mixedcos

Percentage Accurate: 66.8% → 99.3%

Time: 12.0s

Alternatives: 11

Speedup: 24.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x\_m \cdot 2\right)}{s\_m}}{x\_m \cdot c\_m}}{s\_m \cdot \left(x\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 5e-79)
     (/ (/ 1.0 t_0) t_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5e-79) {
		tmp = (1.0 / t_0) / t_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) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 5d-79) then
        tmp = (1.0d0 / t_0) / t_0
    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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 5e-79) {
		tmp = (1.0 / t_0) / t_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):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 5e-79:
		tmp = (1.0 / t_0) / t_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)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 5e-79)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = 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
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 5e-79)
		tmp = (1.0 / t_0) / t_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_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5e-79], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], 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. Split input into 2 regimes

2. if x < 4.99999999999999999e-79

   1. Initial program 68.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 80.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unswap-sqr N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      8. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      17. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      3. *-lowering-*.f64 80.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 80.7%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(x \cdot s\right) \cdot \color{blue}{c}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(s \cdot x\right) \cdot c\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(s \cdot x\right), \color{blue}{c}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot s\right), c\right)\right) \]

       5. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), c\right)\right) \]

   11. Applied egg-rr 82.4%

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

   if 4.99999999999999999e-79 < x

   1. Initial program 67.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 84.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 84.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unswap-sqr N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      8. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      17. *-lowering-*.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 98.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot \color{blue}{s}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(\left(c \cdot x\right) \cdot s\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(\left(c \cdot x\right), \color{blue}{s}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot c\right), s\right)\right) \]

      6. *-lowering-*.f64 98.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), s\right)\right) \]

   8. Applied egg-rr 98.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), s\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot s}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{c}\right), s\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(x \cdot c\right)}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{c}\right), s\right)\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\cos \left(2 \cdot x\right)}{s}}{x \cdot c}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, c\right)}, s\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{s}\right), \left(x \cdot c\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, c\right)}, s\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), s\right), \left(x \cdot c\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, c\right), s\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), s\right), \left(x \cdot c\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), s\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), s\right), \left(x \cdot c\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), s\right)\right) \]

      9. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), s\right), \mathsf{*.f64}\left(x, c\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{c}\right), s\right)\right) \]

   10. Applied egg-rr 99.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 2.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{x\_m \cdot \left(\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 2.1e-76)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m 2.0)) (* x_m (* (* c_m s_m) (* x_m (* c_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.1e-76) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * 2.0)) / (x_m * ((c_m * s_m) * (x_m * (c_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) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 2.1d-76) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * 2.0d0)) / (x_m * ((c_m * s_m) * (x_m * (c_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.1e-76) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * 2.0)) / (x_m * ((c_m * s_m) * (x_m * (c_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):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 2.1e-76:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * 2.0)) / (x_m * ((c_m * s_m) * (x_m * (c_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)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 2.1e-76)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(x_m * Float64(Float64(c_m * s_m) * Float64(x_m * Float64(c_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)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 2.1e-76)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * 2.0)) / (x_m * ((c_m * s_m) * (x_m * (c_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_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.1e-76], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999992e-76

   1. Initial program 68.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 80.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unswap-sqr N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      8. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      17. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      3. *-lowering-*.f64 80.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 80.7%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(x \cdot s\right) \cdot \color{blue}{c}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(s \cdot x\right) \cdot c\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(s \cdot x\right), \color{blue}{c}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot s\right), c\right)\right) \]

       5. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), c\right)\right) \]

   11. Applied egg-rr 82.4%

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

   if 2.09999999999999992e-76 < x

   1. Initial program 67.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 84.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 84.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right), \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)\right), x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right), x\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot s\right)\right), x\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right), x\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), \left(x \cdot \left(c \cdot s\right)\right)\right), x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(s \cdot c\right), \left(x \cdot \left(c \cdot s\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, c\right), \left(x \cdot \left(c \cdot s\right)\right)\right), x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, c\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), x\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, c\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), x\right)\right) \]

      12. *-lowering-*.f64 96.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, c\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), x\right)\right) \]

   6. Applied egg-rr 96.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 0.00152:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 0.00152)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m 2.0)) (* (* x_m c_m) (* x_m (* s_m (* c_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 0.00152) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * 2.0)) / ((x_m * c_m) * (x_m * (s_m * (c_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) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 0.00152d0) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * 2.0d0)) / ((x_m * c_m) * (x_m * (s_m * (c_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 0.00152) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * 2.0)) / ((x_m * c_m) * (x_m * (s_m * (c_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):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 0.00152:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * 2.0)) / ((x_m * c_m) * (x_m * (s_m * (c_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)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 0.00152)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(Float64(x_m * c_m) * Float64(x_m * Float64(s_m * Float64(c_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)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 0.00152)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * 2.0)) / ((x_m * c_m) * (x_m * (s_m * (c_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_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00152], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(s$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0015200000000000001

   1. Initial program 68.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 80.9%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unswap-sqr N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      8. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      17. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      3. *-lowering-*.f64 80.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 80.9%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(x \cdot s\right) \cdot \color{blue}{c}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(s \cdot x\right) \cdot c\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(s \cdot x\right), \color{blue}{c}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot s\right), c\right)\right) \]

       5. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), c\right)\right) \]

   11. Applied egg-rr 82.6%

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

   if 0.0015200000000000001 < x

   1. Initial program 66.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 84.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 84.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left(x \cdot s\right) \cdot \left(\left(c \cdot s\right) \cdot c\right)\right) \cdot x\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot c\right) \cdot x\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot \color{blue}{\left(c \cdot x\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right), \color{blue}{\left(c \cdot x\right)}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\left(x \cdot \left(s \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{c} \cdot x\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(s \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{c} \cdot x\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot s\right)\right)\right), \left(c \cdot x\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot c\right)\right)\right), \left(c \cdot x\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(c \cdot x\right)\right)\right) \]

      12. *-lowering-*.f64 87.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(c, \color{blue}{x}\right)\right)\right) \]

   6. Applied egg-rr 87.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00152:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{x\_m \cdot \left(x\_m \cdot \left(s\_m \cdot \left(c\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 0.001)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m 2.0)) (* x_m (* x_m (* s_m (* c_m (* c_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 0.001) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * 2.0)) / (x_m * (x_m * (s_m * (c_m * (c_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) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 0.001d0) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * 2.0d0)) / (x_m * (x_m * (s_m * (c_m * (c_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 0.001) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * 2.0)) / (x_m * (x_m * (s_m * (c_m * (c_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):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 0.001:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * 2.0)) / (x_m * (x_m * (s_m * (c_m * (c_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)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 0.001)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(x_m * Float64(x_m * Float64(s_m * Float64(c_m * Float64(c_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)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 0.001)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * 2.0)) / (x_m * (x_m * (s_m * (c_m * (c_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_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.001], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(x$95$m * N[(s$95$m * N[(c$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-3

   1. Initial program 68.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 80.9%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unswap-sqr N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

      8. count-2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

      17. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

      3. *-lowering-*.f64 80.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   9. Simplified 80.9%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(x \cdot s\right) \cdot \color{blue}{c}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(s \cdot x\right) \cdot c\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(s \cdot x\right), \color{blue}{c}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot s\right), c\right)\right) \]

       5. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), c\right)\right) \]

   11. Applied egg-rr 82.6%

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

   if 1e-3 < x

   1. Initial program 66.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 84.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 84.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 2.7× speedup

Math

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

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 (* x_m (* c_m s_m)))) (/ (/ (cos (* x_m 2.0)) t_0) t_0)))

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = x_m * (c_m * s_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(x_m * Float64(c_m * s_m))
	return Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / t_0)
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = x_m * (c_m * s_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[(x$95$m * N[(c$95$m * s$95$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 67.9%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 81.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 81.6%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unswap-sqr N/A

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

   5. associate-/r* N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   8. count-2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   10. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   17. *-lowering-*.f64 98.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

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

Alternative 6: 75.6% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;c\_m \leq 7.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= c_m 7.5e-218)
   (/ 1.0 (* x_m (* x_m (* c_m (* s_m (* c_m s_m))))))
   (/ 1.0 (* x_m (* s_m (* x_m (* c_m (* c_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 (c_m <= 7.5e-218) {
		tmp = 1.0 / (x_m * (x_m * (c_m * (s_m * (c_m * s_m)))));
	} else {
		tmp = 1.0 / (x_m * (s_m * (x_m * (c_m * (c_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 (c_m <= 7.5d-218) then
        tmp = 1.0d0 / (x_m * (x_m * (c_m * (s_m * (c_m * s_m)))))
    else
        tmp = 1.0d0 / (x_m * (s_m * (x_m * (c_m * (c_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 (c_m <= 7.5e-218) {
		tmp = 1.0 / (x_m * (x_m * (c_m * (s_m * (c_m * s_m)))));
	} else {
		tmp = 1.0 / (x_m * (s_m * (x_m * (c_m * (c_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 c_m <= 7.5e-218:
		tmp = 1.0 / (x_m * (x_m * (c_m * (s_m * (c_m * s_m)))))
	else:
		tmp = 1.0 / (x_m * (s_m * (x_m * (c_m * (c_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 (c_m <= 7.5e-218)
		tmp = Float64(1.0 / Float64(x_m * Float64(x_m * Float64(c_m * Float64(s_m * Float64(c_m * s_m))))));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(s_m * Float64(x_m * Float64(c_m * Float64(c_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 (c_m <= 7.5e-218)
		tmp = 1.0 / (x_m * (x_m * (c_m * (s_m * (c_m * s_m)))));
	else
		tmp = 1.0 / (x_m * (s_m * (x_m * (c_m * (c_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[c$95$m, 7.5e-218], N[(1.0 / N[(x$95$m * N[(x$95$m * N[(c$95$m * N[(s$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(s$95$m * N[(x$95$m * N[(c$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 7.50000000000000011e-218

   1. Initial program 68.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 81.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 81.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 73.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\left(s \cdot c\right) \cdot s\right) \cdot \color{blue}{c}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot s\right), \color{blue}{c}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot s\right)\right), c\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(s \cdot c\right)\right), c\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \left(s \cdot c\right)\right), c\right)\right)\right)\right) \]

      8. *-lowering-*.f64 72.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, c\right)\right), c\right)\right)\right)\right) \]

   9. Applied egg-rr 72.9%

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

   if 7.50000000000000011e-218 < c

   1. Initial program 67.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 82.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 82.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 65.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(\left(c \cdot \left(c \cdot s\right)\right) \cdot x\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \left(\left(c \cdot \left(s \cdot c\right)\right) \cdot x\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \left(\left(\left(s \cdot c\right) \cdot c\right) \cdot x\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot x\right)}\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(s \cdot c\right) \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{s}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot \left(c \cdot x\right)\right), \color{blue}{s}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(c \cdot x\right) \cdot \left(s \cdot c\right)\right), s\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot \left(s \cdot c\right)\right), s\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot \left(s \cdot c\right)\right)\right), s\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot \left(c \cdot s\right)\right)\right), s\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(c \cdot \left(c \cdot s\right)\right)\right), s\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(c \cdot \left(s \cdot c\right)\right)\right), s\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right), s\right)\right)\right) \]

      15. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, c\right)\right)\right), s\right)\right)\right) \]

   9. Applied egg-rr 67.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 7.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 81.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 81.6%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unswap-sqr N/A

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

   5. associate-/r* N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   8. count-2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   10. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   17. *-lowering-*.f64 98.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 98.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   3. *-lowering-*.f64 76.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

9. Simplified 76.8%

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

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(x \cdot s\right) \cdot \color{blue}{c}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(s \cdot x\right) \cdot c\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(s \cdot x\right), \color{blue}{c}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot s\right), c\right)\right) \]

    5. *-lowering-*.f64 78.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), c\right)\right) \]

11. Applied egg-rr 78.1%

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

12. Final simplification 78.1%

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

Alternative 8: 79.2% accurate, 24.1× speedup

Math

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

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))) (* x_m (* c_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))) / (x_m * (c_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))) / (x_m * (c_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))) / (x_m * (c_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))) / (x_m * (c_m * s_m))

Julia

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 81.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 81.6%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unswap-sqr N/A

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

   5. associate-/r* N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   8. count-2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   10. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   17. *-lowering-*.f64 98.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 98.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{c \cdot \left(s \cdot x\right)}\right)}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(c \cdot \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

   3. *-lowering-*.f64 76.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right) \]

9. Simplified 76.8%

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

10. Final simplification 76.8%

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

Alternative 9: 78.7% accurate, 24.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 67.9%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 81.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 81.6%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 69.9%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\left(s \cdot c\right) \cdot s\right) \cdot \color{blue}{c}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(s \cdot c\right) \cdot s\right), \color{blue}{c}\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(c \cdot s\right)\right), c\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(s \cdot \left(s \cdot c\right)\right), c\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \left(s \cdot c\right)\right), c\right)\right)\right)\right) \]

   8. *-lowering-*.f64 68.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, c\right)\right), c\right)\right)\right)\right) \]

9. Applied egg-rr 68.0%

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

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot \left(s \cdot c\right)\right) \cdot c\right)}\right)\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(s \cdot c\right)}\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)\right)\right)\right) \]

    7. swap-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot \left(s \cdot c\right)\right), \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot s\right)\right), \left(x \cdot \left(s \cdot c\right)\right)\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\left(x \cdot c\right) \cdot s\right), \left(\color{blue}{x} \cdot \left(s \cdot c\right)\right)\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(s \cdot \left(x \cdot c\right)\right), \left(\color{blue}{x} \cdot \left(s \cdot c\right)\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \left(x \cdot c\right)\right), \left(\color{blue}{x} \cdot \left(s \cdot c\right)\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, c\right)\right), \left(x \cdot \left(s \cdot c\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, c\right)\right), \left(x \cdot \left(c \cdot \color{blue}{s}\right)\right)\right)\right) \]

    15. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, c\right)\right), \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, c\right)\right), \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, c\right)\right), \mathsf{*.f64}\left(s, \color{blue}{\left(x \cdot c\right)}\right)\right)\right) \]

    18. *-lowering-*.f64 78.2%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, c\right)\right), \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{c}\right)\right)\right)\right) \]

11. Applied egg-rr 78.2%

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

Alternative 10: 72.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 81.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 81.6%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 69.9%

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

Alternative 11: 30.8% accurate, 34.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m) :precision binary64 (/ (/ (/ -2.0 (* s_m s_m)) c_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 ((-2.0 / (s_m * s_m)) / c_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 = (((-2.0d0) / (s_m * s_m)) / c_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 ((-2.0 / (s_m * s_m)) / c_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 ((-2.0 / (s_m * s_m)) / c_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(-2.0 / Float64(s_m * s_m)) / c_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 = ((-2.0 / (s_m * s_m)) / c_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[(-2.0 / N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]

Derivation

1. Initial program 67.9%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 81.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 81.6%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unswap-sqr N/A

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

   5. associate-/r* N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}\right), \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(\color{blue}{x} \cdot \left(c \cdot s\right)\right)\right) \]

   8. count-2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   10. count-2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(c \cdot s\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot s\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(c \cdot s\right)}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{c}\right)\right)\right) \]

   17. *-lowering-*.f64 98.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right) \]

6. Applied egg-rr 98.0%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(x \cdot \left(c \cdot \color{blue}{s}\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(\left(x \cdot c\right) \cdot \color{blue}{s}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \left(\left(c \cdot x\right) \cdot s\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(\left(c \cdot x\right), \color{blue}{s}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot c\right), s\right)\right) \]

   6. *-lowering-*.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), s\right)\right) \]

8. Applied egg-rr 97.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}}{x}\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, c\right), s\right)\right) \]
10. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{c \cdot s} + \frac{1}{c \cdot s}\right), x\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, c\right)}, s\right)\right) \]

11. Simplified 61.5%

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

12. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
13. Step-by-step derivation

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. associate-/r* N/A

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

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{{s}^{2} \cdot c}\right), \color{blue}{c}\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{c \cdot {s}^{2}}\right), c\right) \]

    7. associate-/l/ N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-2}{{s}^{2}}}{c}\right), c\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{{s}^{2}}\right), c\right), c\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left({s}^{2}\right)\right), c\right), c\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(s \cdot s\right)\right), c\right), c\right) \]

    11. *-lowering-*.f64 27.2%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(s, s\right)\right), c\right), c\right) \]

14. Simplified 27.2%

    \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{s \cdot s}}{c}}{c}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))