mixedcos

Percentage Accurate: 66.2% → 97.5%

Time: 12.4s

Alternatives: 16

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x\_m \cdot x\_m\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{\cos \left(x\_m \cdot 2\right)}{x\_m}}{c\_m}}{s\_m}}{x\_m \cdot c\_m}}{s\_m}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* x_m (* s_m c_m))))
   (if (<= x_m 2.5e-5)
     (/ (+ 1.0 (* -2.0 (* x_m x_m))) (* t_0 t_0))
     (/ (/ (/ (/ (/ (cos (* x_m 2.0)) x_m) c_m) s_m) (* x_m c_m)) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = ((((cos((x_m * 2.0)) / x_m) / c_m) / s_m) / (x_m * c_m)) / s_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (s_m * c_m)
    if (x_m <= 2.5d-5) then
        tmp = (1.0d0 + ((-2.0d0) * (x_m * x_m))) / (t_0 * t_0)
    else
        tmp = ((((cos((x_m * 2.0d0)) / x_m) / c_m) / s_m) / (x_m * c_m)) / s_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = ((((Math.cos((x_m * 2.0)) / x_m) / c_m) / s_m) / (x_m * c_m)) / s_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = x_m * (s_m * c_m)
	tmp = 0
	if x_m <= 2.5e-5:
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0)
	else:
		tmp = ((((math.cos((x_m * 2.0)) / x_m) / c_m) / s_m) / (x_m * c_m)) / s_m
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(x_m * Float64(s_m * c_m))
	tmp = 0.0
	if (x_m <= 2.5e-5)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x_m * x_m))) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(Float64(x_m * 2.0)) / x_m) / c_m) / s_m) / Float64(x_m * 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 = x_m * (s_m * c_m);
	tmp = 0.0;
	if (x_m <= 2.5e-5)
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	else
		tmp = ((((cos((x_m * 2.0)) / x_m) / c_m) / s_m) / (x_m * c_m)) / s_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.5e-5], N[(N[(1.0 + N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000012e-5

   1. Initial program 68.9%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 49.8%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. swap-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l/ N/A

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. associate-*r* N/A

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

      18. associate-*r* N/A

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

   9. Applied egg-rr 76.6%

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

   if 2.50000000000000012e-5 < x

   1. Initial program 55.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.8%

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

   3. Simplified 81.8%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. count-2 N/A

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

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

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

      11. count-2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

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

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

   8. Applied egg-rr 88.1%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. count-2 N/A

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

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

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

      9. count-2 N/A

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

      10. *-lowering-*.f64 90.9%

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

   10. Applied egg-rr 90.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{c}}{s}}{x \cdot c}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x\_m \cdot x\_m\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{x\_m \cdot c\_m}}{s\_m}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* x_m (* s_m c_m))))
   (if (<= x_m 2.5e-5)
     (/ (+ 1.0 (* -2.0 (* x_m x_m))) (* t_0 t_0))
     (/ (/ (/ (cos (* x_m 2.0)) t_0) (* x_m c_m)) s_m))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = ((cos((x_m * 2.0)) / t_0) / (x_m * c_m)) / s_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (s_m * c_m)
    if (x_m <= 2.5d-5) then
        tmp = (1.0d0 + ((-2.0d0) * (x_m * x_m))) / (t_0 * t_0)
    else
        tmp = ((cos((x_m * 2.0d0)) / t_0) / (x_m * c_m)) / s_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = ((Math.cos((x_m * 2.0)) / t_0) / (x_m * c_m)) / s_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = x_m * (s_m * c_m)
	tmp = 0
	if x_m <= 2.5e-5:
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0)
	else:
		tmp = ((math.cos((x_m * 2.0)) / t_0) / (x_m * c_m)) / s_m
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(x_m * Float64(s_m * c_m))
	tmp = 0.0
	if (x_m <= 2.5e-5)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x_m * x_m))) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / Float64(x_m * 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 = x_m * (s_m * c_m);
	tmp = 0.0;
	if (x_m <= 2.5e-5)
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	else
		tmp = ((cos((x_m * 2.0)) / t_0) / (x_m * c_m)) / s_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.5e-5], N[(N[(1.0 + N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000012e-5

   1. Initial program 68.9%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 49.8%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. swap-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l/ N/A

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. associate-*r* N/A

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

      18. associate-*r* N/A

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

   9. Applied egg-rr 76.6%

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

   if 2.50000000000000012e-5 < x

   1. Initial program 55.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.8%

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

   3. Simplified 81.8%

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

      1. associate-/r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. count-2 N/A

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

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

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

      11. count-2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

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

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

   8. Applied egg-rr 88.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot c}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(x\_m \cdot 2\right)}{x\_m}\\ \mathbf{if}\;c\_m \leq 10^{-94}:\\ \;\;\;\;\frac{t\_0}{\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{s\_m}}{\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (/ (cos (* x_m 2.0)) x_m)))
   (if (<= c_m 1e-94)
     (/ t_0 (* (* s_m c_m) (* x_m (* s_m c_m))))
     (/ (/ t_0 s_m) (* (* s_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)) / x_m;
	double tmp;
	if (c_m <= 1e-94) {
		tmp = t_0 / ((s_m * c_m) * (x_m * (s_m * c_m)));
	} else {
		tmp = (t_0 / s_m) / ((s_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) :: tmp
    t_0 = cos((x_m * 2.0d0)) / x_m
    if (c_m <= 1d-94) then
        tmp = t_0 / ((s_m * c_m) * (x_m * (s_m * c_m)))
    else
        tmp = (t_0 / s_m) / ((s_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)) / x_m;
	double tmp;
	if (c_m <= 1e-94) {
		tmp = t_0 / ((s_m * c_m) * (x_m * (s_m * c_m)));
	} else {
		tmp = (t_0 / s_m) / ((s_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)) / x_m
	tmp = 0
	if c_m <= 1e-94:
		tmp = t_0 / ((s_m * c_m) * (x_m * (s_m * c_m)))
	else:
		tmp = (t_0 / s_m) / ((s_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 = Float64(cos(Float64(x_m * 2.0)) / x_m)
	tmp = 0.0
	if (c_m <= 1e-94)
		tmp = Float64(t_0 / Float64(Float64(s_m * c_m) * Float64(x_m * Float64(s_m * c_m))));
	else
		tmp = Float64(Float64(t_0 / s_m) / Float64(Float64(s_m * 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)) / x_m;
	tmp = 0.0;
	if (c_m <= 1e-94)
		tmp = t_0 / ((s_m * c_m) * (x_m * (s_m * c_m)));
	else
		tmp = (t_0 / s_m) / ((s_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[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[c$95$m, 1e-94], N[(t$95$0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / s$95$m), $MachinePrecision] / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 9.9999999999999996e-95

   1. Initial program 65.1%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 85.2%

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

   3. Simplified 85.2%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. count-2 N/A

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

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

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

      6. count-2 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 96.0%

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

   6. Applied egg-rr 96.0%

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

   if 9.9999999999999996e-95 < c

   1. Initial program 66.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 84.8%

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

   3. Simplified 84.8%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

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

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

      8. count-2 N/A

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

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

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

      10. count-2 N/A

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

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

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-commutative N/A

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

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

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

      17. *-lowering-*.f64 95.1%

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

   6. Applied egg-rr 95.1%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 10^{-94}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{s}}{\left(s \cdot c\right) \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{x\_m}}{\left(s\_m \cdot c\_m\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* x_m (* s_m c_m))))
   (if (<= x_m 5.2e-22)
     (/ 1.0 (* t_0 t_0))
     (/ (/ (cos (* x_m 2.0)) x_m) (* (* s_m c_m) t_0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 5.2e-22) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = (cos((x_m * 2.0)) / x_m) / ((s_m * c_m) * t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (s_m * c_m)
    if (x_m <= 5.2d-22) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = (cos((x_m * 2.0d0)) / x_m) / ((s_m * c_m) * t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 5.2e-22) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / x_m) / ((s_m * c_m) * t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = x_m * (s_m * c_m)
	tmp = 0
	if x_m <= 5.2e-22:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = (math.cos((x_m * 2.0)) / x_m) / ((s_m * c_m) * t_0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(x_m * Float64(s_m * c_m))
	tmp = 0.0
	if (x_m <= 5.2e-22)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / x_m) / Float64(Float64(s_m * c_m) * t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = x_m * (s_m * c_m);
	tmp = 0.0;
	if (x_m <= 5.2e-22)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = (cos((x_m * 2.0)) / x_m) / ((s_m * c_m) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5.2e-22], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision] / N[(N[(s$95$m * c$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2e-22

   1. Initial program 67.5%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 76.9%

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

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 86.8%

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

   9. Applied egg-rr 86.8%

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

   if 5.2e-22 < x

   1. Initial program 60.7%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 83.9%

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

   3. Simplified 83.9%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. count-2 N/A

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

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

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

      6. count-2 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 95.6%

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

   6. Applied egg-rr 95.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x\_m \cdot x\_m\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{x\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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 (* x_m (* s_m c_m))))
   (if (<= x_m 2.5e-5)
     (/ (+ 1.0 (* -2.0 (* x_m x_m))) (* t_0 t_0))
     (/ (cos (* x_m 2.0)) (* x_m (* (* x_m c_m) (* s_m (* s_m c_m))))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = cos((x_m * 2.0)) / (x_m * ((x_m * c_m) * (s_m * (s_m * c_m))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (s_m * c_m)
    if (x_m <= 2.5d-5) then
        tmp = (1.0d0 + ((-2.0d0) * (x_m * x_m))) / (t_0 * t_0)
    else
        tmp = cos((x_m * 2.0d0)) / (x_m * ((x_m * c_m) * (s_m * (s_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 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / (x_m * ((x_m * c_m) * (s_m * (s_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 = x_m * (s_m * c_m)
	tmp = 0
	if x_m <= 2.5e-5:
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0)
	else:
		tmp = math.cos((x_m * 2.0)) / (x_m * ((x_m * c_m) * (s_m * (s_m * c_m))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(x_m * Float64(s_m * c_m))
	tmp = 0.0
	if (x_m <= 2.5e-5)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x_m * x_m))) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(x_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(s_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 = x_m * (s_m * c_m);
	tmp = 0.0;
	if (x_m <= 2.5e-5)
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	else
		tmp = cos((x_m * 2.0)) / (x_m * ((x_m * c_m) * (s_m * (s_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[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.5e-5], N[(N[(1.0 + N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000012e-5

   1. Initial program 68.9%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 49.8%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. swap-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l/ N/A

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. associate-*r* N/A

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

      18. associate-*r* N/A

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

   9. Applied egg-rr 76.6%

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

   if 2.50000000000000012e-5 < x

   1. Initial program 55.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.8%

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

   3. Simplified 81.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 84.9%

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

   6. Applied egg-rr 84.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x\_m \cdot x\_m\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{s\_m \cdot \left(x\_m \cdot \left(\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

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 (* x_m (* s_m c_m))))
   (if (<= x_m 2.5e-5)
     (/ (+ 1.0 (* -2.0 (* x_m x_m))) (* t_0 t_0))
     (/ (cos (* x_m 2.0)) (* s_m (* x_m (* (* s_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 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = cos((x_m * 2.0)) / (s_m * (x_m * ((s_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) :: tmp
    t_0 = x_m * (s_m * c_m)
    if (x_m <= 2.5d-5) then
        tmp = (1.0d0 + ((-2.0d0) * (x_m * x_m))) / (t_0 * t_0)
    else
        tmp = cos((x_m * 2.0d0)) / (s_m * (x_m * ((s_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 = x_m * (s_m * c_m);
	double tmp;
	if (x_m <= 2.5e-5) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / (s_m * (x_m * ((s_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 = x_m * (s_m * c_m)
	tmp = 0
	if x_m <= 2.5e-5:
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0)
	else:
		tmp = math.cos((x_m * 2.0)) / (s_m * (x_m * ((s_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 = Float64(x_m * Float64(s_m * c_m))
	tmp = 0.0
	if (x_m <= 2.5e-5)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x_m * x_m))) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(s_m * Float64(x_m * Float64(Float64(s_m * 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 = x_m * (s_m * c_m);
	tmp = 0.0;
	if (x_m <= 2.5e-5)
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / (t_0 * t_0);
	else
		tmp = cos((x_m * 2.0)) / (s_m * (x_m * ((s_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[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.5e-5], N[(N[(1.0 + N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(x$95$m * N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000012e-5

   1. Initial program 68.9%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 49.8%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. swap-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l/ N/A

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. associate-*r* N/A

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

      18. associate-*r* N/A

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

   9. Applied egg-rr 76.6%

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

   if 2.50000000000000012e-5 < x

   1. Initial program 55.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 81.8%

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

   3. Simplified 81.8%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-lowering-*.f64 84.9%

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

   6. Applied egg-rr 84.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x \cdot x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + -2 \cdot \left(x\_m \cdot x\_m\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* x_m (* s_m c_m))) (t_1 (* t_0 t_0)))
   (if (<= x_m 8.5e+67) (/ (+ 1.0 (* -2.0 (* x_m x_m))) t_1) (/ 1.0 t_1))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	double t_1 = t_0 * t_0;
	double tmp;
	if (x_m <= 8.5e+67) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / t_1;
	} else {
		tmp = 1.0 / t_1;
	}
	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 = x_m * (s_m * c_m)
    t_1 = t_0 * t_0
    if (x_m <= 8.5d+67) then
        tmp = (1.0d0 + ((-2.0d0) * (x_m * x_m))) / t_1
    else
        tmp = 1.0d0 / t_1
    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 = x_m * (s_m * c_m);
	double t_1 = t_0 * t_0;
	double tmp;
	if (x_m <= 8.5e+67) {
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / t_1;
	} else {
		tmp = 1.0 / t_1;
	}
	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 = x_m * (s_m * c_m)
	t_1 = t_0 * t_0
	tmp = 0
	if x_m <= 8.5e+67:
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / t_1
	else:
		tmp = 1.0 / t_1
	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(x_m * Float64(s_m * c_m))
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (x_m <= 8.5e+67)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(x_m * x_m))) / t_1);
	else
		tmp = Float64(1.0 / t_1);
	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 = x_m * (s_m * c_m);
	t_1 = t_0 * t_0;
	tmp = 0.0;
	if (x_m <= 8.5e+67)
		tmp = (1.0 + (-2.0 * (x_m * x_m))) / t_1;
	else
		tmp = 1.0 / t_1;
	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[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 8.5e+67], N[(N[(1.0 + N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000038e67

   1. Initial program 69.2%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0

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

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

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

   7. Simplified 49.2%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. swap-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l/ N/A

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

      9. associate-*r* N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. associate-*r* N/A

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

      18. associate-*r* N/A

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

   9. Applied egg-rr 74.1%

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

   if 8.50000000000000038e67 < x

   1. Initial program 49.7%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 65.4%

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

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.5%

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

   9. Applied egg-rr 67.5%

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

Alternative 8: 73.5% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;c\_m \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(x\_m \cdot \left(s\_m \cdot \left(c\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= c_m 2.2e-94)
   (/ 1.0 (* x_m (* x_m (* s_m (* c_m (* s_m c_m))))))
   (/ 1.0 (* x_m (* s_m (* s_m (* x_m (* c_m c_m))))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (c_m <= 2.2e-94) {
		tmp = 1.0 / (x_m * (x_m * (s_m * (c_m * (s_m * c_m)))));
	} else {
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (c_m <= 2.2d-94) then
        tmp = 1.0d0 / (x_m * (x_m * (s_m * (c_m * (s_m * c_m)))))
    else
        tmp = 1.0d0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (c_m <= 2.2e-94) {
		tmp = 1.0 / (x_m * (x_m * (s_m * (c_m * (s_m * c_m)))));
	} else {
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if c_m <= 2.2e-94:
		tmp = 1.0 / (x_m * (x_m * (s_m * (c_m * (s_m * c_m)))))
	else:
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (c_m <= 2.2e-94)
		tmp = Float64(1.0 / Float64(x_m * Float64(x_m * Float64(s_m * Float64(c_m * Float64(s_m * c_m))))));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(s_m * Float64(s_m * Float64(x_m * Float64(c_m * c_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (c_m <= 2.2e-94)
		tmp = 1.0 / (x_m * (x_m * (s_m * (c_m * (s_m * c_m)))));
	else
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[c$95$m, 2.2e-94], N[(1.0 / N[(x$95$m * N[(x$95$m * N[(s$95$m * N[(c$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(s$95$m * N[(s$95$m * N[(x$95$m * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.20000000000000001e-94

   1. Initial program 65.1%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 72.7%

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

   if 2.20000000000000001e-94 < c

   1. Initial program 66.6%

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 84.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 71.1%

      \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot x\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left({s}^{2} \cdot x\right) \cdot \color{blue}{{c}^{2}}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(s \cdot s\right) \cdot x\right) \cdot {c}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot {c}^{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \left({c}^{2} \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot {c}^{2}\right) \cdot x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \color{blue}{\left({c}^{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot x\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left({c}^{2}\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(c \cdot c\right), x\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 69.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), x\right)\right)\right)\right)\right) \]

   10. Simplified 69.0%

       \[\leadsto \frac{1}{x \cdot \color{blue}{\left(s \cdot \left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(s \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;c\_m \leq 3.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

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
 (if (<= c_m 3.4e-149)
   (/ 1.0 (* c_m (* c_m (* s_m (* (* x_m x_m) s_m)))))
   (/ 1.0 (* x_m (* s_m (* s_m (* x_m (* c_m c_m))))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (c_m <= 3.4e-149) {
		tmp = 1.0 / (c_m * (c_m * (s_m * ((x_m * x_m) * s_m))));
	} else {
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (c_m <= 3.4d-149) then
        tmp = 1.0d0 / (c_m * (c_m * (s_m * ((x_m * x_m) * s_m))))
    else
        tmp = 1.0d0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (c_m <= 3.4e-149) {
		tmp = 1.0 / (c_m * (c_m * (s_m * ((x_m * x_m) * s_m))));
	} else {
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if c_m <= 3.4e-149:
		tmp = 1.0 / (c_m * (c_m * (s_m * ((x_m * x_m) * s_m))))
	else:
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (c_m <= 3.4e-149)
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(s_m * Float64(Float64(x_m * x_m) * s_m)))));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(s_m * Float64(s_m * Float64(x_m * Float64(c_m * c_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (c_m <= 3.4e-149)
		tmp = 1.0 / (c_m * (c_m * (s_m * ((x_m * x_m) * s_m))));
	else
		tmp = 1.0 / (x_m * (s_m * (s_m * (x_m * (c_m * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[c$95$m, 3.4e-149], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(s$95$m * N[(s$95$m * N[(x$95$m * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 3.3999999999999999e-149

   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. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 85.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 72.8%

      \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 64.0%

       \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]

   if 3.3999999999999999e-149 < c

   1. Initial program 66.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 84.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 70.9%

      \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot x\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left({s}^{2} \cdot x\right) \cdot \color{blue}{{c}^{2}}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(s \cdot s\right) \cdot x\right) \cdot {c}^{2}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot {\color{blue}{c}}^{2}\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot {c}^{2}\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(s \cdot \left({c}^{2} \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot \left(s \cdot x\right)\right)}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(s \cdot {c}^{2}\right) \cdot x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(s \cdot \color{blue}{\left({c}^{2} \cdot x\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot x\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left({c}^{2}\right), \color{blue}{x}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\left(c \cdot c\right), x\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 68.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), x\right)\right)\right)\right)\right) \]

   10. Simplified 68.2%

       \[\leadsto \frac{1}{x \cdot \color{blue}{\left(s \cdot \left(s \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 3.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(\left(x \cdot x\right) \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(s \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.0% accurate, 19.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{2}{s\_m}}{s\_m \cdot \left(c\_m \cdot \left(0 - c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 1.45e+20)
   (/ (/ 2.0 s_m) (* s_m (* c_m (- 0.0 c_m))))
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.45e+20) {
		tmp = (2.0 / s_m) / (s_m * (c_m * (0.0 - c_m)));
	} else {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 1.45d+20) then
        tmp = (2.0d0 / s_m) / (s_m * (c_m * (0.0d0 - c_m)))
    else
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 1.45e+20) {
		tmp = (2.0 / s_m) / (s_m * (c_m * (0.0 - c_m)));
	} else {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 1.45e+20:
		tmp = (2.0 / s_m) / (s_m * (c_m * (0.0 - c_m)))
	else:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 1.45e+20)
		tmp = Float64(Float64(2.0 / s_m) / Float64(s_m * Float64(c_m * Float64(0.0 - c_m))));
	else
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 1.45e+20)
		tmp = (2.0 / s_m) / (s_m * (c_m * (0.0 - c_m)));
	else
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 1.45e+20], N[(N[(2.0 / s$95$m), $MachinePrecision] / N[(s$95$m * N[(c$95$m * N[(0.0 - c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e20

   1. Initial program 68.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 86.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   7. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{\left(1 + x \cdot \left(x \cdot -2\right)\right) \cdot \frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. 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 22.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

   10. Simplified 22.7%

       \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
   11. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(\color{blue}{c} \cdot s\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot s\right) \cdot \left(\color{blue}{s} \cdot c\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot c\right) \cdot s\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(\left(c \cdot s\right) \cdot s\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 25.2%

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   12. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\frac{-2}{c}}{\color{blue}{c \cdot \left(s \cdot s\right)}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\frac{-2}{c}}{\left(c \cdot s\right) \cdot \color{blue}{s}} \]

       3. associate-/l/ N/A

          \[\leadsto \frac{\frac{\frac{-2}{c}}{s}}{\color{blue}{c \cdot s}} \]

       4. div-inv N/A

          \[\leadsto \frac{\frac{-2}{c} \cdot \frac{1}{s}}{\color{blue}{c} \cdot s} \]

       5. *-commutative N/A

          \[\leadsto \frac{\frac{1}{s} \cdot \frac{-2}{c}}{\color{blue}{c} \cdot s} \]

       6. associate-*r/ N/A

          \[\leadsto \frac{\frac{\frac{1}{s} \cdot -2}{c}}{\color{blue}{c} \cdot s} \]

       7. associate-/l/ N/A

          \[\leadsto \frac{\frac{1}{s} \cdot -2}{\color{blue}{\left(c \cdot s\right) \cdot c}} \]

       8. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{s} \cdot -2\right)}{\color{blue}{\mathsf{neg}\left(\left(c \cdot s\right) \cdot c\right)}} \]

       9. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{s} \cdot -2\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}} \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{s} \cdot -2\right)\right), \color{blue}{\left(\left(c \cdot s\right) \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(-2 \cdot \frac{1}{s}\right)\right), \left(\left(\color{blue}{c} \cdot s\right) \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

       12. distribute-lft-neg-in N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot \frac{1}{s}\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{1}{s}\right), \left(\left(\color{blue}{c} \cdot s\right) \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{s}\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \left(\left(s \cdot c\right) \cdot \left(\mathsf{neg}\left(\color{blue}{c}\right)\right)\right)\right) \]

       17. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \left(s \cdot \color{blue}{\left(c \cdot \left(\mathsf{neg}\left(c\right)\right)\right)}\right)\right) \]

       18. distribute-rgt-neg-in N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \left(s \cdot \left(\mathsf{neg}\left(c \cdot c\right)\right)\right)\right) \]

       19. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \mathsf{*.f64}\left(s, \color{blue}{\left(\mathsf{neg}\left(c \cdot c\right)\right)}\right)\right) \]

       20. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \mathsf{*.f64}\left(s, \left(0 - \color{blue}{c \cdot c}\right)\right)\right) \]

       21. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \mathsf{*.f64}\left(s, \mathsf{\_.f64}\left(0, \color{blue}{\left(c \cdot c\right)}\right)\right)\right) \]

       22. *-lowering-*.f64 36.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, s\right), \mathsf{*.f64}\left(s, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, \color{blue}{c}\right)\right)\right)\right) \]

   14. Applied egg-rr 36.9%

       \[\leadsto \color{blue}{\frac{\frac{2}{s}}{s \cdot \left(0 - c \cdot c\right)}} \]

   if 1.45e20 < x

   1. Initial program 55.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   7. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(1 + x \cdot \left(x \cdot -2\right)\right) \cdot \frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. 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 33.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

   10. Simplified 33.2%

       \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
   11. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(\color{blue}{c} \cdot s\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot s\right) \cdot \left(\color{blue}{s} \cdot c\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot c\right) \cdot s\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(\left(c \cdot s\right) \cdot s\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 38.0%

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   12. Applied egg-rr 38.0%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{2}{s}}{s \cdot \left(c \cdot \left(0 - c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.1% accurate, 19.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 1.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{t\_0 \cdot \left(0 - c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c\_m \cdot t\_0}\\ \end{array} \end{array} \]

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 (* c_m (* s_m s_m))))
   (if (<= x_m 1.45e+20) (/ 2.0 (* t_0 (- 0.0 c_m))) (/ -2.0 (* c_m 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 * (s_m * s_m);
	double tmp;
	if (x_m <= 1.45e+20) {
		tmp = 2.0 / (t_0 * (0.0 - c_m));
	} else {
		tmp = -2.0 / (c_m * t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * s_m)
    if (x_m <= 1.45d+20) then
        tmp = 2.0d0 / (t_0 * (0.0d0 - c_m))
    else
        tmp = (-2.0d0) / (c_m * t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * s_m);
	double tmp;
	if (x_m <= 1.45e+20) {
		tmp = 2.0 / (t_0 * (0.0 - c_m));
	} else {
		tmp = -2.0 / (c_m * t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * s_m)
	tmp = 0
	if x_m <= 1.45e+20:
		tmp = 2.0 / (t_0 * (0.0 - c_m))
	else:
		tmp = -2.0 / (c_m * t_0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * s_m))
	tmp = 0.0
	if (x_m <= 1.45e+20)
		tmp = Float64(2.0 / Float64(t_0 * Float64(0.0 - c_m)));
	else
		tmp = Float64(-2.0 / Float64(c_m * t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.45e+20)
		tmp = 2.0 / (t_0 * (0.0 - c_m));
	else
		tmp = -2.0 / (c_m * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.45e+20], N[(2.0 / N[(t$95$0 * N[(0.0 - c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(c$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e20

   1. Initial program 68.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 86.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   7. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{\left(1 + x \cdot \left(x \cdot -2\right)\right) \cdot \frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. 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 22.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

   10. Simplified 22.7%

       \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
   11. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(\color{blue}{c} \cdot s\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

       4. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(-2\right)}{\color{blue}{\mathsf{neg}\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{neg}\left(\color{blue}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\right)} \]

       6. sub0-neg N/A

          \[\leadsto \frac{2}{0 - \color{blue}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{2}{0 \cdot 0 - \color{blue}{\left(s \cdot c\right)} \cdot \left(s \cdot c\right)} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(0 \cdot 0 - \left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(0 - \color{blue}{\left(s \cdot c\right)} \cdot \left(s \cdot c\right)\right)\right) \]

       10. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \left(\left(c \cdot s\right) \cdot \left(\color{blue}{s} \cdot c\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \left(c \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot c\right) \cdot s\right)}\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, \left(\left(c \cdot s\right) \cdot s\right)\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right) \]

       18. *-lowering-*.f64 47.9%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{\frac{2}{0 - c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if 1.45e20 < x

   1. Initial program 55.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   7. Simplified 17.8%

      \[\leadsto \color{blue}{\frac{\left(1 + x \cdot \left(x \cdot -2\right)\right) \cdot \frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. 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 33.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

   10. Simplified 33.2%

       \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
   11. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(\color{blue}{c} \cdot s\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot s\right) \cdot \left(\color{blue}{s} \cdot c\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot c\right) \cdot s\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(\left(c \cdot s\right) \cdot s\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 38.0%

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   12. Applied egg-rr 38.0%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(c \cdot \left(s \cdot s\right)\right) \cdot \left(0 - c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.1% accurate, 19.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(0 - c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 4.8e-18)
   (/ 2.0 (* (* s_m c_m) (* s_m (- 0.0 c_m))))
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 4.8e-18) {
		tmp = 2.0 / ((s_m * c_m) * (s_m * (0.0 - c_m)));
	} else {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 4.8d-18) then
        tmp = 2.0d0 / ((s_m * c_m) * (s_m * (0.0d0 - c_m)))
    else
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 4.8e-18) {
		tmp = 2.0 / ((s_m * c_m) * (s_m * (0.0 - c_m)));
	} else {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 4.8e-18:
		tmp = 2.0 / ((s_m * c_m) * (s_m * (0.0 - c_m)))
	else:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 4.8e-18)
		tmp = Float64(2.0 / Float64(Float64(s_m * c_m) * Float64(s_m * Float64(0.0 - c_m))));
	else
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 4.8e-18)
		tmp = 2.0 / ((s_m * c_m) * (s_m * (0.0 - c_m)));
	else
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 4.8e-18], N[(2.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(0.0 - c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999988e-18

   1. Initial program 67.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 85.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   7. Simplified 47.9%

      \[\leadsto \color{blue}{\frac{\left(1 + x \cdot \left(x \cdot -2\right)\right) \cdot \frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. 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 23.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

   10. Simplified 23.3%

       \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
   11. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \frac{-2}{c \cdot s} \cdot \color{blue}{\frac{1}{c \cdot s}} \]

       2. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(c \cdot s\right)} \cdot \frac{\color{blue}{1}}{c \cdot s} \]

       3. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{neg}\left(c \cdot s\right)} \cdot \frac{1}{c \cdot s} \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{neg}\left(s \cdot c\right)} \cdot \frac{1}{c \cdot s} \]

       5. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{neg}\left(s \cdot c\right)} \cdot \frac{1}{s \cdot \color{blue}{c}} \]

       6. frac-times N/A

          \[\leadsto \frac{2 \cdot 1}{\color{blue}{\left(\mathsf{neg}\left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{neg}\left(s \cdot c\right)\right)} \cdot \left(s \cdot c\right)} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\mathsf{neg}\left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)\right)}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(s \cdot c\right)\right), \color{blue}{\left(s \cdot c\right)}\right)\right) \]

       10. sub0-neg N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(0 - s \cdot c\right), \left(\color{blue}{s} \cdot c\right)\right)\right) \]

       11. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(s \cdot c\right)\right), \left(\color{blue}{s} \cdot c\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot s\right)\right), \left(s \cdot c\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, s\right)\right), \left(s \cdot c\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, s\right)\right), \left(c \cdot \color{blue}{s}\right)\right)\right) \]

       15. *-lowering-*.f64 26.4%

           \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right) \]

   12. Applied egg-rr 26.4%

       \[\leadsto \color{blue}{\frac{2}{\left(0 - c \cdot s\right) \cdot \left(c \cdot s\right)}} \]

   if 4.79999999999999988e-18 < x

   1. Initial program 60.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 83.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   3. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

   7. Simplified 25.5%

      \[\leadsto \color{blue}{\frac{\left(1 + x \cdot \left(x \cdot -2\right)\right) \cdot \frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. 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%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

   10. Simplified 30.1%

       \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
   11. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(\color{blue}{c} \cdot s\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot s\right) \cdot \left(\color{blue}{s} \cdot c\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot c\right) \cdot s\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(\left(c \cdot s\right) \cdot s\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       12. *-lowering-*.f64 34.1%

           \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   12. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(s \cdot c\right) \cdot \left(s \cdot \left(0 - c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(s\_m \cdot c\_m\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* x_m (* s_m c_m)))) (/ 1.0 (* t_0 t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	return 1.0 / (t_0 * t_0);
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = x_m * (s_m * c_m)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = x_m * (s_m * c_m);
	return 1.0 / (t_0 * t_0);
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = x_m * (s_m * c_m)
	return 1.0 / (t_0 * t_0)

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(x_m * Float64(s_m * c_m))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = x_m * (s_m * c_m);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

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. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 85.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 85.1%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 72.0%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)\right)\right)\right) \]

   5. unswap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot \left(s \cdot c\right)\right), \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(s \cdot c\right)\right), \left(\color{blue}{x} \cdot \left(s \cdot c\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \left(x \cdot \left(s \cdot c\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left(s \cdot c\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 79.7%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, c\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{c}\right)\right)\right)\right) \]

9. Applied egg-rr 79.7%

   \[\leadsto \color{blue}{\frac{1}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}} \]
10. Add Preprocessing

Alternative 14: 75.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{x\_m \cdot \left(c\_m \cdot \left(\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot s\_m\right)\right)\right)} \end{array} \]

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 (* x_m (* c_m (* (* s_m c_m) (* x_m s_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 1.0 / (x_m * (c_m * ((s_m * c_m) * (x_m * s_m))));
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (x_m * (c_m * ((s_m * c_m) * (x_m * s_m))))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return 1.0 / (x_m * (c_m * ((s_m * c_m) * (x_m * s_m))));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 1.0 / (x_m * (c_m * ((s_m * c_m) * (x_m * s_m))))

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(1.0 / Float64(x_m * Float64(c_m * Float64(Float64(s_m * c_m) * Float64(x_m * s_m)))))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = 1.0 / (x_m * (c_m * ((s_m * c_m) * (x_m * s_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(1.0 / N[(x$95$m * N[(c$95$m * N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 85.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 85.1%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 72.0%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot s\right) \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot s\right) \cdot \left(\left(c \cdot s\right) \cdot \color{blue}{c}\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot s\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(x \cdot s\right) \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{c}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(x \cdot s\right) \cdot \left(s \cdot c\right)\right), \color{blue}{c}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot s\right), \left(s \cdot c\right)\right), c\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \left(s \cdot c\right)\right), c\right)\right)\right) \]

   8. *-lowering-*.f64 75.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, s\right), \mathsf{*.f64}\left(s, c\right)\right), c\right)\right)\right) \]

9. Applied egg-rr 75.1%

   \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(\left(x \cdot s\right) \cdot \left(s \cdot c\right)\right) \cdot c\right)}} \]

10. Final simplification 75.1%

    \[\leadsto \frac{1}{x \cdot \left(c \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot s\right)\right)\right)} \]
11. Add Preprocessing

Alternative 15: 66.5% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(\left(x\_m \cdot x\_m\right) \cdot s\_m\right)\right)\right)} \end{array} \]

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(1.0 / Float64(c_m * Float64(c_m * Float64(s_m * Float64(Float64(x_m * 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[(1.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 85.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 85.1%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, s\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 72.0%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(c \cdot c\right) \cdot \left(\color{blue}{{s}^{2}} \cdot {x}^{2}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left({s}^{2} \cdot {x}^{2}\right)}\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(\left(s \cdot s\right) \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \color{blue}{\left({x}^{2}\right)}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 64.9%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

10. Simplified 64.9%

    \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]

11. Final simplification 64.9%

    \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(\left(x \cdot x\right) \cdot s\right)\right)\right)} \]
12. Add Preprocessing

Alternative 16: 31.1% accurate, 34.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)} \end{array} \]

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 (/ -2.0 (* c_m (* c_m (* s_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 * (c_m * (s_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 * (c_m * (s_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 * (c_m * (s_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 * (c_m * (s_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(-2.0 / Float64(c_m * Float64(c_m * Float64(s_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 * (c_m * (s_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[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(2 \cdot x\right), \color{blue}{\left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)}\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot x\right)\right), \left(\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({\color{blue}{c}}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(x \cdot \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left({s}^{2} \cdot \color{blue}{x}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({c}^{2} \cdot \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left({c}^{2} \cdot s\right) \cdot \color{blue}{s}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \color{blue}{\left({c}^{2} \cdot s\right)}\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(\left(c \cdot c\right) \cdot s\right)\right)\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \color{blue}{\left(c \cdot s\right)}\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 85.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

3. Simplified 85.1%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

7. Simplified 42.0%

   \[\leadsto \color{blue}{\frac{\left(1 + x \cdot \left(x \cdot -2\right)\right) \cdot \frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
9. 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 25.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, s\right)\right), \mathsf{*.f64}\left(c, \color{blue}{s}\right)\right) \]

10. Simplified 25.1%

    \[\leadsto \color{blue}{\frac{\frac{-2}{c \cdot s}}{c \cdot s}} \]
11. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(\color{blue}{c} \cdot s\right)} \]

    3. *-commutative N/A

       \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left(\left(c \cdot s\right) \cdot \left(\color{blue}{s} \cdot c\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot c\right)\right)}\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \left(c \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{s}\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \color{blue}{\left(\left(s \cdot c\right) \cdot s\right)}\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(\left(c \cdot s\right) \cdot s\right)\right)\right) \]

    10. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

    12. *-lowering-*.f64 28.1%

        \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

12. Applied egg-rr 28.1%

    \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))