mixedcos

Percentage Accurate: 65.8% → 98.4%

Time: 8.6s

Alternatives: 14

Speedup: 9.0×

• 31.5% 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: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{t\_0}{\left(c\_m \cdot {\left(s\_m \cdot x\_m\right)}^{2}\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{c\_m \cdot x\_m}}{s\_m}}{\left(c\_m \cdot x\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

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 (* 2.0 x_m))))
   (if (<= x_m 5e-145)
     (/ t_0 (* (* c_m (pow (* s_m x_m) 2.0)) c_m))
     (/ (/ (/ t_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) {
	double t_0 = cos((2.0 * x_m));
	double tmp;
	if (x_m <= 5e-145) {
		tmp = t_0 / ((c_m * pow((s_m * x_m), 2.0)) * c_m);
	} else {
		tmp = ((t_0 / (c_m * x_m)) / s_m) / ((c_m * x_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) :: tmp
    t_0 = cos((2.0d0 * x_m))
    if (x_m <= 5d-145) then
        tmp = t_0 / ((c_m * ((s_m * x_m) ** 2.0d0)) * c_m)
    else
        tmp = ((t_0 / (c_m * x_m)) / s_m) / ((c_m * x_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((2.0 * x_m));
	double tmp;
	if (x_m <= 5e-145) {
		tmp = t_0 / ((c_m * Math.pow((s_m * x_m), 2.0)) * c_m);
	} else {
		tmp = ((t_0 / (c_m * x_m)) / s_m) / ((c_m * x_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((2.0 * x_m))
	tmp = 0
	if x_m <= 5e-145:
		tmp = t_0 / ((c_m * math.pow((s_m * x_m), 2.0)) * c_m)
	else:
		tmp = ((t_0 / (c_m * x_m)) / s_m) / ((c_m * x_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(2.0 * x_m))
	tmp = 0.0
	if (x_m <= 5e-145)
		tmp = Float64(t_0 / Float64(Float64(c_m * (Float64(s_m * x_m) ^ 2.0)) * c_m));
	else
		tmp = Float64(Float64(Float64(t_0 / Float64(c_m * x_m)) / s_m) / Float64(Float64(c_m * x_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((2.0 * x_m));
	tmp = 0.0;
	if (x_m <= 5e-145)
		tmp = t_0 / ((c_m * ((s_m * x_m) ^ 2.0)) * c_m);
	else
		tmp = ((t_0 / (c_m * x_m)) / s_m) / ((c_m * x_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[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 5e-145], N[(t$95$0 / N[(N[(c$95$m * N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision] / N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999998e-145

   1. Initial program 60.7%

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

      1. lift-*.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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.0

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 81.3

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

   4. Applied rewrites 81.3%

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

   if 4.9999999999999998e-145 < x

   1. Initial program 72.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.6

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot {\left(s \cdot x\right)}^{2}\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}{s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)) (t_1 (* (* s_m x_m) c_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -4e-38)
     (/ (/ (fma -2.0 (* x_m x_m) 1.0) t_0) t_0)
     (/ (/ 1.0 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 = (c_m * x_m) * s_m;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-38) {
		tmp = (fma(-2.0, (x_m * x_m), 1.0) / t_0) / t_0;
	} else {
		tmp = (1.0 / 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(Float64(c_m * x_m) * s_m)
	t_1 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-38)
		tmp = Float64(Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / t_0) / t_0);
	else
		tmp = Float64(Float64(1.0 / 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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-38], N[(N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] / 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))) < -3.9999999999999998e-38

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 92.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 88.8

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.1

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

   6. Applied rewrites 96.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 41.6

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

   9. Applied rewrites 41.6%

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

   if -3.9999999999999998e-38 < (/.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 64.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 97.9

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

   4. Applied rewrites 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 96.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 83.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{\left(\left(c\_m \cdot c\_m\right) \cdot s\_m\right) \cdot \left(\left(s\_m \cdot x\_m\right) \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -4e-38)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* (* (* c_m c_m) s_m) (* (* s_m x_m) x_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 ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-38) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_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(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-38)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(Float64(Float64(c_m * c_m) * s_m) * Float64(Float64(s_m * x_m) * x_m)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	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[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-38], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * N[(N[(s$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $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))) < -3.9999999999999998e-38

   1. Initial program 65.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 65.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. unpow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

   6. Applied rewrites 77.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 41.1

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

   9. Applied rewrites 41.1%

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

   if -3.9999999999999998e-38 < (/.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 64.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 97.9

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

   4. Applied rewrites 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 96.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 83.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(c \cdot c\right) \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ t_1 := \cos \left(2 \cdot x\_m\right)\\ \mathbf{if}\;c\_m \leq 3.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{c\_m \cdot x\_m}}{s\_m}}{\left(c\_m \cdot x\_m\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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)) (t_1 (cos (* 2.0 x_m))))
   (if (<= c_m 3.6e-185)
     (/ (/ (/ t_1 (* c_m x_m)) s_m) (* (* c_m x_m) s_m))
     (/ (/ t_1 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 t_1 = cos((2.0 * x_m));
	double tmp;
	if (c_m <= 3.6e-185) {
		tmp = ((t_1 / (c_m * x_m)) / s_m) / ((c_m * x_m) * s_m);
	} else {
		tmp = (t_1 / 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) :: t_1
    real(8) :: tmp
    t_0 = (s_m * x_m) * c_m
    t_1 = cos((2.0d0 * x_m))
    if (c_m <= 3.6d-185) then
        tmp = ((t_1 / (c_m * x_m)) / s_m) / ((c_m * x_m) * s_m)
    else
        tmp = (t_1 / 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 t_1 = Math.cos((2.0 * x_m));
	double tmp;
	if (c_m <= 3.6e-185) {
		tmp = ((t_1 / (c_m * x_m)) / s_m) / ((c_m * x_m) * s_m);
	} else {
		tmp = (t_1 / 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
	t_1 = math.cos((2.0 * x_m))
	tmp = 0
	if c_m <= 3.6e-185:
		tmp = ((t_1 / (c_m * x_m)) / s_m) / ((c_m * x_m) * s_m)
	else:
		tmp = (t_1 / 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(Float64(s_m * x_m) * c_m)
	t_1 = cos(Float64(2.0 * x_m))
	tmp = 0.0
	if (c_m <= 3.6e-185)
		tmp = Float64(Float64(Float64(t_1 / Float64(c_m * x_m)) / s_m) / Float64(Float64(c_m * x_m) * s_m));
	else
		tmp = Float64(Float64(t_1 / 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;
	t_1 = cos((2.0 * x_m));
	tmp = 0.0;
	if (c_m <= 3.6e-185)
		tmp = ((t_1 / (c_m * x_m)) / s_m) / ((c_m * x_m) * s_m);
	else
		tmp = (t_1 / 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[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 3.6e-185], N[(N[(N[(t$95$1 / N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision] / N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 3.5999999999999998e-185

   1. Initial program 63.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.5

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.5

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 3.5999999999999998e-185 < c

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.7

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 95.9

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

   6. Applied rewrites 95.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 96.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 3.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot x}}{s}}{\left(c \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ t_1 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;c\_m \leq 3.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m + x\_m\right)}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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)) (t_1 (* (* c_m x_m) s_m)))
   (if (<= c_m 3.6e-185)
     (/ (/ (cos (+ x_m x_m)) t_1) t_1)
     (/ (/ (cos (* 2.0 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 t_1 = (c_m * x_m) * s_m;
	double tmp;
	if (c_m <= 3.6e-185) {
		tmp = (cos((x_m + x_m)) / t_1) / t_1;
	} else {
		tmp = (cos((2.0 * 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) :: t_1
    real(8) :: tmp
    t_0 = (s_m * x_m) * c_m
    t_1 = (c_m * x_m) * s_m
    if (c_m <= 3.6d-185) then
        tmp = (cos((x_m + x_m)) / t_1) / t_1
    else
        tmp = (cos((2.0d0 * 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 t_1 = (c_m * x_m) * s_m;
	double tmp;
	if (c_m <= 3.6e-185) {
		tmp = (Math.cos((x_m + x_m)) / t_1) / t_1;
	} else {
		tmp = (Math.cos((2.0 * 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
	t_1 = (c_m * x_m) * s_m
	tmp = 0
	if c_m <= 3.6e-185:
		tmp = (math.cos((x_m + x_m)) / t_1) / t_1
	else:
		tmp = (math.cos((2.0 * 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(Float64(s_m * x_m) * c_m)
	t_1 = Float64(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (c_m <= 3.6e-185)
		tmp = Float64(Float64(cos(Float64(x_m + x_m)) / t_1) / t_1);
	else
		tmp = Float64(Float64(cos(Float64(2.0 * x_m)) / 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;
	t_1 = (c_m * x_m) * s_m;
	tmp = 0.0;
	if (c_m <= 3.6e-185)
		tmp = (cos((x_m + x_m)) / t_1) / t_1;
	else
		tmp = (cos((2.0 * 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[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[c$95$m, 3.6e-185], N[(N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 3.5999999999999998e-185

   1. Initial program 63.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.5

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.5

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if 3.5999999999999998e-185 < c

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.7

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 95.9

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

   6. Applied rewrites 95.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 96.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 3.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m + x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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)) (t_1 (* (* s_m x_m) c_m)))
   (if (<= x_m 2e-144) (/ (/ 1.0 t_1) t_1) (/ (/ (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;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 2e-144) {
		tmp = (1.0 / t_1) / t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (c_m * x_m) * s_m
    t_1 = (s_m * x_m) * c_m
    if (x_m <= 2d-144) then
        tmp = (1.0d0 / t_1) / t_1
    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 = (c_m * x_m) * s_m;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 2e-144) {
		tmp = (1.0 / t_1) / t_1;
	} 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 = (c_m * x_m) * s_m
	t_1 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 2e-144:
		tmp = (1.0 / t_1) / t_1
	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(Float64(c_m * x_m) * s_m)
	t_1 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 2e-144)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	else
		tmp = Float64(Float64(cos(Float64(x_m + x_m)) / 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 = (c_m * x_m) * s_m;
	t_1 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 2e-144)
		tmp = (1.0 / t_1) / t_1;
	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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 2e-144], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-144

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.8

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.3

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

   6. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 76.4%

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

   if 1.9999999999999999e-144 < x

   1. Initial program 72.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.6

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.7

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 8.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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)) (t_1 (* (* s_m x_m) c_m)))
   (if (<= x_m 8.8e-72)
     (/ (/ 1.0 t_1) t_1)
     (/ (cos (* 2.0 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;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 8.8e-72) {
		tmp = (1.0 / t_1) / t_1;
	} else {
		tmp = cos((2.0 * 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) :: t_1
    real(8) :: tmp
    t_0 = (c_m * x_m) * s_m
    t_1 = (s_m * x_m) * c_m
    if (x_m <= 8.8d-72) then
        tmp = (1.0d0 / t_1) / t_1
    else
        tmp = cos((2.0d0 * 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 = (c_m * x_m) * s_m;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 8.8e-72) {
		tmp = (1.0 / t_1) / t_1;
	} else {
		tmp = Math.cos((2.0 * 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 = (c_m * x_m) * s_m
	t_1 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 8.8e-72:
		tmp = (1.0 / t_1) / t_1
	else:
		tmp = math.cos((2.0 * 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(Float64(c_m * x_m) * s_m)
	t_1 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 8.8e-72)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	else
		tmp = Float64(cos(Float64(2.0 * 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 = (c_m * x_m) * s_m;
	t_1 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 8.8e-72)
		tmp = (1.0 / t_1) / t_1;
	else
		tmp = cos((2.0 * 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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 8.8e-72], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8000000000000001e-72

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.9

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.0

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 96.5

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

   6. Applied rewrites 96.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 95.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.6%

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

   if 8.8000000000000001e-72 < x

   1. Initial program 73.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;s\_m \leq 1.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(c\_m \cdot c\_m\right) \cdot s\_m\right) \cdot \left(\left(s\_m \cdot x\_m\right) \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (<= s_m 1.8e+64)
     (/ (cos (+ x_m x_m)) (* (* (* c_m c_m) s_m) (* (* s_m x_m) x_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 (s_m <= 1.8e+64) {
		tmp = cos((x_m + x_m)) / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_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 (s_m <= 1.8d+64) then
        tmp = cos((x_m + x_m)) / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_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 (s_m <= 1.8e+64) {
		tmp = Math.cos((x_m + x_m)) / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_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 s_m <= 1.8e+64:
		tmp = math.cos((x_m + x_m)) / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_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(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (s_m <= 1.8e+64)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(c_m * c_m) * s_m) * Float64(Float64(s_m * x_m) * x_m)));
	else
		tmp = Float64(Float64(1.0 / 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 (s_m <= 1.8e+64)
		tmp = cos((x_m + x_m)) / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_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[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[s$95$m, 1.8e+64], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * N[(N[(s$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 1.80000000000000007e64

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. unpow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 N/A

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

   6. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2 N/A

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

      4. lower-+.f64 73.7

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

   8. Applied rewrites 73.7%

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

   if 1.80000000000000007e64 < s

   1. Initial program 53.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2}} \cdot \left({s}^{2} \cdot {c}^{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left({c}^{2} \cdot \color{blue}{{s}^{2}}\right)} \]

      15. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

      19. lower-*.f64 97.8

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

      9. lower-/.f64 97.9

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

      15. lower-*.f64 96.1

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{x \cdot \left(c \cdot s\right)}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

      20. *-commutative N/A

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

      21. lower-*.f64 97.9

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

   6. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot x\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot 2\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot 2\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(c \cdot x\right) \cdot s}\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{c \cdot \left(x \cdot s\right)}\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(x \cdot s\right) \cdot c}\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(x \cdot s\right)\right) \cdot c}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(x \cdot s\right)\right) \cdot c}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\left(\mathsf{neg}\left(\color{blue}{s \cdot x}\right)\right) \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot x\right)} \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot x\right)} \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      19. lower-neg.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(-s\right)} \cdot x\right) \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\mathsf{neg}\left(\color{blue}{\left(c \cdot x\right) \cdot s}\right)} \]

   8. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot 2\right)}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\left(\left(-s\right) \cdot x\right) \cdot c}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{1}}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\left(\left(-s\right) \cdot x\right) \cdot c} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.9%

       \[\leadsto \frac{\frac{\color{blue}{1}}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\left(\left(-s\right) \cdot x\right) \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot c\right) \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-58}:\\ \;\;\;\;\frac{1}{\left(\left(c\_m \cdot c\_m\right) \cdot s\_m\right) \cdot \left(\left(s\_m \cdot x\_m\right) \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c\_m \cdot x\_m\right) \cdot x\_m}}{\left(c\_m \cdot s\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

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
 (if (<= x_m 1e-58)
   (/ 1.0 (* (* (* c_m c_m) s_m) (* (* s_m x_m) x_m)))
   (/ (/ 1.0 (* (* c_m x_m) x_m)) (* (* c_m s_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 tmp;
	if (x_m <= 1e-58) {
		tmp = 1.0 / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_m));
	} else {
		tmp = (1.0 / ((c_m * x_m) * x_m)) / ((c_m * s_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) :: tmp
    if (x_m <= 1d-58) then
        tmp = 1.0d0 / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_m))
    else
        tmp = (1.0d0 / ((c_m * x_m) * x_m)) / ((c_m * s_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 tmp;
	if (x_m <= 1e-58) {
		tmp = 1.0 / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_m));
	} else {
		tmp = (1.0 / ((c_m * x_m) * x_m)) / ((c_m * s_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):
	tmp = 0
	if x_m <= 1e-58:
		tmp = 1.0 / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_m))
	else:
		tmp = (1.0 / ((c_m * x_m) * x_m)) / ((c_m * s_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)
	tmp = 0.0
	if (x_m <= 1e-58)
		tmp = Float64(1.0 / Float64(Float64(Float64(c_m * c_m) * s_m) * Float64(Float64(s_m * x_m) * x_m)));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(c_m * x_m) * x_m)) / Float64(Float64(c_m * s_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)
	tmp = 0.0;
	if (x_m <= 1e-58)
		tmp = 1.0 / (((c_m * c_m) * s_m) * ((s_m * x_m) * x_m));
	else
		tmp = (1.0 / ((c_m * x_m) * x_m)) / ((c_m * s_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_] := If[LessEqual[x$95$m, 1e-58], N[(1.0 / N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * N[(N[(s$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(c$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(c$95$m * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-58

   1. Initial program 62.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      3. lower-*.f64 62.5

         \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \cdot {c}^{2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot {c}^{2}} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2}} \cdot \left({s}^{2} \cdot {c}^{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right)} \]

      14. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left({c}^{2} \cdot \color{blue}{{s}^{2}}\right)} \]

      15. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

      16. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

      19. lower-*.f64 97.0

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(s \cdot c\right)}\right)}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(s \cdot c\right)}\right)}^{2}} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2} \cdot {\left(s \cdot c\right)}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(s \cdot c\right)}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left(\left(\left(s \cdot c\right) \cdot c\right) \cdot s\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left(s \cdot \left(\left(s \cdot c\right) \cdot c\right)\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left({x}^{2} \cdot s\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(s \cdot {x}^{2}\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      14. pow2 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      15. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(s \cdot x\right) \cdot x\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(x \cdot s\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(x \cdot s\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      18. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)}} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      22. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot s\right) \cdot x\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      23. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot s\right) \cdot x\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      24. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(x \cdot s\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      25. *-commutative N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(s \cdot x\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

      26. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(s \cdot x\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   6. Applied rewrites 71.2%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.2%

      \[\leadsto \frac{\color{blue}{1}}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)} \]

   if 1e-58 < x

   1. Initial program 70.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      2. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{1}{{x}^{2}}}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{{x}^{2}}}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{x}^{2}}}{c}}{c \cdot {s}^{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{x}^{2}}}{c}}{c \cdot {s}^{2}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{{x}^{2}}}{c}}}{c \cdot {s}^{2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{1}{\color{blue}{x \cdot x}}}{c}}{c \cdot {s}^{2}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{x}}}{c}}{c \cdot {s}^{2}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{x}}}{c}}{c \cdot {s}^{2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{x}}}{x}}{c}}{c \cdot {s}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{c \cdot \color{blue}{\left(s \cdot s\right)}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(c \cdot s\right) \cdot s}} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right) \cdot s}} \]

      16. lower-*.f64 61.1

          \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\left(s \cdot c\right) \cdot s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.1%

      \[\leadsto \frac{\frac{1}{\left(x \cdot x\right) \cdot c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.8%

      \[\leadsto \frac{\frac{1}{\left(c \cdot x\right) \cdot x}}{\left(s \cdot \color{blue}{c}\right) \cdot s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-58}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot c\right) \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot x\right) \cdot x}}{\left(c \cdot s\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

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))) (/ (/ 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;
	return (1.0 / 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 = (s_m * x_m) * c_m
    code = (1.0d0 / 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 = (s_m * x_m) * c_m;
	return (1.0 / 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 = (s_m * x_m) * c_m
	return (1.0 / 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(Float64(s_m * x_m) * c_m)
	return Float64(Float64(1.0 / 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 = (s_m * x_m) * c_m;
	tmp = (1.0 / 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[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.6%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. lower-*.f64 64.6

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \cdot {c}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot {c}^{2}} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2}} \cdot \left({s}^{2} \cdot {c}^{2}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left({c}^{2} \cdot \color{blue}{{s}^{2}}\right)} \]

   15. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

   16. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

   19. lower-*.f64 97.3

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

   9. lower-/.f64 97.4

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

   15. lower-*.f64 95.3

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{x \cdot \left(c \cdot s\right)}} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

   18. associate-*r* N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

   20. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

   21. lower-*.f64 97.5

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

6. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(2 \cdot x\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot 2\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot 2\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(c \cdot x\right) \cdot s}\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{c \cdot \left(x \cdot s\right)}\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\mathsf{neg}\left(\color{blue}{\left(x \cdot s\right) \cdot c}\right)}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(x \cdot s\right)\right) \cdot c}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\mathsf{neg}\left(x \cdot s\right)\right) \cdot c}}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\left(\mathsf{neg}\left(\color{blue}{s \cdot x}\right)\right) \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot x\right)} \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot x\right)} \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   19. lower-neg.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(-s\right)} \cdot x\right) \cdot c}}{\mathsf{neg}\left(\left(c \cdot x\right) \cdot s\right)} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(x \cdot 2\right)}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\mathsf{neg}\left(\color{blue}{\left(c \cdot x\right) \cdot s}\right)} \]

8. Applied rewrites 96.4%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot 2\right)}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\left(\left(-s\right) \cdot x\right) \cdot c}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\color{blue}{1}}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\left(\left(-s\right) \cdot x\right) \cdot c} \]
10. Step-by-step derivation

11. Applied rewrites 75.4%

    \[\leadsto \frac{\frac{\color{blue}{1}}{\left(\left(-s\right) \cdot x\right) \cdot c}}{\left(\left(-s\right) \cdot x\right) \cdot c} \]

12. Final simplification 75.4%

    \[\leadsto \frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c} \]
13. Add Preprocessing

Alternative 11: 78.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\_m\right) \cdot x\_m\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

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 s_m) x_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 = (c_m * s_m) * x_m;
	return (1.0 / 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 * s_m) * x_m
    code = (1.0d0 / 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 * s_m) * x_m;
	return (1.0 / 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 * s_m) * x_m
	return (1.0 / 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(Float64(c_m * s_m) * x_m)
	return Float64(Float64(1.0 / 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 * s_m) * x_m;
	tmp = (1.0 / 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[(N[(c$95$m * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.6%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   2. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{1}{{x}^{2}}}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{{x}^{2}}}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{x}^{2}}}{c}}{c \cdot {s}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{x}^{2}}}{c}}{c \cdot {s}^{2}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{{x}^{2}}}{c}}}{c \cdot {s}^{2}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{x \cdot x}}}{c}}{c \cdot {s}^{2}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{x}}}{c}}{c \cdot {s}^{2}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{x}}}{c}}{c \cdot {s}^{2}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{x}}}{x}}{c}}{c \cdot {s}^{2}} \]

   12. unpow2 N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{c \cdot \color{blue}{\left(s \cdot s\right)}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(c \cdot s\right) \cdot s}} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right) \cdot s}} \]

   16. lower-*.f64 62.3

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]

5. Applied rewrites 62.3%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\left(s \cdot c\right) \cdot s}} \]
6. Step-by-step derivation

7. Applied rewrites 62.3%

   \[\leadsto \frac{\frac{1}{\left(x \cdot x\right) \cdot c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]
8. Step-by-step derivation

9. Applied rewrites 67.0%

   \[\leadsto \frac{\frac{1}{\left(c \cdot x\right) \cdot x}}{\left(s \cdot \color{blue}{c}\right) \cdot s} \]

11. Applied rewrites 76.9%

    \[\leadsto -\frac{\frac{-1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]

12. Final simplification 76.9%

    \[\leadsto \frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\left(c \cdot s\right) \cdot x} \]
13. Add Preprocessing

Alternative 12: 77.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

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))) (/ (/ 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 = (c_m * x_m) * s_m;
	return (1.0 / 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 = (1.0d0 / 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 (1.0 / 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 (1.0 / 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(Float64(c_m * x_m) * s_m)
	return Float64(Float64(1.0 / 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 = (1.0 / 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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.6%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. lower-*.f64 64.6

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \cdot {c}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot {c}^{2}} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2}} \cdot \left({s}^{2} \cdot {c}^{2}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left({c}^{2} \cdot \color{blue}{{s}^{2}}\right)} \]

   15. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

   16. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

   19. lower-*.f64 97.3

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

   9. lower-/.f64 97.4

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

   15. lower-*.f64 95.3

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{x \cdot \left(c \cdot s\right)}} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

   18. associate-*r* N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

   20. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

   21. lower-*.f64 97.5

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

6. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\color{blue}{1}}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s} \]
8. Step-by-step derivation

9. Applied rewrites 77.2%

   \[\leadsto \frac{\frac{\color{blue}{1}}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s} \]
10. Add Preprocessing

Alternative 13: 77.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{1}{\left(s\_m \cdot x\_m\right) \cdot c\_m}}{\left(c\_m \cdot x\_m\right) \cdot s\_m} \end{array} \]

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 (* (* s_m x_m) c_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 / ((s_m * x_m) * c_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 / ((s_m * x_m) * c_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 / ((s_m * x_m) * c_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 / ((s_m * x_m) * c_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(Float64(1.0 / Float64(Float64(s_m * x_m) * c_m)) / Float64(Float64(c_m * 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 / ((s_m * x_m) * c_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[(N[(1.0 / N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. lower-*.f64 64.6

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \cdot {c}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot {c}^{2}} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2}} \cdot \left({s}^{2} \cdot {c}^{2}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left({c}^{2} \cdot \color{blue}{{s}^{2}}\right)} \]

   15. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

   16. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

   19. lower-*.f64 97.3

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}} \]

   9. lower-/.f64 97.4

      \[\leadsto \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(c \cdot s\right)}}}{x \cdot \left(c \cdot s\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot s}}}{x \cdot \left(c \cdot s\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

   15. lower-*.f64 95.3

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot x\right)} \cdot s}}{x \cdot \left(c \cdot s\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{x \cdot \left(c \cdot s\right)}} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{x \cdot \color{blue}{\left(c \cdot s\right)}} \]

   18. associate-*r* N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]

   20. *-commutative N/A

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

   21. lower-*.f64 97.5

       \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\color{blue}{\left(c \cdot x\right)} \cdot s} \]

6. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{c \cdot \left(s \cdot x\right)}}}{\left(c \cdot x\right) \cdot s} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{c \cdot \left(s \cdot x\right)}}}{\left(c \cdot x\right) \cdot s} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(s \cdot x\right) \cdot c}}}{\left(c \cdot x\right) \cdot s} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(s \cdot x\right) \cdot c}}}{\left(c \cdot x\right) \cdot s} \]

   4. lower-*.f64 74.6

      \[\leadsto \frac{\frac{1}{\color{blue}{\left(s \cdot x\right)} \cdot c}}{\left(c \cdot x\right) \cdot s} \]

9. Applied rewrites 74.6%

   \[\leadsto \frac{\color{blue}{\frac{1}{\left(s \cdot x\right) \cdot c}}}{\left(c \cdot x\right) \cdot s} \]
10. Add Preprocessing

Alternative 14: 67.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{1}{\left(\left(c\_m \cdot c\_m\right) \cdot s\_m\right) \cdot \left(\left(s\_m \cdot x\_m\right) \cdot x\_m\right)} \end{array} \]

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 c_m) s_m) (* (* s_m x_m) x_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 * c_m) * s_m) * ((s_m * x_m) * x_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 * c_m) * s_m) * ((s_m * x_m) * x_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 * c_m) * s_m) * ((s_m * x_m) * x_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 * c_m) * s_m) * ((s_m * x_m) * x_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(Float64(Float64(c_m * c_m) * s_m) * Float64(Float64(s_m * x_m) * x_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 * c_m) * s_m) * ((s_m * x_m) * x_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[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * N[(N[(s$95$m * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.6%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. lower-*.f64 64.6

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \cdot {c}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot {c}^{2}} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2}} \cdot \left({s}^{2} \cdot {c}^{2}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left({c}^{2} \cdot \color{blue}{{s}^{2}}\right)} \]

   15. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

   16. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

   19. lower-*.f64 97.3

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(s \cdot c\right)}\right)}^{2}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(s \cdot c\right)}\right)}^{2}} \]

   6. unpow-prod-down N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2} \cdot {\left(s \cdot c\right)}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(s \cdot c\right)}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left(\left(\left(s \cdot c\right) \cdot c\right) \cdot s\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left(s \cdot \left(\left(s \cdot c\right) \cdot c\right)\right)}} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left({x}^{2} \cdot s\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(s \cdot {x}^{2}\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   14. pow2 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   15. associate-*l* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(s \cdot x\right) \cdot x\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   16. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(x \cdot s\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(x \cdot s\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   18. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   20. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)}} \]

   21. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot s\right)\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   22. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot s\right) \cdot x\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   23. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot s\right) \cdot x\right)} \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   24. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(x \cdot s\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   25. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(s \cdot x\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

   26. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(s \cdot x\right)} \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)} \]

6. Applied rewrites 73.5%

   \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)} \]
8. Step-by-step derivation

9. Applied rewrites 63.0%

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(s \cdot x\right) \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot s\right)} \]

10. Final simplification 63.0%

    \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot s\right) \cdot \left(\left(s \cdot x\right) \cdot x\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024285 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))