr 50.5%

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

   9. metadata-eval 50.5%

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

   10. metadata-eval 50.5%

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

   11. swap-sqr 50.5%

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

   12. *-commutative 50.5%

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

   13. *-commutative 50.5%

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

   14. sqrt-unprod 35.6%

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

   15. add-sqr-sqrt 66.9%

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

   16. *-un-lft-identity 66.9%

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

   17. associate-*r* 67.5%

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

4. Applied egg-rr 82.1%

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

   1. frac-times 82.3%

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

   2. *-un-lft-identity 82.3%

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

   3. pow-prod-down 98.3%

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

   4. associate-*r* 97.1%

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

   5. unpow2 97.1%

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

   6. associate-/r* 97.4%

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

6. Applied egg-rr 97.8%

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

   1. associate-/r* 97.8%

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

   2. div-inv 97.9%

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

   3. *-commutative 97.9%

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

8. Applied egg-rr 97.9%

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

   1. un-div-inv 97.8%

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

10. Applied egg-rr 97.8%

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

11. Final simplification 97.8%

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

Alternative 7: 78.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{\frac{1}{{\left(x_m \cdot \left(s_m \cdot c_m\right)\right)}^{-2}}} \end{array} \]

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 (/ 1.0 (pow (* x_m (* s_m c_m)) -2.0))))

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) {
	return 1.0 / (1.0 / pow((x_m * (s_m * c_m)), -2.0));
}

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
    code = 1.0d0 / (1.0d0 / ((x_m * (s_m * c_m)) ** (-2.0d0)))
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 / (1.0 / Math.pow((x_m * (s_m * c_m)), -2.0));
}

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):
	return 1.0 / (1.0 / math.pow((x_m * (s_m * c_m)), -2.0))

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)
	return Float64(1.0 / Float64(1.0 / (Float64(x_m * Float64(s_m * c_m)) ^ -2.0)))
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)
	tmp = 1.0 / (1.0 / ((x_m * (s_m * c_m)) ^ -2.0));
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_] := N[(1.0 / N[(1.0 / N[Power[N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

3. Taylor expanded in x around 0 61.2%

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

   1. associate-/r* 60.8%

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

   2. *-commutative 60.8%

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

   3. unpow2 60.8%

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

   4. unpow2 60.8%

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

   5. swap-sqr 70.7%

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

   6. unpow2 70.7%

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

   7. associate-/r* 71.1%

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

   8. unpow2 71.1%

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

   9. unpow2 71.1%

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

   10. swap-sqr 81.6%

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

   11. unpow2 81.6%

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

   12. *-commutative 81.6%

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

5. Simplified 81.6%

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

   1. add-sqr-sqrt 81.6%

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

   2. sqrt-div 81.6%

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

   3. metadata-eval 81.6%

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

   4. sqrt-pow1 57.9%

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

   5. metadata-eval 57.9%

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

   6. associate-*r* 57.8%

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

   7. *-commutative 57.8%

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

   8. associate-*r* 57.1%

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

   9. pow1 57.1%

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

   10. sqrt-div 57.1%

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

   11. metadata-eval 57.1%

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

   12. sqrt-pow1 80.4%

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

   13. metadata-eval 80.4%

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

   14. associate-*r* 80.9%

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

   15. *-commutative 80.9%

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

   16. associate-*r* 80.9%

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

   17. pow1 80.9%

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

7. Applied egg-rr 80.9%

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

   1. frac-2neg 80.9%

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

   2. metadata-eval 80.9%

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

   3. frac-2neg 80.9%

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

   4. metadata-eval 80.9%

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

   5. frac-times 81.0%

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

   6. metadata-eval 81.0%

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

   7. *-commutative 81.0%

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

   8. distribute-rgt-neg-in 81.0%

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

   9. *-commutative 81.0%

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

   10. *-commutative 81.0%

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

   11. distribute-rgt-neg-in 81.0%

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

   12. *-commutative 81.0%

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

9. Applied egg-rr 81.0%

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

    1. /-rgt-identity 81.0%

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

    2. clear-num 81.0%

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

    3. swap-sqr 70.9%

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

    4. sqr-neg 70.9%

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

    5. swap-sqr 81.0%

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

    6. *-commutative 81.0%

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

    7. *-commutative 81.0%

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

    8. *-commutative 81.0%

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

    9. associate-*r* 80.4%

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

    10. *-commutative 80.4%

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

    11. associate-/l/ 80.4%

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

    12. *-commutative 80.4%

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

    13. associate-*r* 81.6%

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

    14. *-commutative 81.6%

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

    15. un-div-inv 81.6%

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

    16. inv-pow 81.6%

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

    17. inv-pow 81.6%

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

    18. pow-sqr 81.6%

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

11. Applied egg-rr 82.1%

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

12. Final simplification 82.1%

    \[\leadsto \frac{1}{\frac{1}{{\left(x \cdot \left(s \cdot c\right)\right)}^{-2}}} \]
13. Add Preprocessing

Alternative 8: 78.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ {\left(\frac{x_m}{\frac{-1}{s_m \cdot c_m}}\right)}^{-2} \end{array} \]

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
 (pow (/ x_m (/ -1.0 (* s_m c_m))) -2.0))

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) {
	return pow((x_m / (-1.0 / (s_m * c_m))), -2.0);
}

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
    code = (x_m / ((-1.0d0) / (s_m * c_m))) ** (-2.0d0)
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.pow((x_m / (-1.0 / (s_m * c_m))), -2.0);
}

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):
	return math.pow((x_m / (-1.0 / (s_m * c_m))), -2.0)

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)
	return Float64(x_m / Float64(-1.0 / Float64(s_m * c_m))) ^ -2.0
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)
	tmp = (x_m / (-1.0 / (s_m * c_m))) ^ -2.0;
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_] := N[Power[N[(x$95$m / N[(-1.0 / N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 71.8%

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

   1. associate-/r* 71.3%

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

   2. associate-*l* 71.3%

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

   3. unpow2 71.3%

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

   4. sqr-neg 71.3%

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

   5. unpow2 71.3%

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

   6. *-commutative 71.3%

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

   7. *-commutative 71.3%

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

   8. associate-/r* 71.8%

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

   9. cos-neg 71.8%

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

   10. *-commutative 71.8%

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

   11. distribute-rgt-neg-in 71.8%

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

   12. metadata-eval 71.8%

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

   13. associate-*r* 73.0%

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

   14. *-commutative 73.0%

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

   15. unpow2 73.0%

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

   16. sqr-neg 73.0%

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

   17. associate-*l* 79.1%

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

   18. associate-*r* 80.2%

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

3. Simplified 66.9%

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

5. Taylor expanded in x around inf 66.9%

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

   1. associate-/r* 66.4%

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

   2. *-commutative 66.4%

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

   3. unpow2 66.4%

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

   4. unpow2 66.4%

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

   5. swap-sqr 81.1%

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

   6. unpow2 81.1%

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

   7. associate-/r* 81.6%

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

   8. *-commutative 81.6%

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

   9. unpow2 81.6%

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

   10. unpow2 81.6%

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

   11. swap-sqr 97.1%

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

   12. unpow2 97.1%

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

   13. associate-*r* 97.5%

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

   14. *-commutative 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in x around 0 61.2%

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

   1. associate-*r* 61.6%

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

   2. *-commutative 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 72.4%

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

   6. unpow2 72.4%

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

   7. swap-sqr 82.1%

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

   8. associate-*r* 80.9%

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

   9. associate-*r* 81.0%

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

   10. unpow2 81.0%

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

   11. /-rgt-identity 81.0%

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

   12. unpow2 81.0%

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

   13. associate-/l* 81.0%

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

   14. associate-/l* 80.9%

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

   15. associate-*l/ 80.9%

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

   16. unpow-1 80.9%

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

   17. unpow-1 80.9%

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

   18. pow-sqr 81.0%

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

   19. metadata-eval 81.0%

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

10. Simplified 81.6%

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

    1. /-rgt-identity 81.6%

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

    2. frac-2neg 81.6%

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

    3. *-commutative 81.6%

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

    4. associate-*r* 81.0%

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

    5. distribute-lft-neg-in 81.0%

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

    6. add-sqr-sqrt 35.5%

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

    7. sqrt-unprod 72.9%

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

    8. sqr-neg 72.9%

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

    9. sqrt-unprod 45.4%

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

    10. add-sqr-sqrt 81.0%

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

    11. *-commutative 81.0%

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

    12. associate-*l* 82.1%

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

    13. metadata-eval 82.1%

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

12. Applied egg-rr 82.1%

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

    1. associate-/l* 82.1%

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

14. Simplified 82.1%

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

15. Final simplification 82.1%

    \[\leadsto {\left(\frac{x}{\frac{-1}{s \cdot c}}\right)}^{-2} \]
16. Add Preprocessing

Alternative 9: 37.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. associate-/r* 71.3%

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

   2. *-commutative 71.3%

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

   3. associate-*r* 66.4%

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

   4. unpow2 66.4%

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

   5. associate-/r* 66.9%

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

   6. add-sqr-sqrt 28.7%

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

   7. sqrt-unprod 50.5%

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

   8. swap-sqr 50.5%

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

   9. metadata-eval 50.5%

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

   10. metadata-eval 50.5%

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

   11. swap-sqr 50.5%

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

   12. *-commutative 50.5%

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

   13. *-commutative 50.5%

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

   14. sqrt-unprod 35.6%

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

   15. add-sqr-sqrt 66.9%

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

   16. *-un-lft-identity 66.9%

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

   17. associate-*r* 67.5%

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

4. Applied egg-rr 82.1%

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

   1. frac-times 82.3%

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

   2. *-un-lft-identity 82.3%

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

   3. pow-prod-down 98.3%

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

   4. associate-*r* 97.1%

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

   5. unpow2 97.1%

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

   6. associate-/r* 97.4%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in x around 0 80.4%

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

9. Applied egg-rr 39.9%

   \[\leadsto \frac{\color{blue}{0 - \frac{\frac{1}{c}}{x \cdot s}}}{s \cdot \left(c \cdot x\right)} \]
10. Step-by-step derivation

    1. neg-sub0 39.9%

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

    2. associate-/r* 39.9%

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

    3. *-commutative 39.9%

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

    4. associate-*r* 39.9%

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

    5. *-commutative 39.9%

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

    6. distribute-neg-frac 39.9%

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

    7. metadata-eval 39.9%

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

11. Simplified 39.9%

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

12. Final simplification 39.9%

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

Alternative 10: 78.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. associate-/r* 71.3%

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

   2. associate-*l* 71.3%

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

   3. unpow2 71.3%

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

   4. sqr-neg 71.3%

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

   5. unpow2 71.3%

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

   6. *-commutative 71.3%

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

   7. *-commutative 71.3%

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

   8. associate-/r* 71.8%

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

   9. cos-neg 71.8%

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

   10. *-commutative 71.8%

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

   11. distribute-rgt-neg-in 71.8%

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

   12. metadata-eval 71.8%

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

   13. associate-*r* 73.0%

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

   14. *-commutative 73.0%

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

   15. unpow2 73.0%

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

   16. sqr-neg 73.0%

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

   17. associate-*l* 79.1%

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

   18. associate-*r* 80.2%

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

3. Simplified 66.9%

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

5. Taylor expanded in x around inf 66.9%

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

   1. associate-/r* 66.4%

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

   2. *-commutative 66.4%

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

   3. unpow2 66.4%

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

   4. unpow2 66.4%

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

   5. swap-sqr 81.1%

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

   6. unpow2 81.1%

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

   7. associate-/r* 81.6%

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

   8. *-commutative 81.6%

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

   9. unpow2 81.6%

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

   10. unpow2 81.6%

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

   11. swap-sqr 97.1%

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

   12. unpow2 97.1%

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

   13. associate-*r* 97.5%

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

   14. *-commutative 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in x around 0 61.2%

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

   1. associate-*r* 61.6%

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

   2. *-commutative 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 72.4%

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

   6. unpow2 72.4%

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

   7. swap-sqr 82.1%

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

   8. associate-*r* 80.9%

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

   9. associate-*r* 81.0%

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

   10. unpow2 81.0%

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

   11. /-rgt-identity 81.0%

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

   12. unpow2 81.0%

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

   13. associate-/l* 81.0%

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

   14. associate-/l* 80.9%

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

   15. associate-*l/ 80.9%

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

   16. unpow-1 80.9%

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

   17. unpow-1 80.9%

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

   18. pow-sqr 81.0%

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

   19. metadata-eval 81.0%

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

10. Simplified 81.6%

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

    1. metadata-eval 81.6%

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

    2. pow-sqr 81.6%

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

    3. inv-pow 81.6%

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

    4. inv-pow 81.6%

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

    5. associate-/r* 81.6%

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

    6. frac-times 80.4%

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

    7. *-un-lft-identity 80.4%

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

    8. *-commutative 80.4%

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

    9. associate-*r* 79.2%

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

    10. *-commutative 79.2%

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

    11. associate-*l* 80.4%

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

    12. *-commutative 80.4%

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

12. Applied egg-rr 80.4%

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

13. Final simplification 80.4%

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

Alternative 11: 80.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. associate-/r* 71.3%

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

   2. *-commutative 71.3%

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

   3. associate-*r* 66.4%

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

   4. unpow2 66.4%

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

   5. associate-/r* 66.9%

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

   6. add-sqr-sqrt 28.7%

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

   7. sqrt-unprod 50.5%

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

   8. swap-sqr 50.5%

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

   9. metadata-eval 50.5%

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

   10. metadata-eval 50.5%

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

   11. swap-sqr 50.5%

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

   12. *-commutative 50.5%

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

   13. *-commutative 50.5%

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

   14. sqrt-unprod 35.6%

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

   15. add-sqr-sqrt 66.9%

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

   16. *-un-lft-identity 66.9%

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

   17. associate-*r* 67.5%

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

4. Applied egg-rr 82.1%

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

   1. frac-times 82.3%

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

   2. *-un-lft-identity 82.3%

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

   3. pow-prod-down 98.3%

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

   4. associate-*r* 97.1%

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

   5. unpow2 97.1%

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

   6. associate-/r* 97.4%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in x around 0 80.4%

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

8. Taylor expanded in s around 0 81.6%

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

9. Final simplification 81.6%

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

Alternative 12: 78.8% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* 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(x_m * Float64(s_m * c_m))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = 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[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 71.8%

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

3. Taylor expanded in x around 0 61.2%

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

   1. associate-/r* 60.8%

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

   2. *-commutative 60.8%

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

   3. unpow2 60.8%

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

   4. unpow2 60.8%

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

   5. swap-sqr 70.7%

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

   6. unpow2 70.7%

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

   7. associate-/r* 71.1%

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

   8. unpow2 71.1%

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

   9. unpow2 71.1%

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

   10. swap-sqr 81.6%

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

   11. unpow2 81.6%

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

   12. *-commutative 81.6%

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

5. Simplified 81.6%

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

   1. add-sqr-sqrt 81.6%

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

   2. sqrt-div 81.6%

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

   3. metadata-eval 81.6%

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

   4. sqrt-pow1 57.9%

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

   5. metadata-eval 57.9%

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

   6. associate-*r* 57.8%

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

   7. *-commutative 57.8%

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

   8. associate-*r* 57.1%

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

   9. pow1 57.1%

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

   10. sqrt-div 57.1%

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

   11. metadata-eval 57.1%

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

   12. sqrt-pow1 80.4%

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

   13. metadata-eval 80.4%

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

   14. associate-*r* 80.9%

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

   15. *-commutative 80.9%

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

   16. associate-*r* 80.9%

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

   17. pow1 80.9%

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

7. Applied egg-rr 80.9%

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

   1. un-div-inv 80.9%

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

   2. *-commutative 80.9%

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

   3. *-commutative 80.9%

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

   4. *-commutative 80.9%

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

   5. associate-*l* 80.9%

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

   6. *-commutative 80.9%

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

   7. *-commutative 80.9%

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

   8. associate-*l* 82.0%

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

   9. *-commutative 82.0%

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

9. Applied egg-rr 82.0%

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

10. Final simplification 82.0%

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

Reproduce

herbie shell --seed 2024013 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))