mixedcos

Percentage Accurate: 66.4% → 98.9%

Time: 8.3s

Alternatives: 11

Speedup: 9.0×

• 31.8% 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: 66.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 1.75 \cdot 10^{+39}:\\ \;\;\;\;\frac{t\_0}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}}{s\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 1.75e+39)
     (/ t_0 (* t_1 t_1))
     (/ (/ t_0 (* (* x_m c_m) (* s_m (* x_m c_m)))) s_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.75e+39) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = (t_0 / ((x_m * c_m) * (s_m * (x_m * c_m)))) / s_m;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 1.75d+39) then
        tmp = t_0 / (t_1 * t_1)
    else
        tmp = (t_0 / ((x_m * c_m) * (s_m * (x_m * c_m)))) / s_m
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m + x_m));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 1.75e+39) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = (t_0 / ((x_m * c_m) * (s_m * (x_m * c_m)))) / s_m;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m + x_m))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 1.75e+39:
		tmp = t_0 / (t_1 * t_1)
	else:
		tmp = (t_0 / ((x_m * c_m) * (s_m * (x_m * c_m)))) / s_m
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 1.75e+39)
		tmp = Float64(t_0 / Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * c_m)))) / s_m);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m + x_m));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 1.75e+39)
		tmp = t_0 / (t_1 * t_1);
	else
		tmp = (t_0 / ((x_m * c_m) * (s_m * (x_m * c_m)))) / s_m;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.75e+39], N[(t$95$0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7500000000000001e39

   1. Initial program 59.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 86.4

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

   4. Applied egg-rr 86.4%

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

      1. count-2 N/A

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

      2. lift-+.f64 86.4

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

   6. Applied egg-rr 86.4%

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

      1. lift-*.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 70.9

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

   8. Applied egg-rr 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. unpow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 94.2

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

   10. Applied egg-rr 97.7%

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

   if 1.7500000000000001e39 < x

   1. Initial program 59.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 94.7

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

   4. Applied egg-rr 94.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   6. Applied egg-rr 89.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+39}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m \cdot -2, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))) (t_1 (/ 1.0 (* c_m (* x_m s_m)))))
   (if (<=
        (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
        -2e-178)
     (/ (fma x_m (* x_m -2.0) 1.0) (* t_0 t_0))
     (* t_1 t_1))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double t_1 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -2e-178) {
		tmp = fma(x_m, (x_m * -2.0), 1.0) / (t_0 * t_0);
	} else {
		tmp = t_1 * t_1;
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	t_1 = Float64(1.0 / Float64(c_m * Float64(x_m * s_m)))
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -2e-178)
		tmp = Float64(fma(x_m, Float64(x_m * -2.0), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(t_1 * t_1);
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-178], N[(N[(x$95$m * N[(x$95$m * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.9999999999999999e-178

   1. Initial program 61.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 90.0

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

   4. Applied egg-rr 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 90.0

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 90.0

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.0

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 90.2

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 90.2

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. associate-*l* N/A

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

      29. lower-*.f64 99.5

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

      30. lift-*.f64 N/A

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

      31. *-commutative N/A

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

      32. lower-*.f64 99.5

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.0

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

   9. Simplified 49.0%

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

   if -1.9999999999999999e-178 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 59.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 71.4

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

   5. Simplified 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

   7. Applied egg-rr 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 76.6

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.5

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

   9. Applied egg-rr 77.5%

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

   11. Applied egg-rr 80.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))))
   (if (<= x_m 5.5e-57)
     (/ 1.0 (* c_m (* (* x_m s_m) (* c_m (* x_m s_m)))))
     (/ (cos (+ x_m x_m)) (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 5.5e-57) {
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	} else {
		tmp = cos((x_m + x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    if (x_m <= 5.5d-57) then
        tmp = 1.0d0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))))
    else
        tmp = cos((x_m + x_m)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 5.5e-57) {
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	} else {
		tmp = Math.cos((x_m + x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 5.5e-57:
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))))
	else:
		tmp = math.cos((x_m + x_m)) / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 5.5e-57)
		tmp = Float64(1.0 / Float64(c_m * Float64(Float64(x_m * s_m) * Float64(c_m * Float64(x_m * s_m)))));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 5.5e-57)
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	else
		tmp = cos((x_m + x_m)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5.5e-57], N[(1.0 / N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000011e-57

   1. Initial program 58.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   5. Simplified 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

   7. Applied egg-rr 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 77.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 79.5

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 79.5

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

   11. Applied egg-rr 79.5%

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

   if 5.50000000000000011e-57 < x

   1. Initial program 62.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 95.6

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

   4. Applied egg-rr 95.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 95.6

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 95.6

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 95.6

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 91.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 91.5

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. associate-*l* N/A

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

      29. lower-*.f64 94.7

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

      30. lift-*.f64 N/A

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

      31. *-commutative N/A

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

      32. lower-*.f64 94.7

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

   6. Applied egg-rr 94.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{if}\;s\_m \leq 7.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{c\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* c_m (* x_m s_m)))))
   (if (<= s_m 7.6e+132)
     (/ (cos (+ x_m x_m)) (* c_m (* (* x_m c_m) (* s_m (* x_m s_m)))))
     (* t_0 t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (s_m <= 7.6e+132) {
		tmp = cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (c_m * (x_m * s_m))
    if (s_m <= 7.6d+132) then
        tmp = cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))))
    else
        tmp = t_0 * t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (s_m <= 7.6e+132) {
		tmp = Math.cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = 1.0 / (c_m * (x_m * s_m))
	tmp = 0
	if s_m <= 7.6e+132:
		tmp = math.cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))))
	else:
		tmp = t_0 * t_0
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(1.0 / Float64(c_m * Float64(x_m * s_m)))
	tmp = 0.0
	if (s_m <= 7.6e+132)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(c_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * s_m)))));
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = 1.0 / (c_m * (x_m * s_m));
	tmp = 0.0;
	if (s_m <= 7.6e+132)
		tmp = cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
	else
		tmp = t_0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s$95$m, 7.6e+132], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(c$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 7.60000000000000012e132

   1. Initial program 58.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 86.0

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

   4. Applied egg-rr 86.0%

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

      1. count-2 N/A

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

      2. lift-+.f64 86.0

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

   6. Applied egg-rr 86.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 71.6

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

   8. Applied egg-rr 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 77.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 77.0

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

   10. Applied egg-rr 77.0%

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

   if 7.60000000000000012e132 < s

   1. Initial program 67.4%

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

   3. 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. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

   7. Applied egg-rr 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 83.8

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 86.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 86.3

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

   9. Applied egg-rr 86.3%

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

   11. Applied egg-rr 91.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 7.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{if}\;s\_m \leq 7.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{c\_m \cdot \left(c\_m \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* c_m (* x_m s_m)))))
   (if (<= s_m 7.5e+132)
     (/ (cos (+ x_m x_m)) (* c_m (* c_m (* (* x_m s_m) (* x_m s_m)))))
     (* t_0 t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (s_m <= 7.5e+132) {
		tmp = cos((x_m + x_m)) / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (c_m * (x_m * s_m))
    if (s_m <= 7.5d+132) then
        tmp = cos((x_m + x_m)) / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))))
    else
        tmp = t_0 * t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = 1.0 / (c_m * (x_m * s_m));
	double tmp;
	if (s_m <= 7.5e+132) {
		tmp = Math.cos((x_m + x_m)) / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = 1.0 / (c_m * (x_m * s_m))
	tmp = 0
	if s_m <= 7.5e+132:
		tmp = math.cos((x_m + x_m)) / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))))
	else:
		tmp = t_0 * t_0
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(1.0 / Float64(c_m * Float64(x_m * s_m)))
	tmp = 0.0
	if (s_m <= 7.5e+132)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(c_m * Float64(c_m * Float64(Float64(x_m * s_m) * Float64(x_m * s_m)))));
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = 1.0 / (c_m * (x_m * s_m));
	tmp = 0.0;
	if (s_m <= 7.5e+132)
		tmp = cos((x_m + x_m)) / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	else
		tmp = t_0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s$95$m, 7.5e+132], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(c$95$m * N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 7.50000000000000017e132

   1. Initial program 58.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 95.5

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

   4. Applied egg-rr 95.5%

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

   6. Applied egg-rr 83.0%

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

   if 7.50000000000000017e132 < s

   1. Initial program 67.4%

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

   3. 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. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 76.9

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

   5. Simplified 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

   7. Applied egg-rr 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 83.8

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 86.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 86.3

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

   9. Applied egg-rr 86.3%

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

   11. Applied egg-rr 91.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 7.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 86.3

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

4. Applied egg-rr 86.3%

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

   1. count-2 N/A

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

   2. lift-+.f64 86.3

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

6. Applied egg-rr 86.3%

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

   1. lift-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 70.6

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

8. Applied egg-rr 70.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. pow2 N/A

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

   16. unpow-prod-down N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. pow2 N/A

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

   22. lift-*.f64 94.0

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

10. Applied egg-rr 97.3%

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

11. Final simplification 97.3%

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

Alternative 7: 79.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 65.3

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

5. Simplified 65.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. swap-sqr N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*l* N/A

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

7. Applied egg-rr 63.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 70.1

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 70.9

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 70.9

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

9. Applied egg-rr 70.9%

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

11. Applied egg-rr 73.7%

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

12. Final simplification 73.7%

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

Alternative 8: 78.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))))
   (if (<= x_m 5.5e-57)
     (/ 1.0 (* c_m (* (* x_m s_m) (* c_m (* x_m s_m)))))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 5.5e-57) {
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    if (x_m <= 5.5d-57) then
        tmp = 1.0d0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 5.5e-57) {
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 5.5e-57:
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 5.5e-57)
		tmp = Float64(1.0 / Float64(c_m * Float64(Float64(x_m * s_m) * Float64(c_m * Float64(x_m * s_m)))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 5.5e-57)
		tmp = 1.0 / (c_m * ((x_m * s_m) * (c_m * (x_m * s_m))));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5.5e-57], N[(1.0 / N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000011e-57

   1. Initial program 58.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   5. Simplified 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

   7. Applied egg-rr 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 77.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.8

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.8

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

   9. Applied egg-rr 77.8%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 79.5

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 79.5

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

   11. Applied egg-rr 79.5%

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

   if 5.50000000000000011e-57 < x

   1. Initial program 62.1%

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

   3. 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. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 56.6

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

   5. Simplified 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 56.6

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. swap-sqr N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 58.7

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

   7. Applied egg-rr 58.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.1% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   18. lower-*.f64 65.3

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

5. Simplified 65.3%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)} \]

   7. associate-*r* N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)} \]

   10. swap-sqr N/A

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(s \cdot c\right)} \cdot x\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

   19. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot s\right) \cdot \left(x \cdot c\right)}} \]

7. Applied egg-rr 63.4%

   \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(c \cdot x\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot x\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot x\right)}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   9. lower-*.f64 70.1

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot x\right)\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \color{blue}{\left(c \cdot x\right)}\right) \cdot \left(s \cdot x\right)\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(s \cdot c\right) \cdot x\right)} \cdot \left(s \cdot x\right)\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(x \cdot \left(s \cdot c\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(x \cdot \left(s \cdot c\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   15. lower-*.f64 70.9

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \color{blue}{\left(s \cdot c\right)}\right) \cdot \left(s \cdot x\right)\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)} \]

   18. lower-*.f64 70.9

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)} \]

9. Applied egg-rr 70.9%

   \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot s\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(x \cdot s\right) \cdot c\right)} \cdot \left(x \cdot s\right)\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(\color{blue}{\left(x \cdot s\right)} \cdot c\right) \cdot \left(x \cdot s\right)\right)} \]

    3. lower-*.f64 72.0

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(x \cdot s\right) \cdot c\right)} \cdot \left(x \cdot s\right)\right)} \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(\color{blue}{\left(x \cdot s\right)} \cdot c\right) \cdot \left(x \cdot s\right)\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(\color{blue}{\left(s \cdot x\right)} \cdot c\right) \cdot \left(x \cdot s\right)\right)} \]

    6. lower-*.f64 72.0

       \[\leadsto \frac{1}{c \cdot \left(\left(\color{blue}{\left(s \cdot x\right)} \cdot c\right) \cdot \left(x \cdot s\right)\right)} \]

11. Applied egg-rr 72.0%

    \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(s \cdot x\right) \cdot c\right)} \cdot \left(x \cdot s\right)\right)} \]

12. Final simplification 72.0%

    \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \]
13. Add Preprocessing

Alternative 10: 76.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{c\_m \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ 1.0 (* c_m (* (* x_m s_m) (* x_m (* c_m s_m))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))));
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))))

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(1.0 / Float64(c_m * Float64(Float64(x_m * s_m) * Float64(x_m * Float64(c_m * s_m)))))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = 1.0 / (c_m * ((x_m * s_m) * (x_m * (c_m * s_m))));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(1.0 / N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

   11. associate-*l* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   18. lower-*.f64 65.3

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

5. Simplified 65.3%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)} \]

   7. associate-*r* N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)} \]

   10. swap-sqr N/A

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(s \cdot c\right)} \cdot x\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

   19. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot s\right) \cdot \left(x \cdot c\right)}} \]

7. Applied egg-rr 63.4%

   \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(c \cdot x\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot x\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot x\right)}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\left(c \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   9. lower-*.f64 70.1

      \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot x\right)\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(s \cdot \color{blue}{\left(c \cdot x\right)}\right) \cdot \left(s \cdot x\right)\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(\left(s \cdot c\right) \cdot x\right)} \cdot \left(s \cdot x\right)\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(x \cdot \left(s \cdot c\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\color{blue}{\left(x \cdot \left(s \cdot c\right)\right)} \cdot \left(s \cdot x\right)\right)} \]

   15. lower-*.f64 70.9

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \color{blue}{\left(s \cdot c\right)}\right) \cdot \left(s \cdot x\right)\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)} \]

   18. lower-*.f64 70.9

       \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)} \]

9. Applied egg-rr 70.9%

   \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot s\right)\right)}} \]

10. Final simplification 70.9%

    \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)} \]
11. Add Preprocessing

Alternative 11: 72.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{c\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ 1.0 (* c_m (* (* x_m c_m) (* s_m (* x_m s_m))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 1.0 / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return 1.0 / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 1.0 / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))))

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(1.0 / Float64(c_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * s_m)))))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = 1.0 / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(1.0 / N[(c$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

   11. associate-*l* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

   14. associate-*r* N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   18. lower-*.f64 65.3

       \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

5. Simplified 65.3%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)} \]

   7. associate-*r* N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)} \]

   10. swap-sqr N/A

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\color{blue}{\left(s \cdot c\right)} \cdot x\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

   19. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot s\right) \cdot \left(x \cdot c\right)}} \]

7. Applied egg-rr 63.4%

   \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot x\right)\right)}} \]

8. Final simplification 63.4%

   \[\leadsto \frac{1}{c \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))