mixedcos

Percentage Accurate: 66.9% → 97.6%

Time: 12.7s

Alternatives: 12

Speedup: 24.1×

• 31.4% 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: 66.9% 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: 97.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 1.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{elif}\;x\_m \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;\frac{t\_0}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{s\_m}}{s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 1.9e-105)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x_m 1.95e+149)
       (/ t_0 (* (* c_m s_m) (* x_m (* x_m (* c_m s_m)))))
       (/ (/ t_0 s_m) (* s_m (* x_m (* c_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 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.9e-105) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.95e+149) {
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	} else {
		tmp = (t_0 / s_m) / (s_m * (x_m * (c_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) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 1.9d-105) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x_m <= 1.95d+149) then
        tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
    else
        tmp = (t_0 / s_m) / (s_m * (x_m * (c_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 = Math.cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.9e-105) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.95e+149) {
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	} else {
		tmp = (t_0 / s_m) / (s_m * (x_m * (c_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 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1.9e-105:
		tmp = (1.0 / t_1) / t_1
	elif x_m <= 1.95e+149:
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
	else:
		tmp = (t_0 / s_m) / (s_m * (x_m * (c_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 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1.9e-105)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x_m <= 1.95e+149)
		tmp = Float64(t_0 / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(x_m * Float64(c_m * s_m)))));
	else
		tmp = Float64(Float64(t_0 / s_m) / Float64(s_m * Float64(x_m * Float64(c_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 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.9e-105)
		tmp = (1.0 / t_1) / t_1;
	elseif (x_m <= 1.95e+149)
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	else
		tmp = (t_0 / s_m) / (s_m * (x_m * (c_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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.9e-105], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 1.95e+149], N[(t$95$0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(s$95$m * N[(x$95$m * N[(c$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.8999999999999999e-105

   1. Initial program 66.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.4%

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

   5. Simplified 73.4%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

      7. associate-/r* N/A

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 80.6%

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

   7. Applied egg-rr 80.6%

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

   if 1.8999999999999999e-105 < x < 1.95e149

   1. Initial program 65.7%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      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. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 84.6%

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

   5. Simplified 84.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-lowering-*.f64 99.4%

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

   7. Applied egg-rr 99.4%

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

   if 1.95e149 < x

   1. Initial program 66.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. *-lowering-*.f64 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. *-rgt-identity N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. swap-sqr N/A

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

      6. unswap-sqr N/A

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

      7. associate-*r* N/A

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

      8. frac-times N/A

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

      9. associate-/r* N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f64 N/A

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

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

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

      15. count-2 N/A

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

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

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

      17. count-2 N/A

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

      18. *-commutative N/A

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

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

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

      20. associate-*r* N/A

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

   6. Applied egg-rr 78.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+149}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 4.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{elif}\;x\_m \leq 3.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{t\_0}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 4.2e-105)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x_m 3.3e+150)
       (/ t_0 (* (* c_m s_m) (* x_m (* x_m (* c_m s_m)))))
       (/ t_0 (* s_m (* s_m (* x_m (* c_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 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4.2e-105) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 3.3e+150) {
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	} else {
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 4.2d-105) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x_m <= 3.3d+150) then
        tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
    else
        tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = Math.cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 4.2e-105) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 3.3e+150) {
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	} else {
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 4.2e-105:
		tmp = (1.0 / t_1) / t_1
	elif x_m <= 3.3e+150:
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))))
	else:
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 4.2e-105)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x_m <= 3.3e+150)
		tmp = Float64(t_0 / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(x_m * Float64(c_m * s_m)))));
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(s_m * Float64(x_m * Float64(c_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 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 4.2e-105)
		tmp = (1.0 / t_1) / t_1;
	elseif (x_m <= 3.3e+150)
		tmp = t_0 / ((c_m * s_m) * (x_m * (x_m * (c_m * s_m))));
	else
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4.2e-105], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 3.3e+150], N[(t$95$0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(s$95$m * N[(s$95$m * N[(x$95$m * N[(c$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.2e-105

   1. Initial program 66.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.4%

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

   5. Simplified 73.4%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

      7. associate-/r* N/A

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

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

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 80.6%

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

   7. Applied egg-rr 80.6%

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

   if 4.2e-105 < x < 3.29999999999999981e150

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      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. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 84.9%

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

   5. Simplified 84.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-lowering-*.f64 99.4%

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

   7. Applied egg-rr 99.4%

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

   if 3.29999999999999981e150 < x

   1. Initial program 64.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. *-lowering-*.f64 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. swap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 77.6%

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

   6. Applied egg-rr 77.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1 + \left(x\_m \cdot x\_m\right) \cdot -2}{t\_1}}{t\_1}\\ \mathbf{elif}\;x\_m \leq 2.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{t\_0}{\left(x\_m \cdot c\_m\right) \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 9.2e-5)
     (/ (/ (+ 1.0 (* (* x_m x_m) -2.0)) t_1) t_1)
     (if (<= x_m 2.8e+110)
       (/ t_0 (* (* x_m c_m) (* (* x_m s_m) (* c_m s_m))))
       (/ t_0 (* s_m (* s_m (* x_m (* c_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 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 9.2e-5) {
		tmp = ((1.0 + ((x_m * x_m) * -2.0)) / t_1) / t_1;
	} else if (x_m <= 2.8e+110) {
		tmp = t_0 / ((x_m * c_m) * ((x_m * s_m) * (c_m * s_m)));
	} else {
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 9.2d-5) then
        tmp = ((1.0d0 + ((x_m * x_m) * (-2.0d0))) / t_1) / t_1
    else if (x_m <= 2.8d+110) then
        tmp = t_0 / ((x_m * c_m) * ((x_m * s_m) * (c_m * s_m)))
    else
        tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = Math.cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 9.2e-5) {
		tmp = ((1.0 + ((x_m * x_m) * -2.0)) / t_1) / t_1;
	} else if (x_m <= 2.8e+110) {
		tmp = t_0 / ((x_m * c_m) * ((x_m * s_m) * (c_m * s_m)));
	} else {
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 9.2e-5:
		tmp = ((1.0 + ((x_m * x_m) * -2.0)) / t_1) / t_1
	elif x_m <= 2.8e+110:
		tmp = t_0 / ((x_m * c_m) * ((x_m * s_m) * (c_m * s_m)))
	else:
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 9.2e-5)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * -2.0)) / t_1) / t_1);
	elseif (x_m <= 2.8e+110)
		tmp = Float64(t_0 / Float64(Float64(x_m * c_m) * Float64(Float64(x_m * s_m) * Float64(c_m * s_m))));
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(s_m * Float64(x_m * Float64(c_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 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 9.2e-5)
		tmp = ((1.0 + ((x_m * x_m) * -2.0)) / t_1) / t_1;
	elseif (x_m <= 2.8e+110)
		tmp = t_0 / ((x_m * c_m) * ((x_m * s_m) * (c_m * s_m)));
	else
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 9.2e-5], N[(N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 2.8e+110], N[(t$95$0 / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(s$95$m * N[(s$95$m * N[(x$95$m * N[(c$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 9.20000000000000001e-5

   1. Initial program 65.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. *-lowering-*.f64 96.4%

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

   4. Applied egg-rr 96.4%

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. count-2 N/A

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

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

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

      7. count-2 N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + -2 \cdot {x}^{2}\right)}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(-2 \cdot {x}^{2}\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot -2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), -2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), -2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right) \]

      5. *-lowering-*.f64 69.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), -2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right) \]

   9. Simplified 69.8%

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

   if 9.20000000000000001e-5 < x < 2.79999999999999987e110

   1. Initial program 71.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

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

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

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

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

      14. *-lowering-*.f64 95.8%

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

   4. Applied egg-rr 95.8%

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

   if 2.79999999999999987e110 < x

   1. Initial program 65.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. *-lowering-*.f64 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. swap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 75.9%

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

   6. Applied egg-rr 75.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1 + \left(x \cdot x\right) \cdot -2}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := \frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{if}\;x\_m \leq 3.45 \cdot 10^{-37}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{elif}\;x\_m \leq 7.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot \left(x\_m \cdot \left(s\_m \cdot \left(c\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (cos (* x_m 2.0))) (t_1 (/ 1.0 (* c_m (* x_m s_m)))))
   (if (<= x_m 3.45e-37)
     (* t_1 t_1)
     (if (<= x_m 7.6e+116)
       (/ t_0 (* x_m (* x_m (* s_m (* c_m (* c_m s_m))))))
       (/ t_0 (* s_m (* s_m (* x_m (* c_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 = cos((x_m * 2.0));
	double t_1 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (x_m <= 3.45e-37) {
		tmp = t_1 * t_1;
	} else if (x_m <= 7.6e+116) {
		tmp = t_0 / (x_m * (x_m * (s_m * (c_m * (c_m * s_m)))));
	} else {
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = 1.0d0 / (c_m * (x_m * s_m))
    if (x_m <= 3.45d-37) then
        tmp = t_1 * t_1
    else if (x_m <= 7.6d+116) then
        tmp = t_0 / (x_m * (x_m * (s_m * (c_m * (c_m * s_m)))))
    else
        tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = Math.cos((x_m * 2.0));
	double t_1 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (x_m <= 3.45e-37) {
		tmp = t_1 * t_1;
	} else if (x_m <= 7.6e+116) {
		tmp = t_0 / (x_m * (x_m * (s_m * (c_m * (c_m * s_m)))));
	} else {
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = math.cos((x_m * 2.0))
	t_1 = 1.0 / (c_m * (x_m * s_m))
	tmp = 0
	if x_m <= 3.45e-37:
		tmp = t_1 * t_1
	elif x_m <= 7.6e+116:
		tmp = t_0 / (x_m * (x_m * (s_m * (c_m * (c_m * s_m)))))
	else:
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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 = cos(Float64(x_m * 2.0))
	t_1 = Float64(1.0 / Float64(c_m * Float64(x_m * s_m)))
	tmp = 0.0
	if (x_m <= 3.45e-37)
		tmp = Float64(t_1 * t_1);
	elseif (x_m <= 7.6e+116)
		tmp = Float64(t_0 / Float64(x_m * Float64(x_m * Float64(s_m * Float64(c_m * Float64(c_m * s_m))))));
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(s_m * Float64(x_m * Float64(c_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 = cos((x_m * 2.0));
	t_1 = 1.0 / (c_m * (x_m * s_m));
	tmp = 0.0;
	if (x_m <= 3.45e-37)
		tmp = t_1 * t_1;
	elseif (x_m <= 7.6e+116)
		tmp = t_0 / (x_m * (x_m * (s_m * (c_m * (c_m * s_m)))));
	else
		tmp = t_0 / (s_m * (s_m * (x_m * (c_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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3.45e-37], N[(t$95$1 * t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 7.6e+116], N[(t$95$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], N[(t$95$0 / N[(s$95$m * N[(s$95$m * N[(x$95$m * N[(c$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4499999999999999e-37

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.4%

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

   5. Simplified 73.4%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

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

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

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

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 3.4499999999999999e-37 < x < 7.5999999999999998e116

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      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. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 93.7%

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

   5. Simplified 93.7%

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

   if 7.5999999999999998e116 < x

   1. Initial program 64.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. *-lowering-*.f64 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

      4. swap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. associate-*r* N/A

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

      12. associate-*l* N/A

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

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

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

      14. associate-*l* N/A

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

      15. associate-*l* N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 79.6%

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

   6. Applied egg-rr 79.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{if}\;x\_m \leq 4.7 \cdot 10^{-37}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{x\_m \cdot \left(x\_m \cdot \left(s\_m \cdot \left(c\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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 (/ 1.0 (* c_m (* x_m s_m)))))
   (if (<= x_m 4.7e-37)
     (* 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 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (x_m <= 4.7e-37) {
		tmp = 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 = 1.0d0 / (c_m * (x_m * s_m))
    if (x_m <= 4.7d-37) then
        tmp = 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 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (x_m <= 4.7e-37) {
		tmp = 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 = 1.0 / (c_m * (x_m * s_m))
	tmp = 0
	if x_m <= 4.7e-37:
		tmp = 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(1.0 / Float64(c_m * Float64(x_m * s_m)))
	tmp = 0.0
	if (x_m <= 4.7e-37)
		tmp = Float64(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 = 1.0 / (c_m * (x_m * s_m));
	tmp = 0.0;
	if (x_m <= 4.7e-37)
		tmp = 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[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4.7e-37], N[(t$95$0 * 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 < 4.7000000000000003e-37

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.4%

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

   5. Simplified 73.4%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

      7. associate-/r* N/A

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

      8. div-inv N/A

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

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

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

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

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 4.7000000000000003e-37 < x

   1. Initial program 66.0%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      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. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 85.1%

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

   5. Simplified 85.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 2.7× speedup

Math

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

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_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 = c_m * (x_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 = c_m * (x_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 = c_m * (x_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(c_m * Float64(x_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 = c_m * (x_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[(c$95$m * N[(x$95$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 66.0%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

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

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

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

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

   9. *-lowering-*.f64 96.9%

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

4. Applied egg-rr 96.9%

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

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

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

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

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

   5. count-2 N/A

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

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

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

   7. count-2 N/A

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

   8. *-commutative N/A

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. associate-*l* N/A

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

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

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

   15. *-lowering-*.f64 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 7: 68.1% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;c\_m \leq 1.12 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{1}{c\_m}}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{c\_m \cdot s\_m}}{c\_m \cdot s\_m}\\ \end{array} \end{array} \]

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 1.12e+101)
   (/ (/ 1.0 c_m) (* c_m (* s_m (* s_m (* x_m x_m)))))
   (/ (/ -2.0 (* c_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 tmp;
	if (c_m <= 1.12e+101) {
		tmp = (1.0 / c_m) / (c_m * (s_m * (s_m * (x_m * x_m))));
	} else {
		tmp = (-2.0 / (c_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) :: tmp
    if (c_m <= 1.12d+101) then
        tmp = (1.0d0 / c_m) / (c_m * (s_m * (s_m * (x_m * x_m))))
    else
        tmp = ((-2.0d0) / (c_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 tmp;
	if (c_m <= 1.12e+101) {
		tmp = (1.0 / c_m) / (c_m * (s_m * (s_m * (x_m * x_m))));
	} else {
		tmp = (-2.0 / (c_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):
	tmp = 0
	if c_m <= 1.12e+101:
		tmp = (1.0 / c_m) / (c_m * (s_m * (s_m * (x_m * x_m))))
	else:
		tmp = (-2.0 / (c_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)
	tmp = 0.0
	if (c_m <= 1.12e+101)
		tmp = Float64(Float64(1.0 / c_m) / Float64(c_m * Float64(s_m * Float64(s_m * Float64(x_m * x_m)))));
	else
		tmp = Float64(Float64(-2.0 / Float64(c_m * 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)
	tmp = 0.0;
	if (c_m <= 1.12e+101)
		tmp = (1.0 / c_m) / (c_m * (s_m * (s_m * (x_m * x_m))));
	else
		tmp = (-2.0 / (c_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_] := If[LessEqual[c$95$m, 1.12e+101], N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(c$95$m * N[(s$95$m * N[(s$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.1199999999999999e101

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 67.5%

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

   5. Simplified 67.5%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. associate-/l/ N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. frac-times N/A

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

      10. associate-/r* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. div-inv N/A

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

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

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

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

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

      16. associate-*r* N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

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

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

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

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

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

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

      22. *-lowering-*.f64 63.4%

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

   7. Applied egg-rr 63.4%

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

   if 1.1199999999999999e101 < c

   1. Initial program 66.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. *-lowering-*.f64 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. count-2 N/A

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

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

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

      7. count-2 N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

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

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

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

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   9. Simplified 39.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. unpow2 N/A

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

       3. unswap-sqr N/A

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

       4. associate-/r* N/A

          \[\leadsto \frac{\frac{-2}{c \cdot s}}{\color{blue}{c \cdot s}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{\frac{\mathsf{neg}\left(2\right)}{c \cdot s}}{c \cdot s} \]

       6. distribute-neg-frac N/A

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

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{2 \cdot 1}{c \cdot s}\right)}{c \cdot s} \]

       8. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \frac{1}{c \cdot s}\right)}{c \cdot s} \]

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

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2 \cdot \frac{1}{c \cdot s}\right)\right), \color{blue}{\left(c \cdot s\right)}\right) \]

       10. associate-*r/ N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2 \cdot 1}{c \cdot s}\right)\right), \left(c \cdot s\right)\right) \]

       11. metadata-eval N/A

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

       12. distribute-neg-frac N/A

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

       13. metadata-eval N/A

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

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

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

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

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

       16. *-lowering-*.f64 52.9%

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

   12. Simplified 52.9%

       \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 79.6% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 68.7%

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

5. Simplified 68.7%

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

   1. associate-/l/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. swap-sqr N/A

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

   7. associate-/r* N/A

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

   8. div-inv N/A

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. associate-*l* N/A

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

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

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

   17. *-lowering-*.f64 75.8%

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

7. Applied egg-rr 75.8%

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

8. Final simplification 75.8%

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

Alternative 9: 79.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 68.7%

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

5. Simplified 68.7%

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

   1. associate-/l/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. swap-sqr N/A

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

   7. associate-/r* N/A

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

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

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

   13. associate-*l* N/A

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

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

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

   15. *-lowering-*.f64 75.8%

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

7. Applied egg-rr 75.8%

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

8. Final simplification 75.8%

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

Alternative 10: 77.7% accurate, 24.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 66.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 68.7%

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

5. Simplified 68.7%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. associate-/l/ N/A

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

   4. *-rgt-identity N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. frac-times N/A

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

   10. associate-/r* N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. div-inv N/A

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

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

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

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

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

   16. associate-*r* N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

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

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

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

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

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

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

   22. *-lowering-*.f64 63.4%

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

7. Applied egg-rr 63.4%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

   3. remove-double-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(x \cdot \frac{1}{\color{blue}{\frac{1}{s \cdot x}}}\right)\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(\frac{x}{\color{blue}{\frac{1}{s \cdot x}}}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1}{s \cdot x}\right)}\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 70.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 70.2%

   \[\leadsto \frac{\frac{1}{c}}{c \cdot \left(s \cdot \color{blue}{\frac{x}{\frac{1}{x \cdot s}}}\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\left(c \cdot s\right) \cdot \color{blue}{\frac{x}{\frac{1}{x \cdot s}}}\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\frac{\left(c \cdot s\right) \cdot x}{\color{blue}{\frac{1}{x \cdot s}}}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\frac{x \cdot \left(c \cdot s\right)}{\frac{\color{blue}{1}}{x \cdot s}}\right)\right) \]

    4. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \color{blue}{\frac{1}{\frac{1}{x \cdot s}}}\right)\right) \]

    5. remove-double-div N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \color{blue}{s}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(\left(x \cdot \left(c \cdot s\right)\right), \color{blue}{\left(x \cdot s\right)}\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(c \cdot s\right)\right), \left(\color{blue}{x} \cdot s\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, s\right)\right), \left(x \cdot s\right)\right)\right) \]

    9. *-lowering-*.f64 72.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(x, \color{blue}{s}\right)\right)\right) \]

11. Applied egg-rr 72.8%

    \[\leadsto \frac{\frac{1}{c}}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot s\right)}} \]

12. Final simplification 72.8%

    \[\leadsto \frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]
13. Add Preprocessing

Alternative 11: 74.3% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{1}{c\_m}}{c\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (/ 1.0 c_m) (* c_m (* s_m (* x_m (* x_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (1.0 / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return (1.0 / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return (1.0 / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))))

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(Float64(1.0 / c_m) / Float64(c_m * Float64(s_m * Float64(x_m * Float64(x_m * s_m)))))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = (1.0 / c_m) / (c_m * (s_m * (x_m * (x_m * s_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(c$95$m * N[(s$95$m * N[(x$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{{x}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{1}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot \color{blue}{x}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}}{\color{blue}{x}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot x}\right), \color{blue}{x}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x}\right), x\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{{c}^{2} \cdot {s}^{2}}\right), x\right), x\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({c}^{2} \cdot {s}^{2}\right)\right), x\right), x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({s}^{2} \cdot {c}^{2}\right)\right), x\right), x\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\left(s \cdot s\right) \cdot {c}^{2}\right)\right), x\right), x\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(s \cdot \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot {c}^{2}\right)\right)\right), x\right), x\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(s \cdot \left(c \cdot c\right)\right)\right)\right), x\right), x\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(\left(s \cdot c\right) \cdot c\right)\right)\right), x\right), x\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \left(c \cdot \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(s \cdot c\right)\right)\right)\right), x\right), x\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \left(c \cdot s\right)\right)\right)\right), x\right), x\right) \]

   18. *-lowering-*.f64 68.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right), x\right), x\right) \]

5. Simplified 68.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{1}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}}{x}}{x}} \]
6. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{\frac{1}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}}{\color{blue}{x \cdot x}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{s}}{c \cdot \left(c \cdot s\right)}}{\color{blue}{x} \cdot x} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]

   4. *-rgt-identity N/A

      \[\leadsto \frac{\frac{1}{s} \cdot 1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(c \cdot \left(c \cdot s\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\frac{1}{s} \cdot 1}{\left(\left(x \cdot x\right) \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{s} \cdot 1}{\left(\left(x \cdot x\right) \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{s} \cdot 1}{\left(\left(\left(x \cdot x\right) \cdot c\right) \cdot s\right) \cdot \color{blue}{c}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{s} \cdot 1}{\left(s \cdot \left(\left(x \cdot x\right) \cdot c\right)\right) \cdot c} \]

   9. frac-times N/A

      \[\leadsto \frac{\frac{1}{s}}{s \cdot \left(\left(x \cdot x\right) \cdot c\right)} \cdot \color{blue}{\frac{1}{c}} \]

   10. associate-/r* N/A

       \[\leadsto \frac{1}{s \cdot \left(s \cdot \left(\left(x \cdot x\right) \cdot c\right)\right)} \cdot \frac{\color{blue}{1}}{c} \]

   11. associate-*l* N/A

       \[\leadsto \frac{1}{\left(s \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)} \cdot \frac{1}{c} \]

   12. *-commutative N/A

       \[\leadsto \frac{1}{c} \cdot \color{blue}{\frac{1}{\left(s \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)}} \]

   13. div-inv N/A

       \[\leadsto \frac{\frac{1}{c}}{\color{blue}{\left(s \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)}} \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{c}\right), \color{blue}{\left(\left(s \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)\right)}\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\color{blue}{\left(s \cdot s\right)} \cdot \left(\left(x \cdot x\right) \cdot c\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{c}\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right) \cdot c\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right) \]

   22. *-lowering-*.f64 63.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 63.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{c}}{c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(\left(s \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

   3. remove-double-div N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(x \cdot \frac{1}{\color{blue}{\frac{1}{s \cdot x}}}\right)\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \left(\frac{x}{\color{blue}{\frac{1}{s \cdot x}}}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1}{s \cdot x}\right)}\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 70.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 70.2%

   \[\leadsto \frac{\frac{1}{c}}{c \cdot \left(s \cdot \color{blue}{\frac{x}{\frac{1}{x \cdot s}}}\right)} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \left(\frac{s \cdot x}{\color{blue}{\frac{1}{x \cdot s}}}\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \left(\frac{x \cdot s}{\frac{\color{blue}{1}}{x \cdot s}}\right)\right)\right) \]

    3. associate-/r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \left(\frac{x \cdot s}{1} \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)\right) \]

    4. /-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \left(\left(x \cdot s\right) \cdot \left(\color{blue}{x} \cdot s\right)\right)\right)\right) \]

    5. swap-sqr N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \left(\left(\left(x \cdot x\right) \cdot s\right) \cdot \color{blue}{s}\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \left(\left(s \cdot \left(x \cdot x\right)\right) \cdot s\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(s \cdot \left(x \cdot x\right)\right), \color{blue}{s}\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot s\right), s\right)\right)\right) \]

    10. associate-*r* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot s\right)\right), s\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot s\right)\right), s\right)\right)\right) \]

    12. *-lowering-*.f64 70.2%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, c\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, s\right)\right), s\right)\right)\right) \]

11. Applied egg-rr 70.2%

    \[\leadsto \frac{\frac{1}{c}}{c \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot s\right)\right) \cdot s\right)}} \]

12. Final simplification 70.2%

    \[\leadsto \frac{\frac{1}{c}}{c \cdot \left(s \cdot \left(x \cdot \left(x \cdot s\right)\right)\right)} \]
13. Add Preprocessing

Alternative 12: 26.3% accurate, 34.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{-2}{c\_m \cdot s\_m}}{c\_m \cdot s\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m) :precision binary64 (/ (/ -2.0 (* c_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) {
	return (-2.0 / (c_m * s_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 = ((-2.0d0) / (c_m * s_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 (-2.0 / (c_m * s_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 (-2.0 / (c_m * s_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(-2.0 / Float64(c_m * s_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 = (-2.0 / (c_m * s_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[(-2.0 / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left(\left({s}^{2} \cdot x\right) \cdot x\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   4. pow-prod-down N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot \left(\color{blue}{x} \cdot x\right)\right)\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(c \cdot s\right)}^{2} \cdot {x}^{\color{blue}{2}}\right)\right) \]

   6. pow-prod-down N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{2}}\right)\right) \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\left(\left(c \cdot s\right) \cdot x\right), \color{blue}{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(c \cdot s\right), x\right), 2\right)\right) \]

   9. *-lowering-*.f64 96.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, s\right), x\right), 2\right)\right) \]

4. Applied egg-rr 96.9%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot s\right) \cdot x}\right), \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\cos \left(x + x\right), \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x + x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\left(\color{blue}{c} \cdot s\right) \cdot x\right)\right) \]

   7. count-2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(x \cdot 2\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \left(c \cdot \left(s \cdot x\right)\right)\right), \left(\left(c \cdot \color{blue}{s}\right) \cdot x\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(c, \left(s \cdot x\right)\right)\right), \left(\left(c \cdot \color{blue}{s}\right) \cdot x\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(\left(c \cdot s\right) \cdot x\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot x\right)}\right)\right) \]

   15. *-lowering-*.f64 98.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, x\right)\right)\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{x}\right)\right)\right) \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

9. Simplified 47.6%

   \[\leadsto \color{blue}{\frac{\left(1 + \left(x \cdot x\right) \cdot -2\right) \cdot \frac{1}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}}{x \cdot x}} \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \frac{-2}{\left(c \cdot c\right) \cdot {\color{blue}{s}}^{2}} \]

    2. unpow2 N/A

       \[\leadsto \frac{-2}{\left(c \cdot c\right) \cdot \left(s \cdot \color{blue}{s}\right)} \]

    3. unswap-sqr N/A

       \[\leadsto \frac{-2}{\left(c \cdot s\right) \cdot \color{blue}{\left(c \cdot s\right)}} \]

    4. associate-/r* N/A

       \[\leadsto \frac{\frac{-2}{c \cdot s}}{\color{blue}{c \cdot s}} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{neg}\left(2\right)}{c \cdot s}}{c \cdot s} \]

    6. distribute-neg-frac N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{2}{c \cdot s}\right)}{\color{blue}{c} \cdot s} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{2 \cdot 1}{c \cdot s}\right)}{c \cdot s} \]

    8. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \frac{1}{c \cdot s}\right)}{c \cdot s} \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(2 \cdot \frac{1}{c \cdot s}\right)\right), \color{blue}{\left(c \cdot s\right)}\right) \]

    10. associate-*r/ N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2 \cdot 1}{c \cdot s}\right)\right), \left(c \cdot s\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{c \cdot s}\right)\right), \left(c \cdot s\right)\right) \]

    12. distribute-neg-frac N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(2\right)}{c \cdot s}\right), \left(\color{blue}{c} \cdot s\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{c \cdot s}\right), \left(c \cdot s\right)\right) \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \left(c \cdot s\right)\right), \left(\color{blue}{c} \cdot s\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \left(c \cdot s\right)\right) \]

    16. *-lowering-*.f64 30.1%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

12. Simplified 30.1%

    \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))