mixedcos

Percentage Accurate: 66.7% → 95.1%

Time: 13.6s

Alternatives: 10

Speedup: 24.1×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% 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: 95.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\right)\\ \mathbf{if}\;x\_m \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{elif}\;x\_m \leq 1.1 \cdot 10^{+231}:\\ \;\;\;\;\frac{t\_0}{s \cdot \left(\left(x\_m \cdot \left(x\_m \cdot c\_m\right)\right) \cdot \left(s \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{c\_m \cdot \left(x\_m \cdot c\_m\right)}}{s \cdot \left(x\_m \cdot s\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s)
 :precision binary64
 (let* ((t_0 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s))))
   (if (<= x_m 1.7e-38)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x_m 1.1e+231)
       (/ t_0 (* s (* (* x_m (* x_m c_m)) (* s c_m))))
       (/ (/ t_0 (* c_m (* x_m c_m))) (* s (* x_m s)))))))

C

x_m = fabs(x);
c_m = fabs(c);
assert(x_m < c_m && c_m < s);
double code(double x_m, double c_m, double s) {
	double t_0 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 1.7e-38) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.1e+231) {
		tmp = t_0 / (s * ((x_m * (x_m * c_m)) * (s * c_m)));
	} else {
		tmp = (t_0 / (c_m * (x_m * c_m))) / (s * (x_m * s));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s)
    if (x_m <= 1.7d-38) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x_m <= 1.1d+231) then
        tmp = t_0 / (s * ((x_m * (x_m * c_m)) * (s * c_m)))
    else
        tmp = (t_0 / (c_m * (x_m * c_m))) / (s * (x_m * s))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
assert x_m < c_m && c_m < s;
public static double code(double x_m, double c_m, double s) {
	double t_0 = Math.cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 1.7e-38) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.1e+231) {
		tmp = t_0 / (s * ((x_m * (x_m * c_m)) * (s * c_m)));
	} else {
		tmp = (t_0 / (c_m * (x_m * c_m))) / (s * (x_m * s));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
[x_m, c_m, s] = sort([x_m, c_m, s])
def code(x_m, c_m, s):
	t_0 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s)
	tmp = 0
	if x_m <= 1.7e-38:
		tmp = (1.0 / t_1) / t_1
	elif x_m <= 1.1e+231:
		tmp = t_0 / (s * ((x_m * (x_m * c_m)) * (s * c_m)))
	else:
		tmp = (t_0 / (c_m * (x_m * c_m))) / (s * (x_m * s))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
x_m, c_m, s = sort([x_m, c_m, s])
function code(x_m, c_m, s)
	t_0 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s))
	tmp = 0.0
	if (x_m <= 1.7e-38)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x_m <= 1.1e+231)
		tmp = Float64(t_0 / Float64(s * Float64(Float64(x_m * Float64(x_m * c_m)) * Float64(s * c_m))));
	else
		tmp = Float64(Float64(t_0 / Float64(c_m * Float64(x_m * c_m))) / Float64(s * Float64(x_m * s)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp_2 = code(x_m, c_m, s)
	t_0 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s);
	tmp = 0.0;
	if (x_m <= 1.7e-38)
		tmp = (1.0 / t_1) / t_1;
	elseif (x_m <= 1.1e+231)
		tmp = t_0 / (s * ((x_m * (x_m * c_m)) * (s * c_m)));
	else
		tmp = (t_0 / (c_m * (x_m * c_m))) / (s * (x_m * s));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s_] := Block[{t$95$0 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.7e-38], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 1.1e+231], N[(t$95$0 / N[(s * N[(N[(x$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(c$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(s * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.7000000000000001e-38

   1. Initial program 68.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.7%

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

   5. Simplified 73.7%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-/r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 1.7000000000000001e-38 < x < 1.09999999999999996e231

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 82.1%

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

   4. Applied egg-rr 82.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. *-lowering-*.f64 86.9%

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

   6. Applied egg-rr 86.9%

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

   if 1.09999999999999996e231 < x

   1. Initial program 52.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. *-lowering-*.f64 88.5%

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

   4. Applied egg-rr 88.5%

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

      1. *-rgt-identity N/A

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

      2. associate-*l* N/A

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

      3. unpow-prod-down N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

   6. Applied egg-rr 94.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+231}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.3% accurate, 2.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s))))
   (if (<= x_m 1e-5)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m 2.0)) (* s (* s (* c_m (* x_m (* x_m c_m)))))))))

C

x_m = fabs(x);
c_m = fabs(c);
assert(x_m < c_m && c_m < s);
double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 1e-5) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * 2.0)) / (s * (s * (c_m * (x_m * (x_m * c_m)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s)
    if (x_m <= 1d-5) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * 2.0d0)) / (s * (s * (c_m * (x_m * (x_m * c_m)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
assert x_m < c_m && c_m < s;
public static double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 1e-5) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * 2.0)) / (s * (s * (c_m * (x_m * (x_m * c_m)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
[x_m, c_m, s] = sort([x_m, c_m, s])
def code(x_m, c_m, s):
	t_0 = c_m * (x_m * s)
	tmp = 0
	if x_m <= 1e-5:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * 2.0)) / (s * (s * (c_m * (x_m * (x_m * c_m)))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
x_m, c_m, s = sort([x_m, c_m, s])
function code(x_m, c_m, s)
	t_0 = Float64(c_m * Float64(x_m * s))
	tmp = 0.0
	if (x_m <= 1e-5)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(s * Float64(s * Float64(c_m * Float64(x_m * Float64(x_m * c_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp_2 = code(x_m, c_m, s)
	t_0 = c_m * (x_m * s);
	tmp = 0.0;
	if (x_m <= 1e-5)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * 2.0)) / (s * (s * (c_m * (x_m * (x_m * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1e-5], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(s * N[(c$95$m * N[(x$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000008e-5

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 74.5%

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

   5. Simplified 74.5%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-/r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 86.4%

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

   7. Applied egg-rr 86.4%

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

   if 1.00000000000000008e-5 < x

   1. Initial program 64.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-lowering-*.f64 80.1%

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

   4. Applied egg-rr 80.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 82.3%

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

   6. Applied egg-rr 82.3%

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

4. Final simplification 85.3%

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

Alternative 3: 94.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\right)\\ \mathbf{if}\;x\_m \leq 1.55 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(\left(x\_m \cdot s\right) \cdot \left(s \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s))))
   (if (<= x_m 1.55e-38)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m 2.0)) (* (* x_m c_m) (* (* x_m s) (* s c_m)))))))

C

x_m = fabs(x);
c_m = fabs(c);
assert(x_m < c_m && c_m < s);
double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 1.55e-38) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * 2.0)) / ((x_m * c_m) * ((x_m * s) * (s * c_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s)
    if (x_m <= 1.55d-38) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * 2.0d0)) / ((x_m * c_m) * ((x_m * s) * (s * c_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
assert x_m < c_m && c_m < s;
public static double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 1.55e-38) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * 2.0)) / ((x_m * c_m) * ((x_m * s) * (s * c_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
[x_m, c_m, s] = sort([x_m, c_m, s])
def code(x_m, c_m, s):
	t_0 = c_m * (x_m * s)
	tmp = 0
	if x_m <= 1.55e-38:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * 2.0)) / ((x_m * c_m) * ((x_m * s) * (s * c_m)))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
x_m, c_m, s = sort([x_m, c_m, s])
function code(x_m, c_m, s)
	t_0 = Float64(c_m * Float64(x_m * s))
	tmp = 0.0
	if (x_m <= 1.55e-38)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(Float64(x_m * c_m) * Float64(Float64(x_m * s) * Float64(s * c_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp_2 = code(x_m, c_m, s)
	t_0 = c_m * (x_m * s);
	tmp = 0.0;
	if (x_m <= 1.55e-38)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * 2.0)) / ((x_m * c_m) * ((x_m * s) * (s * c_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.55e-38], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(N[(x$95$m * s), $MachinePrecision] * N[(s * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999991e-38

   1. Initial program 68.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.7%

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

   5. Simplified 73.7%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-/r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 1.54999999999999991e-38 < x

   1. Initial program 62.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

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

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

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

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

      14. *-lowering-*.f64 94.2%

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

   4. Applied egg-rr 94.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(s \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\right)\\ \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{x\_m \cdot \left(x\_m \cdot \left(s \cdot \left(c\_m \cdot \left(s \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s))))
   (if (<= x_m 2.2e-35)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (* x_m 2.0)) (* x_m (* x_m (* s (* c_m (* s c_m)))))))))

C

x_m = fabs(x);
c_m = fabs(c);
assert(x_m < c_m && c_m < s);
double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 2.2e-35) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m * 2.0)) / (x_m * (x_m * (s * (c_m * (s * c_m)))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s)
    if (x_m <= 2.2d-35) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m * 2.0d0)) / (x_m * (x_m * (s * (c_m * (s * c_m)))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
assert x_m < c_m && c_m < s;
public static double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (x_m * s);
	double tmp;
	if (x_m <= 2.2e-35) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m * 2.0)) / (x_m * (x_m * (s * (c_m * (s * c_m)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
[x_m, c_m, s] = sort([x_m, c_m, s])
def code(x_m, c_m, s):
	t_0 = c_m * (x_m * s)
	tmp = 0
	if x_m <= 2.2e-35:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m * 2.0)) / (x_m * (x_m * (s * (c_m * (s * c_m)))))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
x_m, c_m, s = sort([x_m, c_m, s])
function code(x_m, c_m, s)
	t_0 = Float64(c_m * Float64(x_m * s))
	tmp = 0.0
	if (x_m <= 2.2e-35)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(x_m * Float64(x_m * Float64(s * Float64(c_m * Float64(s * c_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp_2 = code(x_m, c_m, s)
	t_0 = c_m * (x_m * s);
	tmp = 0.0;
	if (x_m <= 2.2e-35)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m * 2.0)) / (x_m * (x_m * (s * (c_m * (s * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.2e-35], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(x$95$m * N[(s * N[(c$95$m * N[(s * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.19999999999999994e-35

   1. Initial program 68.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/r* N/A

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

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

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

      6. associate-/r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.9%

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

   5. Simplified 73.9%

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

      1. associate-/l/ N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-/r* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

      15. associate-*l* N/A

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

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

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

      17. *-lowering-*.f64 85.9%

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

   7. Applied egg-rr 85.9%

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

   if 2.19999999999999994e-35 < x

   1. Initial program 62.8%

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

   3. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 88.5%

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

   5. Simplified 88.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp = code(x_m, c_m, s)
	t_0 = s * (x_m * c_m);
	tmp = (cos((x_m * 2.0)) / t_0) / t_0;
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

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

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

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

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

   9. *-lowering-*.f64 97.7%

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

4. Applied egg-rr 97.7%

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

   1. *-rgt-identity N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

   11. *-rgt-identity N/A

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

   12. associate-*l* N/A

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

   13. associate-/l/ N/A

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

   14. associate-/r* N/A

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

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

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

6. Applied egg-rr 94.1%

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

8. Applied egg-rr 98.2%

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

Alternative 6: 78.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

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

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

   6. associate-/r* N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 72.9%

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

5. Simplified 72.9%

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

   1. associate-/l/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/r* N/A

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

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

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

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

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

   12. associate-*l* N/A

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

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

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

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

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

   15. associate-*l* N/A

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

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

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

   17. *-lowering-*.f64 81.7%

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

7. Applied egg-rr 81.7%

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

8. Final simplification 81.7%

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

Alternative 7: 78.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

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

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

   6. associate-/r* N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 72.9%

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

5. Simplified 72.9%

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

   1. associate-/l/ N/A

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

   2. pow2 N/A

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

   3. associate-/r* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. pow2 N/A

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

   7. pow-prod-down N/A

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

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

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-*l* N/A

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

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

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

   19. associate-*r* N/A

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

   20. *-commutative N/A

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

   21. associate-*l* N/A

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

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

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

7. Applied egg-rr 73.5%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unswap-sqr N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 81.6%

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

9. Applied egg-rr 81.6%

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

Alternative 8: 74.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

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

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

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

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

   9. *-lowering-*.f64 97.7%

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*l* N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

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

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

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

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

   20. *-lowering-*.f64 77.9%

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

7. Simplified 77.9%

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

8. Final simplification 77.9%

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

Alternative 9: 71.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

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

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

   6. associate-/r* N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 72.9%

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

5. Simplified 72.9%

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

   1. associate-/l/ N/A

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

   2. pow2 N/A

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

   3. associate-/r* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. pow2 N/A

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

   7. pow-prod-down N/A

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

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

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-*l* N/A

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

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

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

   19. associate-*r* N/A

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

   20. *-commutative N/A

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

   21. associate-*l* N/A

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

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

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

7. Applied egg-rr 73.5%

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

8. Final simplification 73.5%

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

Alternative 10: 30.2% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

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

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

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

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

   14. *-lowering-*.f64 93.3%

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

4. Applied egg-rr 93.3%

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

5. Taylor expanded in x around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 58.7%

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

7. Simplified 58.7%

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

8. Taylor expanded in x around inf

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 27.4%

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

10. Simplified 27.4%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))